los angeles probabilitY FORUM

Next Session

Thursday, October 2nd, 2025. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Hendrik Weber (Universität Münster): TBA

5pm: Volodymyr Riabov (ISTA): The Zigzag Strategy for Random Band Matrices.   

Abstract: Random band matrices have entries concentrated in a narrow band of width W around the main diagonal, modeling systems with spatially localized interactions.

We consider one-dimensional random band matrices with bandwidth W >> N^½, general variance profile, and arbitrary entry distributions. We establish complete isotropic delocalization, quantum unique ergodicity (eigenstate thermalization), and Wigner-Dyson universality in the bulk of the spectrum. The key technical input is a family of local laws capturing the spatial decay of resolvent entries, established using a combination of Ornstein-Uhlenbeck dynamics and Green function comparison (the Zigzag strategy). Based on joint work with László Erdős.

6pm: Adrian Gonzalez Casanova (Arizona State): TBA

Dinner: TBA

Upcoming Sessions

Thursday, November 6th, 2025. UCLA. Talks in Math Sciences Room 6627.

Speakers: Christian Serio (Stanford), Yeganeh Alimohammadi (USC)

Thursday, December 4th, 2025. USC. Talks in Kaprielian (KAP) 414.

Speakers: Marcus Michelen (UIC), Peter Morfe (Penn State), Peter Kosenko (UBC)

Past Sessions

Thursday, May 1st, 2025. USC. Talks in Kaprielian (KAP) 414.

2-3pm: Informal meet & greet at Annenberg Cafe.

3pm: Louigi Addario-Berry (McGill): Root recovery for infection processes on trees and graphs

Abstract: Fix a graph G and a starting vertex r (the root), and let T be a spanning tree of G grown as follows. Initially, vertex r is infected and all other vertices are susceptible. At constant rate, each infected node infects each neighbouring susceptible node. For s>0, the infection tree T_s is composed with the set of vertices that have been infected by time s, together with the set of edges along which the infection has spread. 

How easy is it to recover the identity of the initially infected individual r if one is given access to T_s for some (large) s? 

I will discuss recent work on this problem when G is a random graph with a fixed degree sequence, or G is a supercritical Bienaymé tree, or is the Poisson-weighted infinite tree (in which case the problem must be modified slightly). 

Based on joint work with Sasha Bell, Tasmin Chu, Théodore Conrad-Frenkiel, Catherine Fontaine, Robin Khanfir, Louis-Pierre Langevin, and Simone Têtu. 

4pm: Markus Heydenreich (Universität Augsburg): Preferential attachment trees with vertex death

Abstract: Preferential attachment models are a popular class of random graphs that have received a wealth of attention in the last decades and are often used to model evolving networks. In such models, new vertices are added to the graph sequentially and new vertices are more likely to make connections with existing vertices that have a large degree.

In recent work, we study a general preferential attachment model where vertices can both be added but can also be `killed’. Such killed vertices can no longer make new connections, whereas `alive’ vertices continue to make new connections. This models evolving networks that can both increase as well as decrease in size.
We focus on `persistence of the maximum degree’: are the oldest alive vertices also the ones with largest degree? We uncover a novel regime in which killing of vertices makes such persistence entirely impossible. 

This is based on joint work with Bas Lodewijks. 

5pm: Miklos Racz (Northwestern): The largest common subtree of uniform attachment trees

Abstract: Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Céline Kerriou, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006. 

Thursday, April 10th, 2025. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Frank den Hollander (Leiden): Interplay of Vertex and Edge Dynamics for Dense Random Graphs

Abstract: We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same colour or not. The resulting double dynamics is an example of co-evolution.

We prove that, in the limit as the graph size tends to infinity and the graph becomes dense, the graph process converges, in a suitable path topology, to a limiting Markov process that lives on a certain subset of the space of coloured graphons. In the limit, the density of each vertex colour evolves according to a Fisher-Wright diffusion driven by the density of the edges, while the underlying edge connectivity structure evolves according to a stochastic flow whose drift depends on the densities of the two vertex colours. 

Joint work with Siva Athreya (ICTS Bangalore) and Adrian Rollin (NU Singapore).

5pm: Remco van der Hofstad (TUE): Random cluster models on random graphs

Abstract: On locally tree-like random graphs, we relate the random cluster model with external magnetic fields and q\geq 2 to Ising models with vertex-dependent external fields, extending work of Bencs, Borbenyi and Csikvary (2023). The fact that one can formulate general random cluster models in terms of two-spin ferromagnetic Ising models is quite interesting in its own right. However, in the general setting, the external fields are both positive and negative, which is mathematically unexplored territory. Interestingly, due to the reformulation as a two-spin model, we can show that the Bethe partition function, which is believed to have the same pressure per particle, is always a lower bound on the graph pressure per particle. We further investigate special cases in which the external fields do always have the same sign.

