PAST SEMINAR ABSTRACT

FALL 2023

Sep 6  @  Penn: Hao Shen (Wisconsin-Madison)

Stochastic quantization of Yang-Mills in 2D and 3D

Quantum Yang-Mills model is a type of quantum field theory with gauge symmetry. The rigorous construction of quantum Yang-Mills is a central problem in mathematical physics. Stochastic quantization formulates the problem as stochastic dynamics, which can be studied using tools from analysis, PDE and stochastic PDE. We will discuss stochastic quantization of Yang-Mills on the 2 and 3 dimensional tori. To this end we need to address a number of questions, such as the construction of a singular orbit space, together with a class gauge invariant observables (singular holonomies or Wilson loops), solving a stochastic PDE using regularity structures, and projecting the solution to the orbit space. Mostly based on joint work with Chandra, Chevyrev and Hairer.

Sep 12  @  Temple: Luke Peilen (Temple)

Statistical mechanics of Log and Riesz interactions

We study the statistical mechanics of the log gas, an interacting particle system with applications in random matrix theory and statistical physics, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on a novel next order energy quantity that are valid down to microscopic length scales. Simultaneously, we exhibit a control on fluctuations of linear statistics that is also valid down to microscopic scales. Using these local laws, we exhibit for the first time a CLT at arbitrary mesoscales, improving upon previous results of Bekerman and Lodhia.

The methods we use are suitable for generalization to higher dimensional Riesz interactions; we will discuss some generalizations of the above approach and partial results for the Riesz gas in higher dimensions.

Sep 19  @  Penn: Daniel Slonim  (UVA)

Random Walks in (Dirichlet) Random Environments with Jumps on Z

We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

Sep 26  @  Temple: Yier Lin (UChicago)

Large deviations of the KPZ equation and most probable shapes

The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments.

This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

Oct 3  @  Penn: Tomas Berggren (MIT)

Geometry of the doubly periodic Aztec dimer model

Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions -- known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

Oct 10  @  Temple: Joshua McGinnis (UPenn)

A rigorous approximation of a certain random Fermi-Pasta-Ulam-Tsingou (FPUT) lattice by the Korteweg-De Vries (KdV) equation

We review recent results regarding the rigorous approximation of 1D and 2D disordered (random, independent masses and/or springs) harmonic lattices by effective wave equations in the long wave limit. In this linear setting, we show the homogenization argument and highlight the tools used from probability theory to control the stochastic error terms such as the Law of the Iterated Logarithm and Hoeffding’s inequality. With our discussion of the linear problem serving as a springboard, we then present a new result regarding the approximation of an FPUT lattice with random masses by a KdV equation. Specifically, we are able to bound the approximation error in terms of the small parameter from the long wave scaling in an almost sure sense. In our theorem, we require a technical condition on the random masses, which we call transparency. Our proof relies on the incorporation of an auto-regressive process into an approximating ansatz, which itself is approximated by solutions to the KdV equation. We discuss the role of the auto-regressive process as well as the condition of transparency in the proof and give numerical evidence supporting the result. We conclude by discussing open questions such as the apparent lack of KdV dynamics in an FPUT lattice with independent, random masses.

Oct 17  @  Penn: Adrián González-Casanova (UC-Berkeley)

Sample duality 

Heuristically, two processes are dual if one can find a function to study one process by using the other. Sampling duality is a duality which uses a duality function S(n,x) of the form "what is the probability that all the members of a sample of size n are of a certain type, given that the number (or frequency) of that type of individuals is x". Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss examples for which this technique is useful, including an application to the Simple Exclusion Process with reservoirs. The last part of the lecture is based in recent joint work with Simone Floriani https://arxiv.org/abs/2307.02481

Oct 24  @  Temple: Cooper Boniece (Drexel)

An iterative approach to estimating integrated volatility

The quadratic variation of a semimartingale plays an important role in a variety of applications, particularly so in financial econometrics, where it is closely linked to volatility.  It contains information pertaining to both continuous and discontinuous path behavior of the underlying process, and separating its continuous and discontinuous parts based on high-frequency observations is a problem that has been tackled through a variety of approaches to-date.

However, despite the favorable asymptotic statistical properties of many of these approaches, their use in practice requires heuristic selection of tuning parameters that can greatly impact their estimation performance.

In this talk, I will discuss some recent work concerning an iterative approach that circumvents the "tuning problem."

This is based on joint work with J. E. Figueroa-López and Y. Han.

Oct 31  @  Penn: Yuchen Wu (Penn)

Fundamental Limits of Low-Rank Matrix Estimation: Information-Theoretic and Computational Perspectives

Many statistical estimation problems can be reduced to the reconstruction of a low-rank n×d matrix when observed through a noisy channel. While tremendous positive results have been established, relatively few works focus on understanding the fundamental limitations of the proposed models and algorithms. Understanding such limitations not only provides practitioners with guidance on algorithm selection, but also spurs the development of cutting-edge methodologies. In this talk, I will present some recent progress in this direction from two perspectives in the context of low-rank matrix estimation. From an information-theoretic perspective, I will give an exact characterization of the limiting minimum estimation error. Our results apply to the high-dimensional regime n,d→∞ and d/n→∞ (or d/n→0) and generalize earlier works that focus on the proportional asymptotics n,d→∞, d/n→δ∈(0,∞). From an algorithmic perspective, large-dimensional matrices are often processed by iterative algorithms like power iteration and gradient descent, thus encouraging the pursuit of understanding the fundamental limits of these approaches. We introduce a class of general first order methods (GFOM), which is broad enough to include the aforementioned algorithms and many others. I will describe the asymptotic behavior of any GFOM, and provide a sharp characterization of the optimal error achieved by the GFOM class.

Nov 7  @  Temple: Pax Kivimae (Courant)

Relative Instability and Concentration of Equilibria in Non-Gradient Dynamics

A classical picture in the theory of complex high-dimensional random functions is that an exponentially large number of critical points causes the gradient dynamics of the function to become slow and "glassy", becoming trapped in local minima. In non-gradient dynamics however, another case is possible. Here, one may have an exponentially large number of equilibria. but have none that are stable, leading to an endless cycle of wandering around saddles. This is believed to occur when the strength of the non-gradient terms is brought past a certain point, a phenomenon coined Ben Arous, Fyodorov, and Khoruzhenko as the relative-absolute instability transition, and since predicted to occur in a variety of models.

We confirm such a transition occurs in the case of the asymmetric p-spin model, the first such rigorous confirmation of the existence of this transition in any model. To do so, we demonstrate concentration of the quenched complexity of stable and general equilibria around their annealed values. Our methods rely on generalizing the recent framework of Ben Arous, Bourgade, and McKenna on the Kac-Rice formula to the non-relaxational case, as well as a computation of moments of the characteristic polynomial of the elliptic ensemble.

Nov 14  @  Penn: Arka Adhikari (Stanford)

Spectral Gap Estimates for Mixed p-Spin Models at High Temperature

We consider general mixed p-spin mean field spin glass models and provide a method to prove that the spectral gap of the Dirichlet form associated with the Gibbs measure is of order one at sufficiently high temperature. Our proof is based on an iteration scheme relating the spectral gap of the $N$-spin system to that of suitably conditioned subsystems.

Based on joint Work w/ C. Brennecke, C. Xu, and H-T Yau.

Nov 28  @  Temple: Alejandro Ramírez (NYU Shanghai)

KPZ fluctuations in the planar stochastic heat equation

We consider Wick-ordered solutions to the planar stochastic heat equation, corresponding to a Skorokhod interpretation in the Duhamel integral representation of the equation. We prove that the fluctuations far from the center are given by the stochastic heat equation. This talk is based on a joint work with Jeremy Quastel and Balint Virag.

Dec 5  @  Penn DRL 4C2: Zongrui Yang (Columbia)

Stationary measures for integrable polymers on a strip

We prove that the stationary measures for the geometric last passage percolation (LPP) and log-gamma polymer models on a diagonal strip are given by the marginals of objects we call two-layer Gibbs measures. Taking an intermediate disorder limit of the log-gamma polymer stationary measure, we recover the conjectural description of the open KPZ equation stationary measure for all choices of boundary parameters. This is a joint work with Guillaume Barraquand and Ivan Corwin.

Spring 2023

Jan 24  @  Temple: Yuri Bakhtin (Courant)

Rare transitions in noisy heteroclinic networks

We study white noise perturbations of planar dynamical systems with heteroclinic networks in the limit of vanishing noise. We show that the probabilities of transitions between various cells that the network tessellates the plane into decay as powers of the noise magnitude, and we describe the underlying mechanism. A metastability picture emerges, with a hierarchy of time scales and clusters of accessibility, similar to the classical Freidlin-Wentzell picture but with shorter transition times. We discuss applications of our results to homogenization problems and to the invariant distribution asymptotics. At the core of our results are local limit theorems for exit distributions obtained via methods of Malliavin calculus. Joint work with Hong-Bin Chen and Zsolt Pajor-Gyulai.

Jan 31  @  Upenn: Dor Elboim (Princeton)

Infinite cycles in the interchange process in five dimensions

In the interchange process on a graph G=(V, E), distinguished particles are placed on the vertices of G with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta: V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles.

We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $\beta$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions.

In our proof, we study a self-interacting random walk called the cyclic time random walk.  Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.

This is a joint work with Allan Sly.

Feb 7  @  Temple: Kavita Ramanan (Brown)

Mean-field games: asymptotics and refined convergence results

A mean-field game is a game with a continuum of players,  describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. We first study the asymptotic behavior of Nash equilibria in static games with a large number of agents. In particular, we establish law of large number limits and large deviation principles for the set of Nash equilibria and discuss applications to congestion games and the price of anarchy. Then we discuss stochastic differential games, which are often understood via the so-called "master equation", which is an infinite-dimensional PDE for the value function. We will show how analysis of sufficiently smooth solutions to the master equation play a role in analyzing large deviation principles for mean-field games. This is based on joint works with Francois Delarue and Daniel Lacker.

Feb 14  @  Penn: Yu Gu (UMD)

KPZ on a large torus

I will present the recent work with Tomasz Komorowski and Alex Dunlap in which we derived optimal variance bounds on the solution to the KPZ equation on a large torus, in certain regimes where the size of the torus increases with time. We only use stochastic calculus and I will try to give a heuristic explanation of the 2/3 and 1/3 exponents in the 1+1 KPZ universality class.

Feb 21  @  Temple: Alex Dunlap (Courant)

The nonlinear stochastic heat equation in the critical dimension

I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and earlier joint work with Yu Gu.Liouville conformal field theory and the quantum zipper

Feb 28  @  Penn: Konstantin Tikhomirov (CMU)

Functional inequalities on the space of d-regular directed graphs, with applications to mixing

We consider the space of d-regular directed simple graphs, where two graphs are connected whenever there is a simple switching operation transforming one graph to the other. For constant d, we prove optimal bounds on the Modified Log-Sobolev Constant of the associated Markov chain on the space of graphs. This implies that the total variation mixing time of the chain is of order n log(n), which settles an old open problem. Based on joint work with Pierre Youssef.

Mar 14  @  Penn: Morris Ang (Columbia)

Liouville conformal field theory and the quantum zipper

Sheffield showed that conformally welding a \gamma-Liouville quantum gravity (LQG) surface to itself gives a Schramm-Loewner evolution (SLE) curve with parameter \kappa = \gamma^2 as the interface, and Duplantier-Miller-Sheffield proved similar stories for \kappa = 16/\gamma^2 for \gamma-LQG surfaces with boundaries decorated by looptrees of disks or by continuum random trees. We study these dynamics for LQG surfaces coming from Liouville conformal field theory (LCFT). At stopping times depending only on the curve, we give an explicit description of the surface and curve in terms of LCFT and SLE. This has applications to both LCFT and SLE. We prove the boundary BPZ equation for LCFT, which is crucial to solving boundary LCFT. With Yu we prove the reversibility of whole-plane SLE for \kappa ≥ 8 via a novel radial mating-of-trees.

Mar 21  @  Temple: Eren C. Kızıldağ (Columbia)

Algorithmic barriers from intricate geometry in random computational problems

Many computational problems involving randomness exhibit a statistical-to-computational gap (SCG): the best known polynomial-time algorithm performs strictly worse than the existential guarantee. In this talk, we focus on the SCG of the symmetric binary perceptron (SBP), a random constraint satisfaction problem as well as a toy model of a single-layer neural network. We establish that the solution space of the SBP exhibits intricate geometrical features, known as the multi Overlap Gap Property (m-OGP). By leveraging the m-OGP, we obtain nearly sharp hardness guarantees against the class of stable and online algorithms, which capture the best known algorithms for the SBP. Our results mark the first instance of intricate geometry yielding tight algorithmic hardness against classes beyond stable algorithms.

Time permitting, I will discuss how the same program extends also to other models, including (a) discrepancy minimization, and (b) random number partitioning problem. 

Based on joint works with David Gamarnik, Will Perkins, and Changji Xu.