The first example is the Potts model with general external fields on random d-regular graphs. In this case, we show that the pressure per particle in the quenched setting agrees with that of the annealed setting, and verify Assumption 1.4 of Basak, Dembo and Sly (2023). We show that there is a line of values for the external fields where the model displays a first-order phase transition. This completes the identification of the phase diagram of the Potts model on the random d-regular graph. As a second example, we consider the high external field and low temperature phases of the system on locally tree-like graphs with general degree distribution.

This is joint work with Hao Can. See https://arxiv.org/abs/2503.17636 for the preprint.

6pm: Joost Jorritsma (Oxford): Large deviations for the giant in spatial random graphs

Abstract: In (supercritical) Bernoulli bond percolation on $\mathbb{Z}^d$, the proportion of vertices in the largest cluster restricted to a volume-$n$ box converges to $\theta$: the probability that the origin lies in an infinite cluster. The probability that this proportion is smaller than $\theta-\varepsilon$ decays stretched exponentially with exponent strictly smaller than one. The probability that the largest cluster is much larger than expected decays exponentially. Thus, the upper tail decays much faster than the lower tail. 

In this talk, we will see that the discrepancy between the tails is reversed in supercritical spatial random graph models in which the degrees have heavy tails. In particular, we will focus on the soft heavy-tailed Poisson-Boolean model. The lower tail decays stretched exponentially, with an exponent that is determined by the strongest of three competing effects. In contrast, the upper tail decays now polynomially, and thus decays much slower than the lower tail. Time permitting, we will give intuition for the exponent of this polynomial, which is determined by the generating function of the finite cluster-size distribution.

Based on joint work with Julia Komjathy and Dieter Mitsche

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, March 6th, 2025. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Jonas Köppl (WIAS): Spontaneous time-symmetry breaking in interacting particle systems. 

Abstract: We study the long-time behaviour of interacting particle systems on the integer lattice $\mathbb{Z}^d$ with a focus on symmetries. In particular, we are interested in spontaneous symmetry breaking, i.e., situations where there are stationary measures that satisfy less symmetries than the transition rates themselves. Classical examples include the Glauber dynamics for the Ising model, where the global spin-flip symmetry is broken in dimensions $d\geq 2$ and the translation-symmetry is broken in dimension $d\geq3$. In this talk, we discuss the possibility of a spontaneous breaking of the time-symmetry, i.e., whether an interacting particle system with time-homogeneous rates can admit time-periodic behaviour. While this can easily be ruled out in finite volume, the situation is much more delicate in the thermodynamic limit. We provide a no-go theorem for short-range systems under the additional assumption of reversibility in $d=1,2$ and exhibit examples of non-degenerate interacting particle systems which do indeed exhibit spontaneous time-symmetry breaking. The talk is based on joint work with Benedikt Jahnel. 

5pm: Erik Bates (NCSU): Geodesics and approximate geodesics in critical 2D first-passage percolation

Abstract: First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight.  The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic.  Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints.  However, when the edge-weights take the value 0 with probability 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance.  Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor.  I will discuss recent progress on this front (joint work with David Harper, Evan Sorensen, and Xiao Shen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).

6pm: Ryoki Fukushima (Tsukuba University): Number of paths in oriented percolation

Abstract: We consider the number of open paths in a supercritical oriented site percolation. In the finite volume, it is the zero temperature limit of the directed polymer in the Bernoulli environment. Unlike in the positive temperature regime, even the existence of the "free energy" (=growth rate) is a non-trivial problem due to the problem of extinction. It was proved in 2017 by Garet, Gouere and Marchand but fundamental properties of the growth rate, such as the continuity with respect to the percolation parameter, have been unknown. We prove that the free energy of the directed polymer in the Bernoulli environment converges to the growth rate for the number of open paths in super-critical oriented percolation as the temperature tends to zero. Our proof is based on the rate of convergence results which hold uniformly in the temperature. We also prove that the convergence rate is locally uniform in the percolation parameter inside the supercritical phase, which implies that the growth rate depends continuously on the percolation parameter.

Dinner: La Grande Orange Cafe. We will walk there together after the Forum.

Thursday, February 6th, 2025. USC. Talks in Kaprielian (KAP) 414.

2-3pm: Informal meet & greet at Annenberg Cafe.

3pm: Matthew Nicoletti (UC Berkeley): The Doubly Periodic Aztec Diamond Dimer Model: Gaussian Free Field and Discrete Gaussians

Abstract: The dimer model provides a large family of random surface models whose scaling limits exhibit spatial phase separation, and have universal properties such as conformal invariance. The Aztec diamond dimer model in particular has been extensively researched due to its integrability, in order to more precisely understand the various universal behaviors of dimer models. While dimer model height fluctuations have been computed in many cases, until now the exact characterization of height fluctuations of a dimer model with gaseous facets appearing in the bulk has remained open.

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights, which lead to the formation of gaseous facets in the limit shape. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random, lattice-valued rough-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on~$N$, a phenomenon also observed in multi-cut random matrix models.

This is joint work with Tomas Berggren.

4pm: Kunal Chawla (Princeton): The Poisson boundary of hyperbolic groups without moment conditions

Abstract: Given a random walk on a countable group, the Poisson boundary is a measure space which captures all asymptotic events of the markov chain. The Poisson boundary can sometimes be identified with a concrete geometric boundary, but almost all previous results relied strongly on moment conditions of the random walk. I will discuss a technique which allows us to identify the Poisson boundary on any group with hyperbolic properties without moment conditions, new even in the free group case, making progress on a question of Kaimanovich and Vershik. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.

5pm: Axel Saenz (Oregon State): Bethe Ansatz for PushASEP on the ring 

Abstract: Interactive particle systems on the ring provide access to transition between KPZ and Gaussian statistics. In collaboration with J. Li (National Taiwan University), we compute limiting fluctuations for PushASEP on the ring under the relaxation time scale, extending the work of Baik-Liu for periodic TASEP. Our methods are based on the coordinate Bethe Ansatz and contour integral formulas. In the talk, I will describe the results in detail the key ideas for the proofs. The talk will be accessible to a wide audience and no special background is assumed.

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006. 

This meeting is cancelled due to wildfires in LA County.

Thursday, January 9th, 2025. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Yacine Aoun (Geneva): Different flavours of the φ(S)-argument

Abstract: I will explain a method that was developed by Duminil-Copin and Tassion to study the Bernoulli percolation model. After defining and explaining the original idea, we will see some new recent applications.

5pm: Christian Serio (Stanford): Area-tilted Gibbsian line ensembles and SOS level lines

Abstract: The (2+1)D SOS model above a hard wall is a random surface studied in statistical mechanics, among other reasons, to approximate the interface in the 3D Ising model. I will discuss the problem of understanding scaling limits of the level lines of this surface, through the lens of Gibbsian line ensembles with area tilts. These are collections of non-intersecting random walks above a wall, each exponentially tilted by the area underneath it. I will describe results on scaling limits as the domain size and the number of curves grow, in two distinct regimes: the Airy line ensemble of the KPZ universality class in a special integrable case, and a different stationary infinite-volume Gibbs measure constructed by Caputo, Ioffe, and Wachtel in a non-integrable case. This talk is based on joint work with Evgeni Dimitrov, and with Milind Hegde and Yujin Kim.

6pm: Hindy Drillick (Columbia): The Stochastic Six-Vertex Model Speed Process

Abstract: In this talk, I will discuss joint work with Levi Haunschmid-Sibitz where we construct the stochastic six-vertex model speed process. We will first define the stochastic six-vertex model with step initial data and show that a second-class particle started at the origin converges almost surely to an asymptotic speed. We will then generalize this by assigning each particle in the model a different class, allowing us to simultaneously track the speeds of all the particles. The speed process is obtained as the joint limit of these speeds. Along the way, we will develop a stochastic domination result for second- and third-class particles, as well as moderate deviation tail bounds for the stochastic six-vertex model.

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, December 5th, 2024. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Sayan Das (UChicago): Large Deviation Principle for the Directed Landscape

Abstract: The directed landscape is a random directed metric on the plane that arises as the scaling limit of classical metric models in the KPZ universality class. In this talk, I will discuss a functional large deviation principle for the entire random metric and mention certain interesting features of the underlying rate function. If time permits, I will also discuss some applications of our results. Based on a joint work with Duncan Dauvergne and Balint Virag.

5pm: Zhengye Zhou (USC): Convergence of Pfaffian Point Processes to the Airy Line Ensemble

Abstract:  Recent years have seen significant progress in understanding fluctuations in half-space models within the KPZ universality class.  In this talk I will discuss a Pfaffian Schur process, which is a measure on a sequence of partitions, which was introduced by Borodin and Rains as a Pfaffian analog of the determinantal Schur processes introduced by Okounkov and Reshetikhin. The model we investigate arises in a half-quadrant last passage percolation model, which has i.i.d. geometric weights with parameter $\alpha \in (0,1)$ off of the main diagonal and with parameter $c \in (0, \alpha^{-1})$ on the main diagonal. We show that when $c$ is subcritical the line ensembles formed by the parts of our random partitions converge uniformly over compact sets to the Airy line ensemble, and when $c$ is critically scaled with the size of the system the ensemble instead converges to the Airy wanderer line ensemble with a single parameter. 