Mar 28  @  Penn: Wenpin Tang (Columbia)

Phase transition in mean field models

In this talk, I will discuss two mean field models in which a certain phase transition occurs. I first describe McKean-Vlasov equations involving hitting times which arise as the mean field limit of particle systems with annihilation. One such example is the super-cool Stefan problem. It is well known that such a system may have blow-ups. We provide some sufficient conditions on the model data to assure either blow-ups or no blow-ups. In the second part, I will discuss the convergence rate of second order mean-field games to first order ones, motivated from numerical challenges in first order mean field PDEs and the weak noise theory in KPZ universality. When the Hamiltonian and the coupling function have a certain growth, the rate is independent of the dimension; on the other hand, the rate decays in dimension (curse of dimensionality) when the Hamiltonian and the coupling function have small growth. These are based on joint work with Yuming Paul Zhang. 

Apr 4  @  Temple: Yeor Hafouta (UMD)

A Berry-Esseen theorem and Edgeworth expansions for inhomogeneous elliptic Markov chains

We obtain optimal rates in the central limit theorem (CLT) for additive functionals of uniformly elliptic inhomogeneous Markov chains without any assumptions on the growth rates of the variance of the underlying partial sums. (The CLT itself is due to Dobrushin (1956) and it holds in greater generality.)

We will also discuss Edgeworth expansions (i.e., the correction terms in the CLT) of order one for general classes of functionals, which provide a structural characterization of having better than optimal CLT rates.

Finally, for several classes of additive functionals (e.g., Holder continuous), we will provide optimal conditions for Edgeworth expansions of an arbitrary order.

The talk is based on a joint work with Dmitry Dolgopyat.

Apr 11  @  Penn:  Qian Yu (Princeton)

Ising Model on Locally Tree-like Graphs: Uniqueness of Solutions to Cavity Equations

In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models. 

Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid). 

This is a joint work with Yury Polyanskiy. Paper link: https://arxiv.org/abs/2211.15242

Apr 11  @  Penn: Lingfu Zhang (UC Berkeley)

Cutoff profile of the colored ASEP: GOE Tracy-Widom

In this talk, I will discuss the colored Asymmetric Simple Exclusion Process (ASEP) in a finite interval. This Markov chain is also known as the biased card shuffling or random Metropolis scan, and its study dates back to Diaconis-Ram (2000). A total-variation cutoff was proved for this chain a few years ago using hydrodynamic techniques (Labbé-Lacoin, 2016). In this talk, I will explain how to obtain more precise information on its cutoff, specifically to establish the conjectured GOE Tracy-Widom cutoff profile. The proof relies on coupling arguments, as well as symmetries obtained from the Hecke algebra. I will also discuss some related open problems.

Apr 18  @  Temple: Ron Peled (Tel Aviv, IAS and Princeton)

Minimal surfaces in random environment

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of d-dimensional minimal surfaces in a (d+n)-dimensional random environment. Specializing to a model that we term harmonic MSRE, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations.

Joint work with Barbara Dembin, Dor Elboim and Daniel Hadas.

Apr 25  @  Penn: Wai-Tong (Louis) Fan (IU Bloomington)

Stochastic waves on metric graphs and their genealogies

Stochastic reaction-diffusion equations are important models in mathematics and in applied sciences such as spatial population genetics and ecology. These equations describe a quantity (density/concentration of an entity) that evolves over space and time, taking into account random fluctuations. However, for many reaction terms and noises, the solution notion of these equations is still missing in dimension two or above, hindering the study of spatial effect on stochastic dynamics through these equations.

In this talk, I will discuss a new approach, namely, to study these equations on general metric graphs that flexibly parametrize the underlying space. This enables us to not only bypass the ill-posedness issue of these equations in higher dimensions, but also assess the impact of space and stochasticity on the coexistence and the genealogies of interacting populations. We will focus on the computation of the probability of extinction, the quasi-stationary distribution, the asymptotic speed and other long-time behaviors for stochastic reaction-diffusion equations of Fisher-KPP type.  

Fall 2022

Sep 6  @  Penn: Promit Ghosal (MIT)

Fractal Geometry of the KPZ equation

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to the KPZ universality class. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention a work in progress with Jaeyun Yi for the (2+1)-d case.

Sep 13  @  Temple: Matthew Junge (Baruch College)

Ballistic Annihilation

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.

Sep 20  @  Penn: Wei Wu (NYU Shanghai)

A central limit theorem for square ice

In the area of statistical mechanics, an important open question is to show that the height function associated with the square ice model (i.e., planar six vertex model with uniform weights), or equivalently the uniform graph homeomorphisms, converges to a continuum Gaussian free field In the scaling limit, I will review some recent results about this model, including that the single point height function, upon renormalization, converges to a Gaussian random variable. 

Sep 27  @  Temple: Hoi Nguyen (Ohio State)

On roots of random trigonometric polynomials and related models

In this talk, we will discuss various basic statistics of the number of real roots of random trigonometric polynomials, as well as the minimum modulus and the nearest roots statistics to the unit circle of Kac polynomials. We will emphasize the universality aspects of all these problems. Based on joint works with Cook, Do, O. Nguyen, Yakir and Zeitouni. 

Oct 4  @  Penn: Jiaqi Liu (Penn)

Yaglom-type limits for branching Brownian motion with absorption in the slightly subcritical regime

Branching Brownian motion is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we consider a slightly subcritical branching Brownian motion with absorption, where particles move as Brownian motion with drift, undergo dyadic fission at a constant rate, and are killed upon hitting the origin. We are interested in the asymptotic behaviors of the process conditioned on survival up to a large time t as the process approaches criticality. Results like this are called Yaglom type results. Specifically, we will talk about the construction of the Yaglom limit law, Yaglom-type limits for the number of particles and the maximal displacement. Based on joint work with Julien Berestycki, Bastien Mallein and Jason Schweinsberg.

Oct 11  @  Temple: Chris Janjigian (Purdue)

Ergodicity and synchronization of the KPZ equation

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some results-in-progress which show that these distributions are totally ergodic and present some progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. The talk will discuss our coupling of Hopf-Cole solutions, which enables us to study the KPZ equation started from any measurable function valued initial condition. Through this coupling, we give a sharp characterization of when such solutions explode, show that all non-explosive functions become instantaneously continuous, and then study the problem of ergodicity on a natural topology on the space of non-explosive continuous functions (mod constants) in which the equation defines a Feller process. We show that any ergodic stationary distribution on this space is either a Brownian motion with drift or a process of a very peculiar form which will be described in the talk. 

Based on joint works with Tom Alberts, Firas Rassoul-Agha, and Timo Seppäläinen.

Oct 18  @  Penn: Fan Wei (Princeton)

Graph limits and graph homomorphism density inequalities

Graph limits is a recently developed powerful theory in studying large (weighted) graphs from a continuous and analytical perspective. It is particularly useful when studying subgraph homomorphism density, which is closely related to graph property testing, graph parameter estimation, and many central questions in extremal combinatorics. In this talk, we will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when is the random construction close to optimal? We will also show some hardness results for proving general theorems in graph homomorphism density inequalities.

Oct 25  @  Temple:  Lucas Benigni (Université de Montréal)

Optimal delocalization for generalized Wigner matrices

We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

Nov 1  @  Penn: Ahmed Bou-Rabee (Cornell)

Sandpiles

I will introduce the Abelian sandpile model and discuss its large-scale behavior in random environments and on different lattices. There are many open questions.

Nov 8  @  Temple: Şefika Kuzgun (Rochester)

Convergence of densities of spatial averages of the stochastic heat equation

Let u be the solution to the one-dimensional stochastic heat equation driven by a space-time white noise with constant initial condition. The purpose of this talk is to present a recent result on the uniform convergence of the density of the normalized spatial averages of the solution u on an interval [−R,R], as R tends to infinity, to the density of the standard normal distribution, assuming some non-degeneracy and regularity conditions on the diffusion coefficient. These results are based on the combination of Stein's method for normal approximations and Malliavin calculus techniques. This is a joint work with David Nualart.

Nov 29  @  Penn: Yizhe Zhu (UCI)

Non-backtracking spectra of random hypergraphs and community detection

The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL). 

Spring 2022

Apr 26  @  Penn: Antoine Jego (MSRI) 

Multiplicative chaos of the Brownian loop soup

On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity \theta times a loop measure. At criticality (\theta = 1/2), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism). The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when \theta is not equal to 1/2.

Apr 19  @  Temple:  Milind Hegde (Columbia)

Understanding the upper tail behaviour of the KPZ equation via the tangent method

The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, closely connected to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data. 

Apr 12  @  Penn: Amol Aggarwal (Columbia)

Six-Vertex Model and the KPZ Universality Class

In this talk we explain recent results relating the six-vertex model and the Kardar-Parisi-Zhang (KPZ) universality class. In particular, we describe how the six-vertex model can be used to analyze stochastic interacting particle systems, such as asymmetric exclusion processes, and how infinite-volume pure states of the ferroelectric six-vertex model exhibit fluctuations of order N^{1/3}, a characteristic feature of systems in the KPZ universality class.

Apr 5  @  Temple: Kasper Larsen (Rutgers)

Uniqueness in Cauchy problems for diffusive real-valued strict local martingales 

For a real-valued one-dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution. We exemplify our results using quadratic normal volatility models and the two-dimensional Bessel process. Joint work with Umut Cetin (LSE).

Mar 29  @  Temple: Johannes Alt (Courant)

Localization and Delocalization in Erdős–Rényi graphs

We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously. 

This is based on joint works with Raphael Ducatez and Antti Knowles.

Mar 22  @  Temple: Elena Kosygina (Baruch College & The Graduate Center, CUNY)

From generalized Ray-Knight theorems to functional limit theorems for some models of self-interacting random walks on integers

For several models of self-interacting random walks (SIRWs), generalized Ray-Knight theorems for edge local times are a very useful tool for studying the limiting distributions of these walks. Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs studied by Balint Toth (1996), asymptotically free and polynomially self-repelling SIRWs, and discuss recent results (joint work with Thomas Mountford, EPFL, and Jon Peterson, Purdue University) which resolve an open question posed in Toth’s paper. We show that, in the asymptotically free case, the rescaled SIRWs converge to a perturbed Brownian motion (conjectured by Toth), while in the polynomially self-repelling case, the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of perturbed Brownian motions. This negative result was somewhat unexpected. Conjectures on whether there is a suitable limiting process in this case and, if yes, what it might be are welcome.

Mar 15  @  Penn: Hugo Falconet (Courant)

Metric growth dynamics in Liouville quantum gravity

Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with conformal symmetries was introduced in the physics literature by Polyakov in the 80’s and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls. 

Based on a joint work with Julien Dubédat.

Mar 8  @  Temple: Emma Bailey (CUNY)

Large deviation estimates for Selberg’s central limit theorem and applications

Selberg’s celebrated central limit theorem shows that the logarithm of the zeta function at a typical point on the critical line behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent results showing that the Gaussian decay persists in the large deviation regime, at a level on the order of the variance, improving on the best known bounds in that range.  We also present various applications, including on the maximum of the zeta function in short intervals. Whilst the results are number theoretic, the tools used are predominantly probabilistic in nature.  This work is joint with Louis-Pierre Arguin. 

Feb 22  @  Penn: Lingfu Zhang (Princeton)

The local environment of a geodesic in Last-Passage Percolation

In exponential Last-Passage Percolation, each vertex in the 2D lattice is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. This model was first introduced to model fluid flow through a random medium. It is also a central model in the KPZ universality class and related to various natural processes. 

A classical question asks what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present new results on the convergence of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure. In addition, we obtain an explicit description of the limiting ‘environment’. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples.

This is based on joint work with James Martin and Allan Sly.

Feb 15  @  Penn: Xuan Wu (Chicago)

Scaling limits of the Laguerre unitary ensemble 

In this talk, we will discuss the LUE, focusing on the scaling limits. On the hard-edge side, we construct the $\alpha$-Bessel line ensemble for all $\alpha \in \mathbb{N}_0$. This novel Gibbsian line ensemble enjoys the $\alpha$-squared Bessel Gibbs property. Moreover, all $\alpha$-Bessel line ensembles can be naturally coupled together in a Bessel field, which enjoys rich integrable structures. We will also talk about work in progress on the soft-edge side, where we expect to have the Airy field as the scaling limit. This talk is based on joint works with Lucas Benigni, Pei-Ken Hung, and Greg Lawler.

Feb 8  @  Penn: Marianna Russkikh (MIT)

Lozenge Tilings and the Gaussian Free Field on a Cylinder 

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.  

Feb 1  @  Temple: Si Tang (Lehigh)

On convergence of the cavity and Bolthausen’s TAP iterations to the local magnetization

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington-Kirkpatrick model. In the seminal work, Bolthausen introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida-Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called Approximate Message Passing algorithm, a generalization of Bolthausen’s iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated. This is a joint work with Wei-Kuo Chen (University of Minnesota).

Jan 25  @  Penn: Giorgio Cipolloni (Princeton)

Strong Quantum Unique Ergodicity and its Gaussian fluctuations for Wigner matrices

We prove that the eigenvectors of Wigner matrices satisfy the Eigenstate Thermalisation Hypothesis (ETH), which is a strong form of Quantum Unique Ergodicity (QUE) with optimal speed of convergence. Then, using this a priori bound as an input, we analyze the stochastic Eigenstate Equation (SEE) and prove Gaussian fluctuations in the QUE. The main methods behind the above results are: (i) multi-resolvents local laws established via a novel bootstrap scheme; (ii) energy estimates for SEE.

Fall 2021

Dec 7  @  Temple: Louis-Pierre Arguin (CUNY Baruch College)

Large values of the Riemann zeta function in short intervals 

I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables, as well as random matrix theory.

Nov 30  @  Temple: Krishnan Mody (Courant)

Central limit theorem for the characteristic polynomial of general beta-ensembles 

I will discuss recent work with P. Bourgade and M. Pain in which we show that the log-characteristic polynomial for general beta ensembles converges to a log-correlated field in the large-dimension limit. The proof of this result relies on a so-called optimal local law, which I will explain and prove in the Gaussian case. I will explain how the local law is useful, and give an outline of the proof of the log-correlated field. 

Nov 16  @  Penn: Jacopo Borga (Stanford)

The Skew Brownian Permuton

Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, mainly focusing on the case of pattern-avoiding permutations. The main goal of the talk is to present a new family of universal limiting permutons, called skew Brownian permutons. This family includes (as particular cases) some already studied limiting permutons, such as the biased Brownian separable permuton and the Baxter permuton. We also show that some natural families of random constrained permutations converge to some new instances of skew Brownian permutons. The construction of these new limiting objects will lead us to investigate an intriguing connection with some perturbed versions of the Tanaka SDE and the SDEs encoding skew Brownian motions. If time permits, we will present some conjectures on how it should be possible to construct these new limiting permutons directly from the Liouville quantum gravity decorated with two SLE curves.

Nov 9  @  Temple: Christian Houdré (Georgia Tech)

On the limiting shape of Young diagram associated with Markov random words

Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space a totally ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of the associated RSK Young diagrams as a multidimensional Brownian functional. Since the length of the top row of the Young diagrams is also the length of the longest weakly increasing subsequences of $(X_k)_{1\le k \le n}$, the corresponding limiting law follows. We relate our results to a conjecture of Kuperberg by providing, under a cyclic condition, a spectral characterization of the Markov transition matrix precisely characterizing when the limiting shape is the spectrum of the $m \times m$ traceless GUE. For each $m \ge 4$, this characterization identifies a proper, non-trivial class of cyclic transition matrices producing such a limiting shape. However, for $m=3$, all cyclic Markov chains have such a limiting shape, a fact previously only known for $m=2$. For $m$ arbitrary, we also study reversible Markov chains and obtain a characterization of symmetric Markov chains for which the limiting shape is the spectrum of the traceless GUE. To finish, we explore, in this general setting, connections between various limiting laws and spectra of Gaussian random matrices, focusing in particular on the relationship between the terminal points of the Brownian motions, the diagonal terms of the random matrix, and the scaling of its off-diagonal terms, a scaling we conjecture to be a function of the spectrum of the covariance matrix governing the Brownian motion.

Joint work with Trevis Litherland.

Nov 2  @  Penn: Joshua Pfeffer (Columbia)

Loewner chains driven by complex Brownian motion

In my talk, I will discuss Loewner chains whose driving functions are complex Brownian motions with general covariance matrices.  This extends the notion of Schramm-Loewner evolution (SLE) by allowing the driving function to be complex-valued and not just real-valued.  We show that these Loewner chains exhibit the same phases (simple, swallowing, and space-filling) as SLE, and we explicitly characterize the values of the covariance matrix corresponding to each phase.  In contrast to SLE, we show that the evolving left hulls are a.s. not generated by curves, and that they a.s. disconnect each fixed point in the plane from infinity before absorbing the point.  

This talk is based on a joint work with Ewain Gwynne.

Oct 26  @  Temple: Jonathon Peterson (Purdue)

Gaussian, stable, and tempered stable limiting distributions for random walks in cooling random environments

Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has an $s$-stable limiting distribution for some $s \in (1,2)$. It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to $s$-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered $s$-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest.

Oct 19  @  Penn: Guillaume Remy (Columbia)

Integrability of boundary Liouville CFT

Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determine by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.

Oct 12  @  Penn: Robert Hough (Stony Brook)

The local limit theorem on nilpotent groups

Alexopoulos proved local limit theorems for measures with a density and lattice measures in the general setting of groups of moderate growth.  On the Heisenberg group, Breuillard's thesis obtained a local limit theorem for general measures subject to a condition on the characteristic function, and asked if this condition can be removed.  I will discuss two new local limit theorems, one joint with Diaconis, that treat local limit theorems on nilpotent Lie groups driven by general measures.  We prove Breuillard's conjecture and also solve a problem of Diaconis and Saloff-Coste on the mixing of the central coordinate in unipotent matrix walks modulo $p$.

Oct 6  @  Penn Mathematics Colloquium: Evita Nestoridi (Princeton) 

Spectral techniques in Markov chain mixing

How many steps does it take to shuffle a deck of $n$ cards, if at each step we pick two cards uniformly at random and swap them? Diaconis and Shahshahani proved that $\frac{1}{2} n log n$ steps are necessary and sufficient to mix the deck. Using the representation theory of the symmetric group, they proved that this random transpositions card shuffle exhibits a sharp transition from being unshuffled to being very well shuffled.  This is called the cutoff phenomenon.  In this talk, I will explain how to use the spectral information of a Markov chain to study cutoff. As an application, I will briefly discuss the random to random card shuffle (joint with M. Bernstein) and the non-backtracking random walk on Ramanujan graphs (joint with P. Sarnak).

Oct 5  @  Temple: David Renfrew (SUNY Binghamton) 

Singularities in the spectrum of random block matrices

We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

Sep 28  @  Penn: Minjae Park (MIT)

Wilson loop expectations as sums over surfaces in 2D

Although lattice Yang-Mills theory on ℤᵈ is easy to rigorously define, the construction of a satisfactory continuum theory on ℝᵈ is a major open problem when d ≥ 3. Such a theory should assign a Wilson loop expectation to each suitable collection ℒ of loops in ℝᵈ. One classical approach is to try to represent this expectation as a sum over surfaces with boundary ℒ. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.

In this talk, we show how to make sense of Yang-Mills integrals as surface sums for d=2, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.

Sep 21  @  Penn: Jiaming Xia (UPenn)

Hamilton-Jacobi equations for statistical inference problems

In this talk, I will first present the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. We compare the limit with the solution to a certain Hamilton-Jacobi equation, following the recent approach by Jean-Christophe Mourrat. The motivation comes from the averaged free energy solving an approximate Hamilton-Jacobi equation. We consider two notions of solutions which are weak solutions and viscosity solutions. The two types of solutions require different treatments and each has its own advantages. At the end of this part, I will show an example of application of our results to a model with i.i.d. entries and symmetric interactions. If time permits, I will talk about the same problem but with a different model, namely, the multi-layer generalized linear model. I will mainly explain the iteration method as an important tool used in our proof. This is based on joint works with Hong-Bin Chen and J.-C. Mourrat, NYU.

Sep 14  @  Temple: Jack Hanson  (CUNY)

Spanning clusters and subcritical connectivity in high-dimensional percolation

In their study of percolation, physicists have proposed ``scaling hypotheses'' relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ -- the probability that the open cluster of the origin has Euclidean diameter at least $n$.

As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.

Sep 7  @  Penn: Fan Yang (UPenn)

Delocalization and quantum diffusion of random band matrices in high dimensions

We consider a Hermitian random band matrix H on the d-dimensional lattice of linear size L. Its entries are independent centered complex Gaussian random variables with variances s_{xy}, that are negligible if |x-y| exceeds the band width W. In dimensions eight or higher, we prove that, as long as W> L^\epsilon for a small constant \epsilon>0, with high probability, most bulk eigenvectors of H are delocalized in the sense that their localization lengths are comparable to L. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of H. Joint work with Horng-Tzer Yau and Jun Yin.

Spring 2020

Mar 3  @  Penn: Jonathan Niles-Weed(Courant)

Estimation of Wasserstein distances in the Spiked Transport Model

We propose a new statistical model, generalizing the spiked covariance model, which formalizes the assumption that two probability distributions differ only on a low-dimensional subspace. We study various probabilistic and statistical features of this model, including the estimation of the Wasserstein distance, which we show can be accomplished by an estimator which avoids the "curse of dimensionality" typically present in high-dimensional problems involving the Wasserstein distance. However, this estimator does not seem possible to compute in polynomial time, and we give evidence that any computationally efficient estimator is bound to suffer from the curse of dimensionality. Our results therefore suggest the existence of a computational-statistical gap.

Joint work with Philippe Rigollet.

Feb 25  @  Temple: Benjamin Landon(MIT)

Universality of extremal eigenvalue statistics of random matrices

The past decade has seen significant progress on the understanding of universality of various eigenvalue statistics of random matrix theory.  However, the behavior of certain ``extremal'' or ``critical'' observables is not fully understood.  Towards the former, we discuss progress  on the universality of the largest gap between consecutive eigenvalues.  With regards to the latter, we discuss the central limit theorem for the eigenvalue counting function, which can be viewed as a linear spectral statistic with critical regularity and has logarithmically growing variance.

Feb 18  @  Penn: Jiaoyang Huang(Institute for Advanced Study)

Extreme eigenvalue distributions of sparse random graphs

I will discuss the extreme eigenvalue distributions of adjacency matrices of sparse random graphs, in particular the Erd{\H o}s-R{\'e}nyi graphs $G(N,p)$ and the random $d$-regular graphs. For Erd{\H o}s-R{\'e}nyi graphs, there is a crossover in the behavior of the extreme eigenvalues. When the average degree $Np$ is much larger than $N^{1/3}$, the extreme eigenvalues have asymptotically Tracy-Widom fluctuations, the same as Gaussian orthogonal ensemble. However, when $N^{2/9}\ll Np\ll N^{1/3}$ the extreme eigenvalues have asymptotically Gaussian fluctuations. The extreme eigenvalues of random $d$-regular graphs are more rigid, we prove on the regime $N^{2/9}\ll d\ll N^{1/3}$ the extremal eigenvalues are concentrated at scale $N^{-2/3}$ and their fluctuations are governed by the Tracy-Widom statistics. Thus, in the same regime of $d$, $52\%$ of all $d$-regular graphs have the second-largest eigenvalue strictly less than $2\sqrt{d-1}$. These are based on joint works with Roland Bauerschmids, Antti Knowles, Benjamin Landon and Horng-Tzer Yau.

Feb 11  @  Temple: Vladislav Kargin (Binghamton, SUNY)

Entropy of ribbon tilings

I will talk about ribbon tilings, which have been originally introduced and studied by Pak and Sheffield. These are a generalization of the domino tilings which, unfortunately, lacks relations to determinants and spanning trees but still retains some of the nice properties of domino tilings. I will explain how ribbon tilings are connected to multidimensional heights and acyclic orientations, and present some results about enumeration of these tilings on simple regions. Joint work with Yinsong Chen.

Feb 4   @  Penn: Jian Song(Shandong University)

Scaling limit of a directed polymer among a Poisson field of independent walks

We consider a directed polymer model in dimension 1+1, where the disorder is given by the occupation field of a Poisson system of independent random walks on Z. In a suitable continuum and weak disorder limit, we show that the family of quenched partition functions of the directed polymer converges to the Stratonovich solution of a multiplicative stochastic heat equation with a Gaussian noise whose space-time covariance is given by the heat kernel.

Jan 28  @  Temple: Konstantin Matetski (Columbia)

The KPZ fixed point

The KPZ universality class is a broad collection of models, which includes directed random polymers, interacting particle systems and random interface growth, characterized by unusual scale of fluctuations which also appear in the random matrix theory. The KPZ fixed point is a scaling invariant Markov process which is the conjectural universal limit of all models in the class. A complete description of the KPZ fixed point was obtained in a joint work with Jeremy Quastel and Daniel Remenik. In this talk I will describe how the KPZ fixed point was derived by solving a special model in the class called TASEP.

Jan 21  @  Penn: Jonathan Weare (Courant)

Fast randomized iterative numerical linear algebra for quantum chemistry (and other applications)

I will discuss a family of recently developed stochastic techniques for linear algebra problems involving very large matrices.  These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.  The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states since the 1970's.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a very partial explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, Sam Greene, and Rob Webber.

Fall 2019

Dec 3  @ Penn: Eyal Lubetzky (Courant)

Maximum of 3D Ising interfaces 

Consider the random surface separating the plus and minus phases, above and below the $xy$-plane, in the low temperature Ising model in dimension $d\geq 3$. Dobrushin (1972) showed that if the inverse-temperature $\beta$ is large enough then this interface is localized: it has $O(1)$ height fluctuations above a fixed point, and its maximum height on a box of side length $n$ is $O_P ( \log n )$.

We study the large deviations of the interface in Dobrushin’s setting, and derive a shape theorem for its ``pillars’’ conditionally on reaching an atypically large height. We use this to obtain a law of large numbers for the maximum height $M_n$ of the interface: $M_n/ \log n$ converges to $c_\beta$ in probability, where $c_\beta$ is given by a large deviation rate in infinite volume. Furthermore, the sequence $(M_n - E[M_n])_{n\geq 1}$ is tight, and even though this sequence does not converge, its subsequential limits satisfy uniform Gumbel tails bounds.

Joint work with Reza Gheissari.