6pm: Yizhe Zhu (USC): Community Detection with the Bethe-Hessian

Abstract: The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborová (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Joint work with Ludovic Stephan.

Dinner: La Grande Orange Cafe. We will walk there together after the Forum.

Thursday, October 31st, 2024. USC. Talks in Kaprielian (KAP) 414.

2-3pm: Informal meet & greet at Annenberg Cafe.

3pm: Diederik van Engelenburg (Lyon): Ends of the uniform spanning tree on unimodular random rooted graphs.

Abstract: This talk is concerned with the geometry of the (wired) Uniform Spanning Tree (UST) on infinite graphs, and in particular with a property called one-endedness, which means that removing a finite subset does not produce multiple infinite clusters. It was conjectured by Aldous and Lyons (2006) that the UST is one-ended on all non-trivial stationary random graphs. For unimodular random rooted graphs, we show that there is an equiva- lence between the one-endedness of the UST and (a) the existence and uniqueness of the potential kernel, (b) existence and unique- ness of the harmonic measure from infinity, and (c) a new anchored Harnack inequality. Building on this equivalence and extending the relation between recurrent potential theory and the UST, we prove the conjecture of Aldous and Lyons for recurrent stationary graphs.

This is based on joint works with Nathanaël Berestycki and Tom Hutchcroft.

4pm: Miriam Gordin (Princeton): Vector-Valued Concentration on the Symmetric Group

Abstract: Existing vector-valued concentration inequalities, such as the classical results of Pisier, are known only in very special settings, such as the Gaussian measure on R^n and the uniform measure on the discrete hypercube {-1,1}^n. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group which goes beyond the product setting of the prior known results. Furthermore, we discuss the implications of this result for the nonembeddability of the symmetric group into Banach spaces of nontrivial Rademacher type, a question of interest in the metric geometry of Banach spaces. We build on prior work of Ivanisvili, van Handel, and Volberg (2020) who proved a vector-valued inequality on the discrete hypercube, resolving a question of Enflo in the metric theory of Banach spaces. This talk is based on joint work with Ramon van Handel.

5pm: Jacob Richey (Alfréd Rényi Institute): Patterns and statistics in shifts of finite type

Abstract: Consider all words over a finite alphabet that avoid a set of forbidden patterns, e.g. binary sequences with no two adjacent 1s. This can be viewed through the lens of ergodic theory, as a dynamical system (a 'shift space'); or combinatorial probability, as an iid sequence or a markov chain conditioned to avoid the forbidden set; or statistical physics, as a thermodynamic limit of a natural Gibbs measure. I will discuss connections between ideas from these worlds in the case where the forbidden set is of size one or two, including new results related to conjugacy of the underlying shifts, their entropies, and the asymptotic density of 1s.

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006. 

Thursday, October 3rd, 2024. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Milind Hegde (Columbia): The scaling limit of the colored asymmetric simple exclusion process

Abstract: In the colored asymmetric simple exclusion process, one places a particle of "color" -k at each integer site k ∈ \Z. Particles attempt to swap places with an adjacent particle: at rate q ∈ [0,1) if they are initially ordered (e.g., 1 then 2) and at rate 1 if ordered in reverse (e.g., 2 then 1); thus the particles tend to get more ordered over time. We will discuss the scaling limit of this process, which lies in the Kardar-Parisi-Zhang universality class. It  is given by the directed landscape, which was first constructed in 2018 by Dauvergne-Ortmann-Virág as limits in a very different setting---of fluctuations of a model of a random directed metric. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) with certain Gibbs or spatial Markov properties will play fundamental roles in our discussion. This is based on joint work with Amol Aggarwal and Ivan Corwin.

5pm: Nikita Gladkov (UCLA): The Bunkbed Conjecture is False

Abstract: We discuss the Bunkbed Conjecture, a long-standing conjecture about connection probabilities in percolation. We describe a recent counterexample to this conjecture, a graph on 7772 vertices, and describe its structure.

6pm: Ken Alexander (USC): Properties of first passage percolation above the (hypothetical) critical dimension

Abstract: It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\ZZ^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $d<d_c$.  We show that at least one of three fundamental properties of FPP known or believed to hold when $\chi>0$ must be false if $\chi=0$.  A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false.  These consequences support the idea that when $\chi=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$.  Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, May 30th, 2024. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Hao Wu (Tsinghua University): Commutation relation for two SLEs

Abstract: In 1999, O. Schramm introduced Schramm—Loewner evolution (SLE) as a non-self-crossing random curve driven by a multiple of Brownian motion using Loewner’s transform. This definition is motivated by a quest to describe the random interfaces in 2D critical lattice models, which satisfy the conformal invariance and domain Markov property. In this talk, we will consider the law of a pair of two SLEs with conformal invariance and domain Markov property, following the framework of Dub'edat’s commutation relation. 

Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE for $\kappa\le 4$: it is either a two-sided radial SLE with spiral of constant spiraling rate or a chordal SLE weighted by a power of the conformal radius of its complement.  Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. However, unlike in the chordal case, the resampling property does not uniquely determine the pair due to the additional degree of freedom in the spiraling rate. We also discuss the semiclassical limit of the commutation relation as $\kappa\to 0$. 

This talk is based on a joint work with Yilin Wang (IHES, France).

5pm: Matthew Dickson (UBC): An Expansion for the Critical Intensity of Random Connection Models

Abstract: Random connection models (RCMs) are random graph models where the vertices are given by a Poisson point process with a given intensity, $\lambda>0$, and the edges exist independently with a probability that depends upon the relative positions of the two vertices in question. These models include ``Poisson blob models", such as the Gilbert disc model. As we vary $\lambda$, we observe a percolation phase transition at a critical intensity $\lambda_c>0$. Finding $\lambda_c$ is only possible in very exceptional cases, so here we use the lace expansion for the RCM (as found by Heydenreich, van der Hofstad, Last, and Matzke 2022) to find a high-dimension asymptotic expansion for the critical intensity. This is based on arXiv:2309.08830 with Markus Heydenreich (Universität Augsburg).

6pm: Gourab Ray (University of Victoria): Uniqueness and CLT for the  Ground State of the Disordered Monomer-Dimer Model on Z^d

Abstract:  We prove that the disordered monomer-dimer model does not admit infinite volume incongruent ground states in Z^d which can be obtained as a limit of finite volume ground states. As an application, we obtain a CLT for the ground state weight for a growing sequence of tori. Our motivation stems from a similar and long standing open questions for the short range Edwards-Anderson spin glass model.

Joint work with Kesav Krishnan.

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, May 2nd, 2024. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Eveliina Peltola (University of Bonn): Large deviations of Dyson Brownian motion and multiple Schramm-Loewner Evolution

Abstract: Schramm-Loewner evolution SLE(k) is a one-parameter family of random curves arising from two-dimensional conformal geometry, which in the limit k->0 converge to deterministic curves (hyperbolic geodesics). In such a large deviations limit, a precise large deviation principle (LDP) has been proven in some cases involving interacting SLE curves. The rate function, termed Loewner energy, plays an important role for the LDP, and also provides rich interplay with algebraic geometry and Teichmüller theory. In this talk, we shall focus on the probabilistic aspects of the LDP for SLE curves and their Loewner driving processes, which are variants of Brownian motion. Specifically, we discuss the chordal, radial, multichordal, and multiradial cases, and the explicit relationship of the latter to Dyson Brownian motion. In particular, we consider an LDP for Dyson Brownian motion, which may also be of independent interest in probability theory and random matrix theory.

Based on joint works with O. Abuzaid, V. Healey, and Y. Wang.

5pm: Jiaxin Zhang (Caltech): Multiple radial SLE(0) and classical Calogero-Sutherland system

Abstract: I will discuss the joint work with Nikolai Makarov on multiple radial SLE(0) system which is the deterministic limit of the multiple radial SLE(\kappa) system.

By constructing the field integral of motion, we show that the traces of the multiple radial SLE(0) system are the horizontal trajectories of an equivalence class of quadratic differentials. The stationary relations establish a connection between the multiple radial SLE(0) systems and enumerative algebraic geometry. 

Our machinery can also be applied to various multiple SLE(0) systems such as multiple radial SLE(0) with spin and multiple chordal SLE(0) with arbitrary screening charges which extend the results in \cite{ABKM20}.

From a Hamiltonian perspective, we prove that the Loewner dynamics with equal growth weight in multiple radial SLE(0) systems are a special type of classical Calogero-Sutherland system. Furthermore, we interpret  $n$ quadratic null vector equations as $n$ commutating Hamiltonian flows along the submanifolds defined as the intersection of level set of the Hamiltonian. 

The whole theory is classical but motivated by conformal field theory. Notably, the field integral motion can be seen heuristically as the classical limit of the screened martingale observable.

6pm: Perla Sousi (Cambridge): Phase transition for the late points of random walk

Abstract: Let X be a simple random walk in $\mathbb{Z}_n^d$ with $d\geq 3$ and let $t_{\rm{cov}}$ be the expected time it takes for X to visit all vertices of the torus. In joint work with Prévost and Rodriguez we study the set $\mathcal{L}_\alpha$ of points that have not been visited by time $\alpha t_{\rm{cov}}$ and prove that it exhibits a phase transition: there exists $\alpha_*$ so that for all $\alpha>\alpha_*$ and all $\epsilon>0$ there exists a coupling between $\mathcal{L}_\alpha$ and two i.i.d. Bernoulli sets $\mathcal{B}^{\pm}$ on the torus with parameters $n^{-(a\pm\epsilon)d}$ with the property that $\mathcal{B}^-\subseteq \mathcal{L}_\alpha\subseteq \mathcal{B}^+$ with probability tending to 1 as $n\to\infty$. When $\alpha\leq \alpha_*$, we prove that there is no such coupling.