Nov 19 @ Temple: Yu Gu (CMU)

The Edwards-Wilkinson limit of the KPZ equation in d>1

In this talk, I will explain some recent work where we prove that in a certain weak disorder regime, the KPZ equation scales to the Edwards-Wilkinson equation in d>1.

Nov 12 @ Penn: Changji Xu (Chicago)

Sharp threshold for the Ising perceptron model

Consider the discrete cube ${-1,1}^N$ and a random collection of half spaces which includes each half space $H(x) := {y in {-1,1}^N: x cdot y geq kappa sqrt{N}}$ for $x in {-1,1}^N$ independently with probability $p$. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold; that is, the probability that the intersection is empty increases quickly from $epsilon$ to $1- epsilon$ when $p$ increases only by a factor of $1 + o(1)$ as $N o infty$.

Nov 5 @ Temple: Michael Damron (Georgia Tech)

Absence of backward infinite paths in first-passage percolation in arbitrary dimension

In first-passage percolation (FPP), one places weights (t_e) on the edges of Z^d and considers the induced metric. Optimizing paths for this metric are called geodesics, and infinite geodesics are infinite paths all whose finite subpaths are geodesics. It is a major open problem to show that in two dimensions, with i.i.d. continuous weights, there are no bigeodesics (doubly-infinite geodesics). In this talk, I will describe work on bigeodesics in arbitrary dimension using "geodesic graph'' measures introduced in '13 in joint work with J. Hanson. Our main result is that these measures are supported on graphs with no doubly-infinite paths, and this implies that bigeodesics cannot be constructed in a translation-invariant manner in any dimension as limits of point-to-hyperplane geodesics. Because all previous works on bigeodesics were for two dimensions and heavily used planarity and coalescence, we must develop new tools based on the mass transport principle. Joint with G. Brito (Georgia Tech) and J. Hanson (CUNY).

Oct 29 @ Penn: Eliran Subag (Courant)

Geometric TAP approach for spherical spin glasses

The celebrated Thouless-Anderson-Palmer approach suggests a way to relate the free energy of a mean-field spin glass model to the solutions of certain self-consistency equations for the local magnetizations. In this talk I will first describe a new geometric approach to define free energy landscapes for general spherical mixed p-spin models and derive from them a generalized TAP representation for the free energy. I will then explain how these landscapes are related to various concepts and problems: the pure states decomposition, ultrametricity, temperature chaos, and optimization of full-RSB models.

Oct 15 @ Penn: Tatyana Shcherbyna(Princeton)

Local regime of random band matrices

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.

Oct 8 @ Temple: Li-Cheng Tsai (Rutgers)

Lower-tail large deviations of the KPZ Equation

Consider the solution of the KPZ equation with the narrow wedge initial condition. We prove the one-point, lower-tail Large Deviation Principle (LDP) of the solution, with time $ t\to\infty $ being the scaling parameter, and with an explicit rate function. This result confirms existing physics predictions. We utilize a formula from Borodin and Gorin (2016) to convert the LDP of the KPZ equation to calculating an exponential moment of the Airy point process, and analyze the latter via stochastic Airy operator and Riccati transform.

Oct 1 @ Penn: Amir Dembo (Stanford)

Dynamics for spherical spin glasses: Disorder dependent initial conditions

In this talk, based on a joint work with Eliran Subag, I will explain how to rigorously derive the integro-differential equations that arise in the thermodynamic limit of the empirical correlation and response functions for Langevin dynamics in mixed spherical p-spin disordered mean-field models. 

I will then compare the large time asymptotic of these equations in case of a uniform (infinite-temperature) starting point, to what one obtains when starting within one of the spherical bands on which the Gibbs measure concentrates at low temperature, commenting on the existence of an aging phenomenon, and on the relations with the recently discovered geometric structure of the Gibbs measures at low temperature.

Sep 24 @ Temple: Axel Saenz-Rodriguez (Virginia)

Stationary Dynamics in Finite Time for the Totally Asymmetric Simple Exclusion Process

The totally asymmetric simple exclusion process (TASEP) is a Markov process that is the prototypical model for transport phenomena in non-equilibrium statistical mechanics. It was first introduced by Spitzer in 1970, and in the last 20 years, it has gained a strong resurgence in the emerging field of "Integrable Probability" due to exact formulas from Johanson in 2000 and Tracy and Widom in 2007 (among other related formulas and results). In particular, these formulas led to great insights regarding fluctuations related to the Tracy-Widom distribution and scalings to the Kardar-Parisi-Zhang (KPZ) stochastic differential equation. 

In this joint work with Leonid Petrov (University of Virginia), we introduce a new and simple Markov Process that maps the distribution of the TASEP at time $t>0$, given step initial time data, to the distribution of the TASEP at some earlier time $t-s>0$. This process "back in time" is closely related to the Hammersley Process introduced by JM Hammersley in 1972, which later found a resurgence in the longest increasing subsequence problem in the work of Aldous and Diaconis in 1995. Hence, we call our process the Backwards Hammersley-type Process (BHP). As an fun application of our results, we have a new proof of the limit shape for the TASEP. The central objects in our constructions and proofs are the Schur point processes and the Yang-Baxter equation for the sl_2 quantum affine Lie algebra.  In this talk, we will discuss the background in more detail and will explain the main ideas behind the constructions and proof. 

September 17 @Penn: Izabella Stuhl(PSU)

Hard-core models in discrete 2D 

Do hard disks in the plane admit a unique Gibbs measure at high density? This is one of the outstanding open problems of statistical mechanics, and it seems natural to approach it by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter $D$ of the hard disks tend to infinity. Unlike most models in statistical physics, we find non-universality and connections to number theory, with different new phenomena arising in the triangular lattice $\mathbb{A}_2$, the square lattice $\mathbb{Z}^2$ and the hexagonal tiling $\mathbb{H}_2$.

In particular, number-theoretic properties of the exclusion diameter $D$ turn out to be important. We analyze high-density hard-core Gibbs measures via Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal density disk configurations which cannot be locally 'improved'. A key finding is that only certain `dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations.

Answers are provided in terms of Eisenstein primes for $\mathbb{A}_2$ and norm equations in the ring $\mathbb{Z}[\sqrt{3}]$ for $\mathbb{Z}^2$. The number of high-density hard-core Gibbs measures grows indefinitely with $D$ but non-monotonically. In $\mathbb{Z}^2$ we analyze the phenomenon of 'sliding' and show it is rare.

This is a joint work with A. Mazel and Y. Suhov.

Sep 10 @Temple: Pierre Yves Gaudreau Lamarre(Princeton)

Semigroups for One-Dimensional Schrödinger Operators with Multiplicative White Noise

In this talk, we are interested in the semigroup theory of continuous one-dimensional random Schrödinger Operators with white noise. We will begin with a brief reminder of the rigorous definition of these operators as well as some of the problems in which they naturally arise. Then, we will discuss the proof of a Feynman-Kac formula describing their semigroups. In closing, we will showcase an application of this new semigroup theory to the study of rigidity (in the sense of Ghosh-Peres) of random Schrödinger eigenvalue point processes.

Some of the results discussed in this talk are joint work with Promit Ghosal (Columbia) and Yuchen Liao (Michigan).

September 03 @Penn: Ewain Gwynne (Cambridge)

Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$

We show that for each $\gamma \in (0,2)$, there is a unique metric associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the Gaussian free field $h$ on a planar domain $U$, there is a unique random metric $D_h = ``e^{\gamma h} (dx^2 + dy^2)"$ on $U$ which is uniquely characterized by a list of natural axioms. 

The $\gamma$-LQG metric can be constructed explicitly as the scaling limit of \emph{Liouville first passage percolation} (LFPP), the random metric obtained by exponentiating a mollified version of the Gaussian free field. Earlier work by Ding, Dub\'edat, Dunlap, and Falconet (2019) showed that LFPP admits non-trivial subsequential limits. We show that the subsequential limit is unique and satisfies our list of axioms. In the case when $\gamma = \sqrt{8/3}$, our metric coincides with the $\sqrt{8/3}$-LQG metric constructed in previous work by Miller and Sheffield. 

Based on four joint papers with Jason Miller, one joint paper with Julien Dubedat, Hugo Falconet, Josh Pfeffer, and Xin Sun, and one joint paper with Josh Pfeffer.

Spring 2019

April 30 @ Temple: Tom Alberts (Utah)

The geometry of the last passage percolation problem

Last passage percolation is a well-studied model in probability theory that is simple to state but notoriously difficult to analyze. In recent years it has been shown to be related to many seemingly unrelated things: longest increasing subsequences in random permutations, eigenvalues of random matrices, and long-time asymptotics of solutions to stochastic partial differential equations. Much of the previous analysis of the last passage model has been made possible through connections with representation theory of the symmetric group that comes about for certain exact choices of the random input into the last passage model. This has the disadvantage that if the random inputs are modified even slightly then the analysis falls apart. In an attempt to generalize beyond exact analysis, recently my collaborator Eric Cator (Radboud University, Nijmegen) and I have started using tools of tropical geometry to analyze the last passage model. The tools we use to this point are purely geometric, but have the potential advantage that they can be used for very general choices of random inputs. I will describe the very pretty geometry of the last passage model and our work to use it to produce probabilistic information. 

April 16 @ Temple: Jessica Lin (McGill)

Stochastic homogenization for reaction-diffusion equations

I will present several results concerning the stochastic homogenization for reaction-diffusion equations. We consider reaction-diffusion equations with nonlinear, heterogeneous, stationary-ergodic reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation. Our arguments rely on analyzing a suitable analogue of “first passage times” for solutions of reaction-diffusion equations. In particular, under these hypotheses, solutions of heterogeneous reaction-diffusion equations with front-like initial data become asymptotically front-like with a deterministic speed. This talk is based on joint work with Andrej Zlatos. 

April 9 @ Temple: Guillaume Dubach (Courant)

Eigenvectors of non-Hermitian random matrices

Eigenvectors of non-Hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables also quantify the stability of the spectrum, and characterize the joint eigenvalue increments under Dyson-type dynamics. Overlaps first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices. 

April 2 @ Temple: Timo Seppalainen (UW-Madison)

Geometry of the corner growth model

The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah). 

March 26 @ Penn: Xiaoming Song (Drexel)

Large deviations for functionals of Gaussian processes

We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-β}$ with $β\in (0,d)$, or  $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i\in (0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed. 

March 19 @ Penn: Fan Yang (UCLA)

Delocalization of random band matrices 

 We consider Hermitian random band matrices $H$ in dimension $d$, where the entries $h_{xy}$, indexed by $x,y \in [1,N]^d$, vanishes if $|x-y|$ exceeds the band width $W$. It is conjectured that a sharp transition of the eigenvalue and eigenvector statistics occurs at a critical band width $W_c$, with $W_c=\sqrt{N}$ in $d=1$, $W_c=\sqrt{\log N}$ in $d=2$, and $W_c=O(1)$ in $d\ge 3$. Recently, Bourgade, Yau and Yin proved the eigenvector delocalization for 1D random band matrices with generally distributed entries and band width $W\gg N^{3/4}$. In this talk, we will show that for $d\ge 2$, the delocalization of eigenvectors in certain averaged sense holds under the condition $W\gg N^{2/(2+d)}$. Based on Joint work with Bourgade, Yau and Yin.

February 26 @Penn: Tiefeng Jiang (Minnesota)

Distances between Random Orthogonal Matrices and Independent Normals

We study the distance between Haar-orthogonal matrices and independent normal random variables.The distance is measured by the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. They appear different features. Optimal rates are obtained. This is a joint work with Yutao Ma.

February 19 @Penn: Duncan Dauvergne (Toronto)

Asymptotic zero distribution of random polynomials

It is well known that the roots of a random polynomial with i.i.d. coefficients tend to concentrate near the unit circle. In particular, the zero measures of such random polynomials converge almost surely to normalized Lebesgue measure on the unit circle if and only if the underlying coefficient distribution satisfies a particular moment condition. In this talk, I will discuss how to generalize this result to random sums of orthogonal (or asymptotically minimal) polynomials.

February 13 @Penn Colloquium: Bálint Virág (Toronto)

Random interfaces and geodesics

Coastlines, the edge of burned paper, the boundary of coffee spots, the game of Tetris: random interfaces surround us. Still, the mathematical theory of the most important case, the "KPZ universality class", has only been cracked very recently. This class is related to traffic models, longest increasing subsequences of random permutations, the RSK correspondence of combinatorics, last passage percolation, integrable systems and the stochastic heat equation. A new random metric in the plane, the "directed landscape" captures the essence of these problems.

February 5 @Penn: Xin Sun (Columbia)

Conformal embedding and percolation on the uniform triangulation

Following Smirnov's proof of Cardy's formula and Schramm's discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice the has been achieved. Smirnorv's proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. I will also explain how this result fits in the the larger picture of random planar maps and Liouville quantum gravity.

January 29 @Temple: Alex Moll (Northeastern)

Fractional Gaussian Fields in Geometric Quantization and the Semi-Classical Analysis of Coherent States

The Born Rule (1926) formalized in von Neumann's spectral theorem (1932) gives a precise definition of the random outcomes of quantum measurements as random variables from the spectral theory of non-random matrices. In [M. 2017], the Born rule provided a way to derive limit shapes and global fractional Gaussian field fluctuations for a large class of point processes from the first principles of geometric quantization and semi-classical analysis of coherent states. Rather than take a point process as a starting point, these point process are realized as auxiliary objects in an analysis that starts instead from a classical Hamiltonian system with possibly infinitely-many degrees of freedom that is not necessarily Liouville integrable. In this talk, we present these results with a focus on the case of one degree of freedom, where the core ideas in the arguments are faithfully represented.

January 22 @Penn: Xinyi Li (Chicago)

One-point function estimates and natural parametrization for loop-erased random walk in three dimensions

In this talk, I will talk about loop-erased random walk (LERW) in three dimensions. I will first give an asymptotic estimate on the probability that 3D LERW passes a given point (commonly referred to as the one-point function). I will then talk about how to apply this estimate to show that 3D LERW as a curve converges to its scaling limit in natural parametrization. If time permits, I will also talk about the asymptotics of non-intersection probabilities of 3D LERW with simple random walk. This is a joint work with Daisuke Shiraishi (Kyoto).

Fall 2018

December 4 @ Temple: Pascal Maillard (CRM)

The algorithmic hardness threshold for continuous random energy models

I will report on recent work with Louigi Addario-Berry on algorithmic hardness for finding low-energy states in the continuous random energy model of Bovier and Kurkova. This model can be regarded as a toy model for strongly correlated random energy landscapes such as the Sherrington--Kirkpatrick model. We exhibit a precise and explicit hardness threshold: finding states of energy above the threshold can be done in linear time, while below the threshold this takes exponential time for any algorithm with high probability. I further discuss what insights this yields for understanding algorithmic hardness thresholds for random instances of combinatorial optimization problems.

November 27 @ Temple: Sunder Sethuraman (Arizona)

Stick breaking processes, clumping, and Markov chain occupation laws

A GEM (Griffiths-Engen-McCloskey) sequence specifies the (random) proportions in splitting a `resource' infinitely many ways. Such sequences form the backbone of `stick breaking' representations of Dirichlet processes used in nonparametric Bayesian statistics. In this talk, we consider the connections between a class of generalized `stick breaking' processes, an intermediate structure via `clumped' GEM sequences, and the occupation laws of certain time-inhomogeneous Markov chains.

November 13 @ Penn: Abram Magner (Purdue)

Inference and Compression Problems on Dynamic Networks

Networks in the real world are dynamic -- nodes and edges are added and removed over time, and time-varying processes (such as epidemics) run on them. In this talk, I will describe mathematical aspects of some of my recent work with collaborators on statistical inference and compression problems that involve this time-varying aspect of networks. I will focus on two related lines of work: (i) network archaeology -- broadly concerning problems of dynamic graph model validation and inference about previous states of a network given a snapshot of its current state, and (ii) structural compression -- for a given graph model, exhibit an efficient algorithm for invertibly mapping network structures (i.e., graph isomorphism types) to bit strings of minimum expected length. For both classes of problems, I give both information-theoretic limits and efficient algorithms for achieving those limits. Finally, I briefly describe some ongoing projects that continue these lines of work.

November 6 @ Penn: Philippe Sosoe (Cornell)

Applications of CLTs and homogenization for Dyson Brownian Motion to Random Matrix Theory

I will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric matrices. No prior knowledge of random matrix theory will be assumed. (Based on joint work with B Landon and HT Yau).

October 30 @ Temple: Anirban Basak (ICTS-TIFR)

Sharp transition of invertibility of sparse random matrices

Consider an $n \times n$ matrix with i.i.d.~Bernoulli($p$) entries. It is well known that for $p= \Omega(1)$, i.e.~$p$ is bounded below by some positive constant, the matrix is invertible with high probability. If $p \ll \frac{\log n}{n}$ then the matrix contains zero rows and columns with high probability and hence it is singular with high probability. In this talk, we will discuss the sharp transition of the invertibility of this matrix at $p =\frac{\log n}{n}$. This phenomenon extends to the adjacency matrices of directed and undirected Erd\H{o}s-R\'{e}nyi graphs, and random bipartite graph. Joint work with Mark Rudelson.

October 23 @ Penn: Julian Gold (Northwestern)

Isoperimetric shapes in supercritical bond percolation

We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ of supercritical bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove a shape theorem for these random graphs, showing that upon rescaling they tend almost surely to a deterministic shape. This limit shape is itself an isoperimetric set for a norm we construct. In addition, we obtain sharp asymptotics for a modification of the Cheeger constant of $\textbf{C}_\infty \cap [-n,n]^d$, settling a conjecture of Benjamini for this modified Cheeger constant. Analogous results are shown for the giant component in dimension two, where we use the original definition of the Cheeger constant, and a more complicated continuum isoperimetric problem emerges as a result.

October 9 @ Temple: Janos Englander (Boulder)

The coin turning walk and its scaling limit.

Given a sequence of numbers p_n ? [0, 1], consider the following experiment. First, we fl ip a fair coin and then, at step n, we turn the coin over to the other side with probability p_n, n > 1, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as n ? 8? We show that a number of phase transitions take place as the turning gets slower (i.e. p_n is getting smaller), leading fi rst to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is p_n = const/n. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws. The critical regime is particularly interesting: when the corresponding random walk is considered, an interesting process emerges as the scaling limit; also, a connection with Polya urns will be mentioned. This is joint work with S. Volkov (Lund) and Z. Wang (Boulder).

October 2 @ Temple: Firas Rassoul-Agha (Utah)

SHIFTED WEIGHTS AND RESTRICTED PATH LENGTH IN FIRST-PASSAGE PERCOLATION

We study standard first-passage percolation via related optimization problems that restrict path length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of geodesic length due to Hammersley, Smythe and Wierman, and Kesten. We study the regularity of the time constant as a function of the shift of weights. For unbounded weights, this function is strictly concave and in case of two or more atoms it has a dense set of singularities. For any weight distribution with an atom at the origin there is a singularity at zero, generalizing a result of Steele and Zhang for Bernoulli FPP. The regularity results are proved by the van den Berg-Kesten modification argument. This is joint work with Arjun Krishnan and Timo Seppalainen

September 25 @ Penn: Julian Sahasrabudhe (Cambridge)

Zeros of polynomials, the distribution of coefficients, and a problem of J.E. Littlewood

While it is an old and fundamental fact that every (nice enough) even function $f : [-\pi,\pi] \rightarrow \mathbb{C}$ may be uniquely expressed as a cosine series \[ f(\theta) = \sum_{r \geq 0 } C_r\cos(r\theta), \] the relationship between the sequence of coefficients $(C_r)_{r \geq 0 }$ and the behavior of the function $f$ remains mysterious in many aspects. We mention two variations on this theme. First a more probabilistic setting: what can be said about a random variable if we constrain the roots of the probability generating function? We then settle on our main topic; a solution to a problem of J.E. Littlewood about the behavior of the zeros of cosine polynomials with coefficients $C_r \in \{0,1\}$.

September 18 @ Temple: Arjun Krishnan (Rochester)

Stationary coalescing walks on the lattice

Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of semi-infinite geodesics in random translation invariant metrics on the lattice; it applies, in particular, to first- and last-passage percolation. We also construct several examples displaying unexpected behaviors. (joint work with Jon Chaika)

September 11 @ Temple: Thomas Leblé (NYU)

The Sine-beta process: DLR equations and applications

One-dimensional log-gases, or Beta-ensembles, are statistical physics toy models finding their incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. We give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and use it to prove the "rigidity" of the process, in the sense of Ghosh-Peres. If time permits, I will mention another application to the study of fluctuations of linear statistics. Joint work with David Dereudre, Adrien Hardy, and Mylene Maida.

September 4 @ Penn: Swee Hong Chan (Cornell)

In between random walk and rotor walk in the square lattice

How much randomness is needed to prove a scaling limit result? In this talk we consider this question for a family of random walks on the square lattice. When the randomness is turned to the maximum, we have the symmetric random walk, which is known to scale to a two-dimensional Brownian motion. When the randomness is turned to zero, we have the rotor walk, for which its scaling limit is an open problem. This talk is about random walks that lie in between these two extreme cases and for which we can prove their scaling limit. This is a joint work with Lila Greco, Lionel Levine, and Boyao Li.

Spring 2018

May 1 @ Penn: Kavita Ramanan (Brown)

Local characterization of dynamics on sparse graphs

Given a sequence of regular graphs G_n whose size goes to infinity, and dynamics that are suitably symmetric, a key question is to understand the limiting dynamics of a typical particle in the system. The case when each G_n is a clique falls under the purview of classical mean-field limits, and it is well known that (under suitable assumptions) the dynamics of a typical particle is governed by a nonlinear Markov process. In this talk, we consider the complementary sparse case when G_n converges in a suitable sense to a countably infinite locally finite graph G, and describe various limit results, both in the setting of diffusions and Markov chains. In particular, when G is a d-regular tree, we obtain an autonomous characterization of the local dynamics of the neighborhood of a typical node. We also obtain a local characterization for the annealed dynamics on a class of Galton-Watson trees. The proofs rely on a certain Markov random field structure of the dynamics on the countably infinite graph G, which may be of independent interest. This is based on various joint works with Ankan Ganguly, Dan Lacker, Mitchell Wortsman and Ruoyu Wu.

Apr 24 @ Penn: Zhenfu Wang (Penn)

Propagation of Chaos via Large Deviation Principle

We present a new method to derive quantitative estimates proving the propagation of chaos for large stochastic or deterministic systems of interacting particles. Our approach requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit; and it can be applied to very singular kernels that are only in negative Sobolev spaces and include the Biot-Savart law for 2D Navier-Stokes and 2D Euler. Joint work with P.-E. Jabin.

Apr 17 @ Penn: Jay Pantone (Dartmouth)

Local Patterns in Chord Diagrams

A chord diagram with n chords is a set of 2n points in a line connected in n pairs. Chord diagrams, sometimes called matchings, play an important role in mathematical biology, knot theory, and combinatorics, and as a result they have been intensely studied by mathematicians, computer scientists, and biologists alike. We examine enumerative properties of families of chord diagrams that avoid local patterns. In particular, we prove that for all k, the generating function for chord diagrams in which every chord has length at least k is D-finite (and therefore the counting sequence satisfies a linear recurrence with polynomial coefficients). We conjecture that a similar but much more general statement is also true. The proof uses several interesting tools: finite state machines, the sieve method, creative telescoping, and D-finite closure properties. We also give examples of local patterns for which experimental evidence suggests that the generating function is non-D-finite, or worse. This is joint work with Peter Doyle and Everett Sullivan.

Apr 3 @ Penn: Josh Rosenberg (Penn)

Quenched survival of Bernoulli percolation on Galton-Watson trees

In this talk I will explore the subject of Bernoulli percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results relating to the behavior of $g$ in the supercritical region. These include an expression for the right derivative of $g$ at criticality in terms of the martingale limit of $T$, a proof that $g$ is infinitely continuously differentiable in the supercritical region, and a proof that $g'$ extends continuously to the boundary of the supercritical region. Allowing for some mild moment constraints on the offspring distribution, each of these results is shown to hold for almost surely every Galton-Watson tree. This is based on joint work with Marcus Michelen and Robin Pemantle.

Mar 27 @ Penn: Hanbaek Lyu (Ohio State)

Double jump phase transition in a random soliton cellular automaton

In this talk, we consider the soliton cellular automaton introduced by Takahashi and Satsuma in 1990 with a random initial configuration. We give multiple constructions of a Young diagram describing various statistics of the system in terms of familiar objects like birth-and-death chains and Galton-Watson forests. Using these ideas, we establish limit theorems showing that if the first $n$ boxes are occupied independently with probability $p\in(0,1)$, then the number of solitons is of order $n$ for all $p$, and the length of the longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$, and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton lengths have the same order as the longest for $p\leq 1/2$, whereas all but the longest have order at most $\log n$ for $p>1/2$. As an application, we obtain scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and decreasing subsequences in a random stack-sortable permutation of length $n$ in terms of random walks and Brownian excursions.

Mar 22 @ Penn: Ira Gessel (Brandeis)

Rational functions with nonnegative power series coefficients

I will talk about rational power series in several variables with nonnegative power series coefficients. An example of such a series is 1/(1-x-y-z+4xyz), whose power series coefficients were proved nonnegative by Szego and Kaluza in 1933. I will discuss several methods for proving nonnegativity and also some conjectures.

Feb 20 @ Penn: Cheyne Homberger (Maryland)

Permuted Packings and Permutation Breadth

The breadth of a permutation p is the minimum value of |i - j| + |p(i) - p(j)|, taken over all relevant i and j. Breadth has important consequences to permutation pattern containment, and connections to plane tiling. In this talk we explore the breadth of random permutations using both probabilistic techniques and combinatorial geometry. In particular, we present the expected breadth of a random permutation, the proportion of permutations with a fixed breadth, and a constructive proof for maximizing unique large patterns in permutations. This talk is based on work with both David Bevan and Bridget Tenner and with Simon Blackburn and Pete Winkler.