Dinner: La Grande Orange Cafe. We will walk there together after the Forum.

Thursday, April 4th, 2024. USC. Talks in Kaprelian (KAP) 414.

2-3pm: Informal meet & greet at Annenberg Cafe.

3pm: Marcin Lis (Vienna University of Technology): On Pfaffians in spin models

Abstract: It is a classical result of Groeneveld, Boel and Kasteleyn that boundary spin correlations functions in Ising models on planar graphs satisfy Pfaffian relations. Here we consider the reverse question, and show that any classical ferromagnetic spin model whose correlation functions satisfy Pfaffian relations must by (up to local simplifications of the graph) an Ising model on a planar graph. Our main tool is a new (coupled) version of the Edwards—Sokal representation of the Ising model applied to two independent copies of the spin model.

Joint work with Diederik van Engelenburg.

4pm: Yujin Kim (NYU): The shape of the front of multidimensional branching Brownian motion

Abstract: The extremal process of branching Brownian motion (BBM)--- i.e., the collection of particles furthest from the origin-- has gained lots of attention in dimension $d = 1$ due to its significance to the universality class of log-correlated fields, as well as to certain PDEs. In recent years, a description of the extrema of BBM in $d > 1$ has been obtained. In this talk, we address the following geometrical question that can only be asked in $d > 1$. Generate a BBM at a large time, and draw the outer envelope of the cloud of particles: what is its shape? Macroscopically, the shape is known to be a sphere; however, we focus on the outer envelope around an extremal point-- the "front" of the BBM. We describe the scaling limit for the front, with scaling exponent 3/2, as an explicit, rotationally-symmetric random surface. Based on joint works with Julien Berestycki, Bastien Mallein, Eyal Lubetzky, and Ofer Zeitouni.

5pm: Moritz Voss (UCLA): Equilibrium in functional stochastic games with mean-field interaction

Abstract: We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact. 

This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at https://ssrn.com/abstract=4470883. 

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006. 7pm.

Thursday, February 29th, 2024. UCLA. Talks in Math Sciences Room 6221.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Marianna Russkikh (Notre Dame): Perfect t-embedding of uniformly weighted Aztec diamond

Abstract: A new type of graph embedding called a t-embedding, was recently introduced and used to prove the convergence of dimer model height fluctuations to a Gaussian Free Field (GFF) in a naturally associated metric, under certain technical assumptions. We study the properties of t-embeddings of uniform Aztec diamond graphs, and in particular utilize the integrability of the “shuffling algorithm” on these graphs to provide a precise asymptotic analysis of t-embeddings and verify the validity of the technical assumptions required for convergence. As a consequence, we complete a new proof of GFF fluctuations for the dimer model height function on the uniformly weighted Aztec diamond.

5pm: Haiyu Huang (UCLA): A Limit Law for the Maximum of Subcritcal DG-model on a Hierarchical Lattice

Abstract: The extremal properties of logarithmically correlated random fields have been a subject of considerable interest in recent years. A picture that has emerged from the analysis of salient examples such as Branching Random Walk and Discrete Gaussian Free Field in Z^2 is that, in these systems, the suitably centered maximum tends in law to a randomly-shifted Gumbel random variable while the associated extremal process tends in law to a decorated Poisson point process with a random intensity measure. In this talk, we will study the hierarchical Discrete Gaussian (DG) model that distinguishes itself from the above by taking only integer values. We will show the same picture holds for the hierarchical DG-model, with the results factor in the discrete nature of the field. The proof will be based on renormalization group techniques with a tight coupling between the hierarchical DG-model and Gaussian Branching Random Walk. The talk is based on joint work with Marek Biskup. 

6pm: Ahmed Bou-Rabee (NYU): Superdiffusion for Brownian motion with random drift

Abstract: A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at this threshold, and recently some progress in this direction has been made by mathematicians. 

I will discuss joint work with Scott Armstrong and Tuomo Kuusi in which we identify and obtain the sharp rate of superdiffusivity. We also establish a quenched invariance principle under this scaling. Our proof is a quantitative renormalization group argument made rigorous by ideas from stochastic homogenization. 