Feb 13 @ Penn: Ewain Gwynne (MIT)

A mating-of-trees approach for graph distances and random walk on random planar maps

I will discuss a general strategy for proving estimates for a certain class of random planar maps, namely, those which can be encoded by a two-dimensional walk with i.i.d. increments via a ``mating-of-trees" type bijection. This class includes the uniform infinite planar triangulation (UIPT) and the infinite-volume limits of random planar maps sampled with probability proportional to the number of spanning trees, bipolar orientations, or Schnyder woods they admit.Using this strategy, we obtain non-trivial estimates for graph distances in certain natural non-uniform random planar maps. We also prove that random walk on the UIPT typically travels graph distance $n^{1/4 + o_n(1)}$ in $n$ units of time and that the spectral dimension of a class of random planar maps (including the UIPT) is a.s. equal to 2---i.e., the return probability to the starting point after $n$ steps is $n^{-1+o(1)}$.Our approach proceeds by way of a strong coupling of the encoding walk for the map with a correlated two-dimensional Brownian motion (Zaitsev, 1998), which allows us to compare our given map with the so-called mated-CRT map constructed from this correlated two-dimensional Brownian. The mated-CRT map is closely related to SLE-decorated Liouville quantum gravity due to results of Duplantier, Miller, and Sheffield (2014). So, we can analyze the mated-CRT map using continuum theory and then transfer to other random planar maps via strong coupling. We expect that this approach will have further applications in the future.Based on various joint works with Nina Holden, Tom Hutchcroft, Jason Miller, and Xin Sun.

Jan 30 @ Penn: David Burstein (Swarthmore)

Tools for constructing graphs with fixed degree sequences

Constructing graphs that resemble their empirically observed counterparts is integral for simulating dynamical processes that occur on networks. Since many real-world networks exhibit degree heterogeneity, we consider some challenges in randomly constructing graphs with a given bi-degree sequence in an unbiased way. In particular, we propose a novel method for the asymptotic enumeration of directed graphs that realize a bi-degree sequence, d, with maximum degree d_max = O(S^{1/2 - tau}) for an arbitrarily small positive number tau, where S is the number of edges; the previous best results allow for d_max = o(S^{1/3} ). Our approach is based on two key steps, graph partitioning and degree preserving switches. The former allows us to relate enumeration results to sequences that are easy to handle, while the latter facilitates expansions based on numbers of shared neighbors of pairs of nodes. We will then discuss the implications of our work in context to other methods, such as Markov Chain Monte Carlo, for generating graphs with a prescribed degree sequence. Joint work with Jonathan Rubin.

Jan 16 @ Penn: Jeffrey Kuan (Columbia)

Algebraic constructions of Markov duality functions

Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues.

Fall 2017

Dec 5 @ Penn: Konstantinos Karatapanis (Penn)

One dimensional system arising in stochastic gradient descent

We consider SDEs of the form dX_t = |X_t|^k/t^gamma dt+1/t^gamma dB_t, for a fixed k in [1,infty). We find the values of gamma in (1/2,1] such that X_t will not converge to the origin with probability 1. Furthermore, we can show that for the rest of the permissible values the process will converge to the origin with some positive probability. The previous results extend for processes that satisfy dX_t = f(X_t)/t^gamma dt+1/t^gamma dB_t, when f(x) is comparable to |x|^k in a neighborhood of the origin. As it is expected, similar results are true for discrete processes satisfying X_{n+1} - X_n =f(X_n)/n^gamma+Y_{n+1}/n^gamma. Here, Y_{n+1} are martingale differences that are almost surely bounded and satisfy E(Y_{n+1}^2| F_n )>delta>0.

Nov 14 @ Penn: Miklos Racz (Princeton)

How fragile are information cascades?

It is well known that sequential decision making may lead to information cascades. If the individuals are choosing between a right and a wrong state, and the initial actions are wrong, then the whole cascade will be wrong. We show that if agents occasionally disregard the actions of others and base their action only on their private information, then wrong cascades can be avoided. Moreover, we obtain the optimal asymptotic rate at which the error probability at time t can go to zero. This is joint work with Yuval Peres, Allan Sly, and Izabella Stuhl.

Nov 7 @ Temple: Indrajit Jana (Temple)

Spectrum of Random Band Matrices

We consider the limiting spectral distribution of matrices of the form $\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth $b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes transform of ESD of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law.

Oct 24 @ Penn: Lisa Hartung (NYU)

Extreme Level Sets of Branching Brownian Motion

We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)

Oct 17 @ Temple: Atilla Yilmaz (NYU)

Homogenization of a class of 1-D nonconvex viscous Hamilton-Jacobi equations with random potential

There are general homogenization results in all dimensions for (inviscid and viscous) Hamilton-Jacobi equations with random Hamiltonians that are convex in the gradient variable. Removing the convexity assumption has proved to be challenging. There was no progress in this direction until two years ago when the 1-D inviscid case was settled positively and several classes of (mostly inviscid) examples for which homogenization holds were constructed as well as a 2-D inviscid counterexample. Methods that were used in the inviscid case are not applicable to the viscous case due to the presence of the diffusion term. In this talk, I will present a new class of 1-D viscous Hamilton-Jacobi equations with nonconvex Hamiltonians for which homogenization holds. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial data have representations involving exponential expectations of controlled Brownian motion in random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in random potential. The proof relies on (i) analyzing the large deviation behavior of the controlled Brownian particle which assumes the role of one of the players in an emergent two-player game, (ii) identifying asymptotically optimal control policies and (iii) constructing correctors which lead to exponential martingales. Based on recent joint work with Elena Kosygina and Ofer Zeitouni.

Oct 10 @ Penn: Sourav Chatterjee (Stanford)

Rigidity of the 3D hierarchical Coulomb gas

The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cube-root of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases.

Oct 3 @ Penn: Stephen Melczer (Penn)

Lattice Path Enumeration, Multivariate Singularity Analysis, and Probability Theory

The problem of enumerating lattice paths with a fixed set of allowable steps and restricted endpoint has a long history dating back at least to the 19th century. For several reasons, much research on this topic over the last decade has focused on two dimensional lattice walks restricted to the first quadrant, whose allowable steps are "small" (that is, each step has coordinates +/- 1, or 0). In this talk we relax some of these conditions and discuss recent work on walks in higher dimensions, with non-small steps, or with weighted steps. Particular attention will be given to the asymptotic enumeration of such walks using representations of the generating functions as diagonals of rational functions, through the theory of analytic combinatorics in several variables. Several techniques from computational and experimental mathematics will be highlighted, and open conjectures of a probabilistic nature will be discussed.

Sep 26 @ Penn: Evita Nestoridi (Princeton)

Cutoff for random to random

Random to random is a card shuffling model that was created to study strong stationary times. Although the mixing time of random to random has been known to be of order n log n since 2002, cutoff had been an open question for many years, and a strong stationary time giving the correct order for the mixing time is still not known. In joint work with Megan Bernstein, we use the eigenvalues of the random to random card shuffling to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at (3 /4) n log n, answering a conjecture of Diaconis.

Sep 19 @ Penn: Marcus Michelen (Penn)

Invasion Percolation on Galton-Watson Trees

Given an infinite rooted tree, how might one sample, nearly uniformly, from the set of paths from the root to infinity? A number of methods have been studied including homesick random walks, or determining the growth rate of the number of self-avoiding paths. Another approach is to use percolation. The model of invasion percolation almost surely induces a measure on such paths in Galton-Watson trees, and we prove that this measure is absolutely continuous with respect to the limit uniform measure; other properties of invasion percolation are proved as well. This work in progress is joint with Robin Pemantle and Josh Rosenberg.

Sep 12 @ Temple: Nicholas Crawford (Technion)

Stability of Phases and Interacting Particle Systems

In this talk, I will discuss recent work with W. de Roeck on the following natural question: Given an interacting particle system are the stationary measures of the dynamics stable to small (extensive) perturbations? In general, there is no reason to believe this is so and one must restrict the class of models under consideration in one way or another. As such, I will focus in this talk on the simplest setting for which one might hope to have a rigorous result: attractive Markov dynamics (without conservation laws) relaxing at an exponential rate to its unique stationary measure in infinite volume. In this case we answer the question affirmatively. As a consequence we show that ferromagnetic Ising Glauber dynamics is stable to small, non-equilibrium perturbations in the entire uniqueness phase of the inverse temperature/external field plane. It is worth highlighting that this application requires new results on the (exponential) rate of relaxation for Glauber dynamics defined with non-zero external field.

Sep 5 @ Penn: Allan Sly (Princeton)

Large Deviations for First Passage Percolation

We establish a large deviation rate function for the upper tail of first passage percolation answering a question of Kesten who established the lower tail in 1986. Moreover, conditional on the large deviation event, we show that the minimal cost path is delocalized, that is it moves linearly far from the straight line path. Joint work with Riddhipratim Basu (Stanford/ICTS) and Shirshendu Ganguly (UC Berkeley)

Spring 2017

May 2 @ Penn: Milan Bradonjic (Rutgers)

Percolation in Weighted Random Connection Model

When modeling the spread of infectious diseases, it is important to incorporate risk behavior of individuals in a considered population. Not only risk behavior, but also the network structure created by the relationships among these individuals as well as the dynamical rules that convey the spread of the disease are the key elements in predicting and better understanding the spread. We propose the weighted random connection model, where each individual of the population is characterized by two parameters: its position and risk behavior. A goal is to model the effect that the probability of transmissions among individuals increases in the individual risk factors, and decays in their Euclidean distance. Moreover, the model incorporates a combined risk behavior function for every pair of the individuals, through which the spread can be directly modeled or controlled. The main results are conditions for the almost sure existence of an infinite cluster in the weighted random connection model. We use results on the random connection model and site percolation in Z^2.

Apr 25 @ Temple: Chris Sinclair (U. Oregon)

An introduction to p-adic electrostatics

We consider the distribution of N p-adic particles with interaction energy proportional to the log of the p-adic distance between two particles. When the particles are constrained to the ring of integers of a local field, the distribution of particles is proportional to a power of the p-adic absolute value of the Vandermonde determinant. This leads to a first question: What is the normalization constant necessary to make this a probability measure? This sounds like a triviality, but this normalization constant as a function of extrinsic variables (like number of particles, or temperature) holds much information about the statistics of the particles. Viewed another way, this normalization constant is a p-adic analog of the now famous Selberg integral. While a closed form for this seems out of reach, I will derive a remarkable identity that may hold the key to unlocking more nuanced information about p-adic electrostatics. Along the way we’ll find an identity for the generating function of probabilities that a degree N polynomial with p-adic integer coefficients split completely. Joint work with Jeff Vaaler.

Apr 11 @ Penn: Patrick Devlin (Rutgers)

Biased random permutations are predictable (proof of an entropy conjecture of Leighton and Moitra)

Suppose F is a random (not necessarily uniform) permutation of {1, 2, ... , n} such that |Prob(F(i) < F(j)) -1/2| > epsilon for all i,j. We show that under this assumption, the entropy of F is at most (1-delta)log(n!), for some fixed delta depending only on epsilon [proving a conjecture of Leighton and Moitra]. In other words, if (for every distinct i,j) our random permutation either noticeably prefers F(i) < F(j) or prefers F(i) > F(j), then the distribution inherently carries significantly less uncertainty (or information) than the uniform distribution. Our proof relies on a version of the regularity lemma, a combinatorial bookkeeping gadget, and a few basic probabilistic ideas. The talk should be accessible for any background, and we will gently recall any relevant notions (e.g., entropy) as needed. Those unhappy with the talk are welcome to form an unruly mob to depose the speaker, and pitchforks and torches will be available for purchase. This is from a recent paper joint with Huseyin Acan and Jeff Kahn.

Apr 4 @ Penn: Tobias Johnson (NYU)

Galton-Watson fixed points, tree automata, and interpretations

Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

Mar 28 @ Temple: Arnab Sen (Minnesota)

Majority dynamics on the infinite 3-regular tree

The majority dynamics on the infinite 3-regular tree can be described as follows. Each vertex of the tree has an i.i.d. Poisson clock attached to it, and when the clock of a vertex rings, the vertex looks at the spins of its three neighbors and flips its spin, if necessary, to come into agreement with majority of its neighbors. The initial spins of the vertices are taken to be i.i.d. Bernoulli random variables with parameter p. In this talk, we will discuss a couple of new results regarding this model. In particular, we will show that the limiting proportion of ‘plus’ spins in the tree is continuous with respect to the initial bias p. A key tool in our argument is the mass transport principle. The talk is based on an ongoing work with M. Damron.

Mar 21 @ Temple: Paul Bourgade (Courant)

Local extrema of random matrices and the Riemann zeta function

Fyodorov, Hiary & Keating have conjectured that the maximum of the characteristic polynomial of random unitary matrices behaves like extremes of log-correlated Gaussian fields. This allowed them to conjecture the typical size of local maxima of the Riemann zeta function along the critical axis. I will first explain the origins of this conjecture, and then outline the proof for the leading order of the maximum, for unitary matrices and the zeta function. This talk is based on a joint works with Arguin, Belius, Radziwill and Soundararajan.

Feb 28 @ Temple: James Melbourne (Delaware)

Bounds on the maximum of the density for certain linear images of independent random variables

We will present a generalization of a theorem of Rogozin that identifies uniform distributions as extremizers of a class of inequalities, and show how the result can reduce specific random variables questions to geometric ones. In particular, by extending "cube slicing" results of K. Ball, we achieve a unification and sharpening of recent bounds on densities achieved as projections of product measures due to Rudelson and Vershynin, and the bounds on sums of independent random variable due to Bobkov and Chistyakov. Time permitting we will also discuss connections with generalizations of the entropy power inequality.

Feb 21 @ Penn: Shirshendu Ganguly (Berkeley)

Large deviation and counting problems in sparse settings

The upper tail problem in the Erd ?os-R Ženyi random graph G ~ Gn,p, where every edge is included independently with probability p, is to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1 + d. The arithmetic analog considers the count of arithmetic progressions in a random subset of Z/nZ, where every element is included independently with probability p. In this talk, I will describe some recent results regarding the solution of the upper tail problem in the sparse setting i.e. where p decays to zero, as n grows to infinity. The solution relies on non-linear large deviation principles developed by Chatterjee and Dembo and more recently by Eldan and solutions to various extremal problems in additive combinatorics.

Feb 14 @ Temple: Mihai Nica (NYU)

Intermediate disorder limits for multi-layer random polymers

The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble. Part of this talk is based on joint work with I. Corwin.

Feb 07 @ Temple: Fabrice Baudoin (U. Conn)

Stochastic areas and Hopf fibrations

We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces ℂℙn and ℂℍn. The characteristic functions of those processes are computed and limit theorems are obtained. For ℂℙn the geometry of the Hopf fibration plays a central role, whereas for ℂℍn it is the anti-de Sitter fibration. This is joint work with Jing Wang (UIUC)

Jan 31 @ Penn: Nina Holden (MIT)

How round are the complementary components of planar Brownian motion?

Consider a Brownian motion W in the complex plane started from 0 and run for time 1. Let A(1), A(2),... denote the bounded connected components of C-W([0,1]). Let R(i) (resp.\ r(i)) denote the out-radius (resp.\ in-radius) of A(i) for i \in N. Our main result is that E[\sum_i R(i)^2|\log R(i)|^\theta ]<\infty for any \theta <1. We also prove that \sum_i r(i)^2|\log r(i)|=\infty almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape. Based on joint work with Serban Nacu, Yuval Peres, and Thomas S. Salisbury.

Jan 24 @ Penn: Charles Burnette(Drexel University)

Abelian Squares and Their Progenies

A polynomial P ∈ C[z1, . . . , zd] is strongly Dd-stable if P has no zeroes in the closed unit polydisc D d . For such a polynomial define its spectral density function as SP (z) = P(z)P(1/z) −1 . An abelian square is a finite string of the form ww0 where w0 is a rearrangement of w. We examine a polynomial-valued operator whose spectral density function’s Fourier coefficients are all generating functions for combinatorial classes of con- strained finite strings over an alphabet of d characters. These classes generalize the notion of an abelian square, and their associated generating functions are the Fourier coefficients of one, and essentially only one, L2 (T d)-valued oper- ator. Integral representations and asymptotic behavior of the coefficients of these generating functions and a combinatorial meaning to Parseval’s equation are given as consequences.

Fall 2016

Dec 06 @ Penn: Hao Shen (Columbia)

Some new scaling limit results on ASEP and Glauber dynamics of spin models

We discuss two scaling limit results for discrete models converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP. We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. A common challenge in the proofs is how to identify the limiting process as the solution to the SPDE, and we will discuss how to overcome the difficulties in the two cases. (Based on joint works with Ivan Corwin and Hendrik Weber)

Nov 29 @ Temple: Jack Hanson (CUNY)

Arm events in invasion percolation

Invasion percolation is a "self-organized critical" distribution on random subgraphs of Z^2, believed to exhibit much of the same behavior as critical percolation models. Self-organization means that this happens spontaneously without tuning some parameter to a critical value. In two dimensions, some aspects of the invasion graph are known to correspond to those in critical models, and some differences are known. We will discuss new results on the probabilities of various "arm events" -- events that connections from the origin to a large distance n are either present or "closed" in the invasion graph. We show that some of these events have probabilities obeying power laws with the same power as in the critical model, while all others differ from the critical model's by a power of n.

Nov 15 @ Penn: Elliot Paquette (Ohio State)

The law of fractional logarithm in the GUE minor process

Consider an infinite array of standard complex normal variables which are independent up to Hermitian symmetry. The eigenvalues of the upper-left NxN submatrices, form what is called the GUE minor process. This largest-eigenvalue process is a canonical example of the Airy process which is connected to many other growth processes. We show that if one lets N vary over all natural numbers, then the sequence of largest eigenvalues satisfies a 'law of fractional logarithm,' in analogy with the classical law of iterated logarithm for simple random walk. This GUE minor process is determinantal, and our proof relies on this. However, we reduce the problem to correlation and decorrelation estimates that must be made about the largest eigenvalues of pairs of GUE matrices, which we hope is useful for other similar problems.

Nov 08 @ Penn: Sébastien Bubeck (Microsoft)

Local max-cut in smoothed polynomial time

The local max-cut problem asks to find a partition of the vertices in a weighted graph such that the cut weight cannot be improved by moving a single vertex (that is the partition is locally optimal). This comes up naturally, for example, in computing Nash equilibrium for the party affiliation game. It is well-known that the natural local search algorithm for this problem might take exponential time to reach a locally optimal solution. We show that adding a little bit of noise to the weights tames this exponential into a polynomial. In particular we show that local max-cut is in smoothed polynomial time (this improves the recent quasi-polynomial result of Etscheid and Roglin). Joint work with Omer Angel, Yuval Peres, and Fan Wei.

Nov 01 @ Penn: Henry Towsner (Penn)

Markov Chains of Exchangeable Structures

The Aldous--Hoover Theorem characterizes arrays of random variables which are exchangeable - that is, the distribution is invariant under permutations of the indices of the array. We consider the extension to exchangeable Markov chains. In order to give a satisfactory classification, we need an extension of the Adous--Hoover Theorem to "relatively exchangeable" arrays, which are only invariant under some permutations. Different families of permutations lead to different characterization theorems, with the crucial distinction coming from a model theoretic characterization of the way finite arrays can be amalgamated.

Oct 25 @ Penn: Alexey Bufetov (MIT)

Asymptotics of stochastic particle systems via Schur generating functions

We will discuss a new approach to the analysis of the global behavior of stochastic discrete particle systems. This approach links the asymptotics of these systems with properties of certain observables related to the Schur symmetric functions. As applications of this method, we prove the Law of Large Numbers and the Central Limit Theorem for various models of random lozenge and domino tilings, non-intersecting random walks, and decompositions of tensor products of representations of unitary groups. Based on joint works with V. Gorin and A. Knizel.

Oct 18 @ Penn: Sanchayan Sen (Eindhoven)

Random discrete structures: Scaling limits and universality

One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}. In other words, the degree exponent determines the universality class the random graph belongs to. More generally, recent research has provided strong evidence to believe that several objects constructed on a wide class of random discrete structures including (a) components under critical percolation, (b) the vacant set left by a random walk, and (c) the minimal spanning tree, viewed as metric measure spaces converge, after scaling the graph distance, to some random fractals, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures. Based on joint work with Shankar Bhamidi, Nicolas Broutin, Remco van der Hofstad, and Xuan Wang.

Oct 11 @ Penn: Louigi Addario-Berry (McGill)

The front location for branching Brownian motion with decay of mass

I will describe joint work with Sarah Penington (Oxford). Consider a standard branching Brownian motion whose particles have varying mass. At time t, if a total mass m of particles have distance less than one from a fixed particle x, then the mass of particle x decays at rate m. The total mass increases via branching events: on branching, a particle of mass m creates two identical mass-m particles. One may define the front of this system as the point beyond which there is a total mass less than one (or beyond which the expected mass is less than one). This model possesses much less independence than standard BBM, and martingales are hard to come by. Nonetheless, it is possible to prove that (in a rather weak sense) the front is at distance ~ c t^{1/3} behind the typical BBM front. At a high level, our argument for this may be described as a proof by contradiction combined with fine estimates on the probability Brownian motion stays in a narrow tube of varying width.

Oct 04 @ Temple: Ramon van Handel (Princeton)

Chaining, interpolation, and convexity

A significant achievement of modern probability theory is the development of sharp connections between the boundedness of random processes and the geometry of the underlying index set. In particular, the generic chaining method of Talagrand provides in principle a sharp understanding of the suprema of Gaussian processes. The multiscale geometric structure that arises in this method is however notoriously difficult to control in any given situation. In this talk, I will exhibit a surprisingly simple but very general geometric construction, inspired by real interpolation of Banach spaces, that is readily amenable to explicit computations and that explains the behavior of Gaussian processes in various interesting situations where classical entropy methods are known to fail.

Sep 27 @ Penn: Amanda Lohss (Drexel)

Corners in Tree-Like Tableaux.

Tree–like tableaux are combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. We have proven both conjectures based on a bijection with permutation tableaux and type–B permutation tableaux. In addition, we have shown that the number of diagonal boxes in symmetric tree–like tableaux is asymptotically normal and that the number of occupied corners in a random tree–like tableau is asymptotically Poisson. This extends earlier results of Aval, Boussicault, Nadeau, and Laborde Zubieta, respectively.

Sep 20 @ Temple: Wei Wu (NYU)

Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and $\Phi^4$ models for $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in $d=4$, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

Sep 13 @ Penn: Yuri Kifer (Hebrew University)

An Introduction to Limit Theorems for Nonconventional Sums

I'll survey some of the series results on limit theorems for nonconventional sums of the form \[ \sum_{n=1}^NF(X_n,X_{2n},...,X_{\ell n}) \] and more general ones, where $\{ X_n\}$ is a sequence of random variables with sufficiently weak dependence.

Sep 06 @ Penn: Jian Ding (Chicago)

Random planar metrics of Gaussian free fields

I will present a few recent results on random planar metrics of two-dimensional discrete Gaussian free fields, including Liouville first passage percolation, the chemical distance for level-set percolation and the electric effective resistance on an associated random network. Besides depicting a fascinating picture for 2D GFF, these metric aspects are closely related to various models of planar random walks.

Spring 2016

May 05 @ Penn: Oren Louidor (Technion)

Aging in a logarithmically correlated potential

We consider a continuous time random walk on the box of side length N in Z^2, whose transition rates are governed by the discrete Gaussian free field h on the box with zero boundary conditions, acting as potential: At inverse temperature \beta, when at site x the walk waits an exponential time with mean \exp(\beta h_x) and then jumps to one of its neighbors chosen uniformly at random. This process can be used to model a diffusive particle in a random potential with logarithmic correlations or alternatively as Glauber dynamics for a spin-glass system with logarithmically correlated energy levels. We show that at any sub-critical temperature and at pre-equilibrium time scales, the walk exhibits aging. More precisely, for any \theta > 0 and suitable sequence of times (t_N), the probability that the walk at time t_N(1+\theta) is within O(1) of where it was at time t_N tends to a non-trivial constant as N \to \infty, whose value can be expressed in terms of the distribution function of the generalized arcsine law. This puts this process in the same aging universality class as many other spin-glass models, e.g. the random energy model. Joint work with Aser Cortines-Peixoto and Adela Svejda.

Apr 26 @ Penn: Josh Rosenberg (Penn)

The frog model with drift on R

This paper considers the following scenario. There is a Poisson process on R with intensity f where 0 \le f(x) \le infty for x \ge 0 and f(x)=0 for x \le 0. The "points" of the process represent sleeping frogs. In addition, there is one active frog initially located at the origin. At time t=0 this frog begins performing Brownian motion with leftward drift C (i.e. its motion is a random process of the form B_t-Ct). Any time an active frog arrives at a point where a sleeping frog is residing, the sleeping frog becomes active and begins performing Brownian motion with leftward drift C, that is independent of the motion of all of the other active frogs. This paper establishes sharp conditions on the intensity function f that determine whether the model is transient (meaning the probability that infinitely many frogs return to the origin is 0), or non-transient (meaning this probability is greater than 0).

Apr 19 @ Penn: Dan Jerison (Cornell)

Markov chain convergence via regeneration

How long does it take for a reversible Markov chain to converge to its stationary distribution? This talk discusses how to get explicit upper bounds on the time to stationarity by identifying a regenerative structure of the chain. I will demonstrate the flexibility of this approach by applying it in two very different cases: Markov chain Monte Carlo estimation on general state spaces, and finite birth and death chains. In the first case, an unusual perspective on the popular ``drift and minorization'' method leads to a simple bound that improves on existing convergence results. In the second case, a hidden connection between reversibility and monotonicity recovers sharp upper bounds on the cutoff window.

Apr 12 @ Penn: Zsolt Pajor-Gyulai (Courant)

Stochastic approach to anomalous diffusion in two dimensional, incompressible, periodic, cellular flows

It is a well known fact that velocity grandients in a flow change the dispersion of a passive tracer. One clear manifestation of this phenomenon is that in systems with homogenization type diffusive long time/large scale behavior, the effective diffusivity often differs greatly from the molecular one. An important aspect of these well known result is that they are only valid on timescales much longer than the inverse molecular diffusivity. We are interested in what happens on shorter timescales (subhomogenization regimes) in a family of two-dimensional incompressible periodic flows that consists only of pockets of recirculations essentially acting as traps and infinite flowlines separating these where significant transport is possible. Our approach is to follow the random motion of a tracer particle and show that under certain scaling it resembles a time-changed Brownian motions. This shows that while the trajectories are still diffusive, the variance grows differently than linear.