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, January 18th, 2024. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Seung-Yeal Ha (Seoul National University): Emergent dynamics of infinitely many Kuramoto oscillators

Abstract: In this talk, we propose an infinite Kuramoto model for a countably infinite set of Kuramoto oscillators and study its emergent dynamics for two classes of network topologies. For a class of symmetric and row (or columm)-summable network topology, we show that a homogeneous ensemble exhibits complete synchronization, and the infinite Kuramoto model can cast as a gradient flow, whereas we obtain a weak synchronization estimate, namely practical synchronization for a heterogeneous ensemble. Unlike with the finite Kuramoto model, phase diameter can be constant for some class of network topologies which is a novel feature of the infinite model. We also consider a second class of network topology (so-called a sender network) in which coupling strengths are proportional to a constant that depends only on sender's index number. For this network topology, we have a better control on emergent dynamics. For a homogeneous ensemble, there are only two possible asymptotic states, complete phase synchrony or bi-cluster configuration in any positive coupling strengths. In contrast, for a heterogeneous ensemble, complete synchronization occurs exponentially fast for a class of initial configuration confined in a quarter arc. This is a joint work with Euntaek Lee (SNU) and Woojoo Shim (Kyungpook National University).

5pm: Matthew Junge (CUNY): The frog model on trees

Abstract: The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on d-ary trees for ten years. I will discuss our progress and what remains.

6pm: Minghao Pan (Caltech): Non-uniqueness of infinite clusters at the uniqueness threshold and subgroup relativization 

Abstract: I will talk about the number of infinite clusters in the Bernoulli bond percolation of a finitely generated Cayley graph of a group. Except for a few cases, we do not know the number of infinite clusters at the uniqueness threshold, and there is not a general conjecture either. We show that if the group has an amenable, normal, subgroup with an exponential growth rate, then the Bernoulli bond percolation has a non-unique infinite cluster. This covers interesting cases like the lamplighter group over a nonamenable group. The main tools we develop, which we call “subgroup relativization”, are then used to study the intersection of an independent random walk on the Cayley graph with an infinite cluster, where we solved a conjecture by Lyons and Schramm (1999).

Dinner: La Grande Orange Cafe, 7:30pm.

Thursday, November 30th, 2023. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Shuangping Li (Stanford): Embedding in high-dimensional random geometric graphs

Abstract: I will discuss a random geometric graph model, where connections between vertices depend on the distances between latent d-dimensional feature vectors. We are especially interested in the high-dimensional case when d is large. Upon observing a graph, our aim is to recover these latent feature vectors (i.e., embedding). We have identified a phase transition phenomenon: when d is significantly larger than nH(p) (with a polylogarithmic term), embedding becomes feasible with high probability using a spectral algorithm. H here represents the entropy function. Conversely, when d is considerably smaller than nH(p), embedding becomes information theoretically impossible. In our proof of the impossibility result, we design a dynamics and show that it can find two distant embeddings that produce the same graph. This is based on joint works with Eric Ma and Tselil Schramm.

5pm: Paata Ivanisvili (UCI): On sharp isoperimetric inequalities on the hypercube

Abstract: I will talk about sharp isoperimetric inequalities on the hypercube.  In the second half of the talk I will present  Talagrand's isoperimetric inequalities for functions taking values in a Banach space having finite cotype. If time permits several different new proofs of recently resolved Talagrand's conjecture will be presented. 

This is joint work with David Beltran and José Ramón Madrid Padilla. 

6pm: Evgeni Dimitrov (USC): Phase transition in the log-gamma polymer measure 

Abstract: The log-gamma polymer is an exactly solvable model for a directed polymer on the Z^2 lattice introduced by Timo Seppalainen in 2012. In its simplest form the model depends on a single positive parameter θ, which governs the background noise of the model. In this talk I will discuss a natural polymer measure on the set of directed lattice path, contained in a large square domain of size N. I will show that for large N the behavior of a typical path undergoes a phase transition, which depends on the value of θ. The talk is based on joint work with Guillaume Barraquand and Ivan Corwin.

Dinner: La Grande Orange Cafe, 7:30pm.

Thursday, November 2nd, 2023. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House. 

4pm: Philippe Sosoe (Cornell): KPZ estimates for ASEP and the Stochastic Six Vertex model

Abstract: The Kardar-Parisi-Zhang (KPZ) universality class of models is characterized by non Gaussian asymptotic fluctuations coming from random matrices. In this talk, I will define the stochastic six vertex model, a specialization of the classical six vertex model (S6V) which is known to lie in the KPZ class. This model can be viewed as a discrete time version of the asymmetric simple exclusion process. Indeed, it converges to ASEP under a certain limit.

I will explain how, using an idea of Emrah-Janjigian-Seppalainen, one derives the analogue of a formula due to Rains in the context of Last Passage Percolation for the height function of the S6V.  I will then deduce several results on fluctuations, including upper tail bounds for the height function and location of a second-class particle in ASEP, from this formula and coupling arguments.