Apr 05 @ Penn: Boris Hanin (MIT)

Nodal Sets of Random Eigenfunctions of the Harmonic Oscillator

Random eigenfunctions of energy E for the isotropic harmonic oscillator in R^d have a U(d) symmetry and are in some ways analogous to random spherical harmonics of fixed degree on S^d, whose nodal sets have been the subject of many recent studies. However, there is a fundamentally new aspect to this ensemble, namely the existence of allowed and forbidden regions. In the allowed region, the Hermite functions behave like spherical harmonics, while in the forbidden region, Hermite functions are exponentially decaying and it is unclear to what extent they oscillate and have zeros. 

The purpose of this talk is to present several results about the expected volume of the zero set of a random Hermite function in both the allowed and forbidden regions as well as in a shrinking tube around the caustic. The results are based on an explicit formula for the scaling limit around the caustic of the fixed energy spectral projector for the isotropic harmonic oscillator. This is joint work with Steve Zelditch and Peng Zhou.

Mar 29 @ Penn: John Pike (Cornell)

Random walks on abelian sandpiles

Given a simple connected graph $G=(V,E)$, the abelian sandpile Markov chain evolves by adding chips to random vertices and then stabilizing according to certain toppling rules. The recurrent states form an abelian group $\Gamma$, the sandpile group of $G$. I will discuss joint work with Dan Jerison and Lionel Levine in which we characterize the eigenvalues and eigenfunctions of the chain restricted to $\Gamma$ in terms of ``multiplicative harmonic functions'' on $V$. We show that the moduli of the eigenvalues are determined up to a constant factor by the lengths of vectors in an appropriate dual Laplacian lattice and use this observation to bound the mixing time of the sandpile chain in terms of the number of vertices and maximum vertex degree of $G$. We also derive a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on $G$.

Mar 22 @ Temple: Christian Benes (CUNY)

The scaling limit of the loop-erased random walk Green's function

We show that the probability that a planar loop-erased random walk passes through a given edge in the interior of a lattice approximation of a simply connected domain converges, as the lattice spacing goes to zero, to a multiple of the SLE(2) Green's function. This is joint work with Greg Lawler and Fredrik Viklund.

Mar 15 @ Temple: Philippe Sosoe (Harvard)

The chemical distance in critical percolation

The chemical distance is the graph distance inside percolation clusters. In the supercritical phase, this distance is known to be linear with exponential probability, enabling a detailed study of processes like random walks on the infinite cluster. By contrast, at the critical point, the distance is known to be longer than Euclidean by some (unknown) power. I will discuss this and some bounds on distance, as well as a result comparing the chemical distance to the size of the lowest crossing. Joint work with Jack Hanson and Michael Damron.

Mar 01 @ Penn: Sumit Mukherjee (Columbia)

Mean field Ising models

In this talk we consider the asymptotics of the log partition function of an Ising model on a sequence of finite but growing graphs/matrices. We give a sufficient condition for the mean field prediction to the log partition function to be asymptotically tight, which in particular covers all regular graphs with degree going to infinity. We show via several examples that our condition is "almost necessary" as well. As application of our result, we derive the asymptotics of the log partition function for approximately regular graphs, and bi-regular bi-partite graphs. We also re-derive asymptotics of the log partition function for a sequence of graphs convering in cut metric. This is joint work with Anirban Basak from Duke University.

Feb 16 @ Temple: Yuri Bakhtin (Courant)

Burgers equation with random forcing

I will talk about the ergodic theory of randomly forced Burgers equation (a basic nonlinear evolution PDE related to fluid dynamics and growth models) in the noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the inviscid case) and polymer measures on one-sided infinite trajectories (in the positive viscosity case). Joint work with Eric Cator, Kostya Khanin, Liying Li.

Feb 09 @ Penn: Nayantara Bhatnagar (Delaware)

Limit Theorems for Monotone Subsequences in Mallows Permutations

The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution. We study the length of the LIS and LDS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation p in S_n is proportional to q^Inv(p) where q is a real parameter and Inv(p) is the number of inversions in p. We determine the typical order of magnitude of the LIS and LDS, large deviation bounds for these lengths and a law of large numbers for the LIS for various regimes of the parameter q. In the regime that q is constant, we make use of the regenerative structure of the permutation to prove a Gaussian CLT for the LIS. This is based on joint work with Ron Peled and with Riddhi Basu.

Feb 02 @ Penn: Erik Slivken (UC Davis)

Bootstrap Percolation on the Hamming Torus

Bootstrap percolation on a graph is a simple to describe yet hard to analyze process on a graph. It begins with some initial configuration (open or closed) on the vertices. At each subsequent step a vertex may change from closed to open if enough of its neighbors are already open. For a random initial configuration where each vertex is open independently with probability p, how does the probability that eventually every vertex will be open change as p varies? The large neighborhood size of the Hamming torus leads to a distinctly different flavor than previous results on the grid and hypercube. We will focus on Hamming tori with high dimension, giving a detailed description of the long term behavior of the process.

Jan 26 @ Penn: Vadim Gorin (MIT)

Largest eigenvalues in random matrix beta-ensembles: structures of the limit

Despite numerous articles devoted to its study, the universal scaling limit for the largest eigenvalues in general beta log-gases remains a mysterious object. I will present two new approaches to such edge scaling limits. The outcomes include a novel scaling limit for the differences between largest eigenvalues in submatrices and a Feynman-Kac type formula for the semigroup spanned by the Stochastic Airy Operator. (based on joint work with M.Shkolnikov)

Fall 2015

Dec 01 @ Penn: Sivak Mkrtchyan (Rochester)

The entropy of Schur-Weyl measures

We will study local and global statistical properties of Young diagrams with respect to a Plancherel-type family of measures called Schur-Weyl measures and use the results to answer a question from asymptotic representation theory. More precisely, we will solve a variational problem to prove a limit-shape result for random Young diagrams with respect to the Schur-Weyl measures and apply the results to obtain logarithmic, order-sharp bounds for the dimensions of certain representations of finite symmetric groups.

Nov 17 @ Penn: Partha Dey (UIUC)

Longest increasing path within the critical strip

Consider a Poisson Point Process of intensity one in the two-dimensional square of side length $n$. In Baik-Deift-Johansson (1999), it was shown that the length of a longest increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later Johansson (2000) showed that all maximal paths lie within the strip of width $n^{2/3+o(1)}$ around the diagonal with high probability. We consider the length $L(n,w)$ of longest increasing paths restricted to lie within a strip of width $w$ around the diagonal and show that when properly centered and scaled it converges to a Gaussian distribution whenever $w \ll n^{2/3}$. We also obtain tight bounds on the expectation and variance of $L(n,w)$ which involves application of BK inequality and approximation of the optimal restricted path by locally optimal unrestricted path. Based on joint work with Matthew Joseph and Ron Peled.

Nov 10 @ Penn: Charles Bordenave (Toulouse)

A new proof of Friedman’s second eigenvalue Theorem and its extensions

It was conjectured by Alon and proved by Friedman that a random d-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most 2 √ ( d − 1) + o(1) with probability tending to one as the size of the graph tends to infinity. We will discuss a new method to prove this statement and give some extensions to random lifts and related models.

Nov 03 @ Penn: Christian Gromoll (UVA)

Fluid limits and queueing policies

There are many different queueing policies discussed in the literature. They tend to be defined in model-specific ways that differ in format from one policy to another, each format suitable for the task at hand (e.g. steady-state derivation, scaling-limit theorem, or proof of some other property). The ad hoc nature of the policy definition often limits the scope of potentially quite general arguments. Moreover, because policies are defined variously, it's difficult to approach classification questions for which the answer presumably spans many policies. In this talk I'll propose a definition of a general queueing policy and discuss exactly what I mean by "general". The setup makes it possible to frame questions about queues in terms of an arbitrary policy and, potentially, to classify policies according to the answer. In this vein, I'll discuss a few results and some ongoing work on proving fluid limit theorems for general policies.

Oct 27 @ Penn: Doug Rizzolo (U Delaware)

Random pattern-avoiding permutations

Abstract: In this talk we will discuss recent results on the structure of random pattern-avoiding permutations. We will focus a surprising connection between random permutations avoiding a fixed pattern of length three and Brownian excursion. For example, this connection lets us describe the shape of the graph of a random 231-avoiding permutation of {1,...,n} as n tends to infinity as well as the asymptotic distribution of fixed points in terms of Brownian excursion. Time permitting, we will discuss work in progress on permutations avoiding longer patterns. This talk is based on joint work with Christopher Hoffman and Erik Slivken. 

Oct 20 @ Penn: Tai Melcher (UVA)

Smooth measures in infinite dimensions

A collection of vector fields on a manifold satisfies H\"{o}rmander's condition if any two points can be connected by a path whose tangent vectors lie in the given collection. It is well known that a diffusion which is allowed to travel only in these directions is smooth, in the sense that its transition probability measure is absolutely continuous with respect to the volume measure and has a strictly positive smooth density. Smoothness results of this kind in infinite dimensions are typically not known, the first obstruction being the lack of an infinite-dimensional volume measure. We will discuss some smoothness results for diffusions in a particular class of infinite-dimensional spaces. This is based on joint work with Fabrice Baudoin, Daniel Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina. 

Oct 06 @ Penn: Leonid Petrov (UVA)

Bethe Ansatz and interacting particle systems

I will describe recent advances in bringing a circle of ideas and techniques around Bethe ansatz and Yang–Baxter relation under the probabilistic roof, which provides new examples of stochastic interacting particle systems, and techniques to solve them. In particular, I plan to discuss a new particle dynamics in continuous inhomogeneous medium with features resembling traffic models, as well as queuing systems. This system has phase transitions (discontinuities in the limit shape) and Tracy-Widom fluctuations (even at the point of the phase transition).

Sep 29 @ Temple: David Belius (Courant)

Branching in log-correlated random fields

This talk will discuss how log-correlated random fields show up in diverse settings, including the study of cover times and random matrix theory. This is explained by the presence of an underlying approximate branching structure in each of the models. I will describe the most basic model of the log-correlated class, namely Branching Random Walk (BRW), where the branching structure is explicit, and explain how to adapt ideas developed in the context of BRW to models where the branching structure is not immediately obvious. 

Sep 24 @ Penn: Steven Heilman (UCLA)

Strong Contraction and Influences in Tail Spaces

We study contraction under a Markov semi-group and influence bounds for functions all of whose low level Fourier coefficients vanish. This study is motivated by the explicit construction of 3-regular expander graphs of Mendel and Naor, though our results have no direct implication for the construction of expander graphs. In the positive direction we prove an L_{p} Poincar\'{e} inequality and moment decay estimates for mean 0 functions and for all 1 \less p \less \infty, proving the degree one case of a conjecture of Mendel and Naor as well as the general degree case of the conjecture when restricted to Boolean functions. In the negative direction, we answer negatively two questions of Hatami and Kalai concerning extensions of the Kahn-Kalai-Linial and Harper Theorems to tail spaces. For example, we construct a function $f\colon\{-1,1\}^{n}\to\{-1,1\}$ whose Fourier coefficients vanish up to level $c \log n$, with all influences bounded by $C \log n/n$ for some constants $0\lessc,C\less \infty$. That is, the Kahn-Kalai-Linial Theorem cannot be improved, even if we assume that the first $c\log n$ Fourier coefficients of the function vanish. This implies there is a phase transition in the largest guaranteed influence of functions $f\colon\{-1,1\}^{n}\to\{-1,1\}$, which occurs when the first $g(n)\log n$ Fourier coefficients vanish and $g(n)\to\infty$ as $n\to\infty$ or $g(n)$ is bounded as $n\to\infty$.. joint with Elchanan Mossel and Krzysztof Oleszkiewicz 

Sep 15 @ Penn: Toby Johnson (USC)

The frog model on trees

Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at some designated vertex wakes up and begins a simple random walk. When it lands on a vertex, the sleeping frog there wakes up and begins its own simple random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model. I'll (mostly) answer a question posed by Serguei Popov in 2003: On an infinite d-ary tree, is the frog model recurrent or transient? That is, is each vertex visited infinitely or finitely often by frogs? The answer is that it depends on d: there's a phase transition between recurrence and transience as d grows. Furthermore, if the system starts with Poi(m) sleeping frogs on each vertex independently, for any d there's a phase transition as m grows. This is joint work with Christopher Hoffman and Matthew Junge. 

Sep 08 @ Penn: Matt Junge (U. Washington)

Splitting hairs (with choice)

Sequentially place n balls into n bins. For each ball, two bins are sampled uniformly and the ball is placed in the emptier of the two. Computer scientists like that this does a much better job of evenly distributing the balls than the "choiceless" version where one places each ball uniformly. Consider the continuous version: Form a random sequence in the unit interval by having the nth term be whichever of two uniformly placed points falls in the larger gap between the previous n-1 points. We confirm the intuition that this sequence is a.s. equidistributed, resolving a conjecture from Itai Benjamini, Pascal Maillard and Elliot Paquette. The history goes back a century to Weyl and more recently to Kakutani.