Joint work with Benjamin Landon.

5pm: Shiping Cao (UW): Convergence of effective resistances on generalized Sierpinski carpets

Abstract: The locally symmetric diffusions, also known as Brownian motions, on generalized Sierpinski carpets were constructed by Barlow and Bass in 1989. On a fixed carpet, by the uniqueness theorem (Barlow-Bass-Kumagai-Teplyaev, 2010), the reflected Brownians motion on level $n$ approximation Euclidean domain, running at speed $\lambda_n\asymp \eta^n$ with $\eta$ being a constant depending on the fractal, converges weakly to the Brownian motion on the Sierpinski carpet as $n$ tends to infinity. In this talk, we show the convergence of $\lambda_n/\eta^n$. We also give a positive answer to a closely related open question of Barlow-Bass (1990) about the convergence of the renormalized effective resistances between two opposite faces of approximation domains. This talk is based on a joint work with Zhen-Qing Chen.

6pm: Lingfu Zhang (Berkeley): Geodesics in Last-Passage Percolation under Large Deviations

Abstract: In KPZ universality, an important family of models arises from 2D last-passage percolation (LPP): in a 2D i.i.d. random field, one considers the geodesic connecting two vertices, which is defined as the up-right path maximizing its weight, i.e., the sum/integral of the random field along it. A characteristic KPZ behavior is the 2/3 geodesic fluctuation exponent, which has been proven for some LPPs with exactly solvable structures. A topic of much recent interest is such models under upper- and lower-tail large deviations, i.e., when the geodesic weight is atypically large or small. In prior works, it was established that the geodesic exponent changes to 1/2 (more localized) and 1 (delocalized) respectively. In this talk, I will describe a further refined picture: the geodesic converges to a Brownian bridge under the upper tail, and a uniformly chosen function from a one-parameter family under the lower tail. I will also discuss the proofs, using a combination of algebraic, geometric, and probabilistic arguments.

This is based on one forthcoming work with Shirshendu Ganguly and Milind Hegde, and one ongoing work with Shirshendu Ganguly.

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. We will walk there together after the Forum.

Thursday, October 5th, 2023. USC. Talks in Kaprelian (KAP) 414.

2-3pm: Informal meet & greet at Annenberg Cafe. 

3pm: Amanda Wilkens (UT Austin): Poisson-Voronoi tessellations and fixed price in higher rank

Abstract: We overview the cost of a group action, which measures how much information is needed to generate its induced orbit equivalence relation, and the ideal Poisson-Voronoi tessellation (IPVT), a new random limit with interesting geometric features. In recent work, we use the IPVT to prove all measure preserving and free actions of a higher rank semisimple Lie group on a standard probability space have cost 1, answering Gaboriau's fixed price question for this class of groups. We sketch a proof, which relies on some simple dynamics of the group action and the definition of a Poisson point process. No prior knowledge on cost, IPVTs, or Lie groups will be assumed. This is joint work with Mikolaj Fraczyk and Sam Mellick.

4pm: Jason Schweinsberg (UCSD): Asymptotics for the site frequency spectrum associated with the genealogy of a birth and death process

Abstract: Consider a birth-death process started from one individual in which each individual gives birth at rate $\lambda$ and dies at rate $\mu$, so that the population size grows at rate $r = \lambda - \mu$.  Lambert (2018) and Harris, Johnston, and Roberts (2020) came up with methods for constructing the exact genealogy of a sample of size $n$ taken from this population at time $T$.  We use the construction of Lambert (2018), which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with this sample.  We also explain how to apply these results to obtain a confidence interval for the growth rate of an exponentially growing tumor.  This is joint work with Kit Curtius, Brian Johnson, and Yubo Shuai.

5pm: Joshua Frisch (UCSD): Poisson Boundaries Without Moment Conditions 

Abstract: The Poisson boundary of a random walk on a (discrete, infinite) group is an object which serves a dual purpose. It serves as a basis for the space of bounded harmonic functions on a group and it serves to quantify the space of possible asymptotic events. Although apriori the Poisson boundary is quite abstract in a number of cases the Poisson boundary can actually be concretely identified allowing us to classify the space of harmonic functions on the group. Until recently almost all nontrivial classifications have heavily relied on moment conditions. I will discuss some more recent examples which no longer require any control on the moments of the step distribution of the random walk. This is based on joint work with Kunal Chawla, Behrang Forghani, Giullio Tiozzo as well as joint work with Eduardo Silva. 

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006, 6:45 pm.

Thursday May 25th, 2023. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House. 

4pm: Quentin Berger (Sorbonne): Scaling limits of disordered systems

Abstract : I will review some recent progress in the study of disordered systems, with a focus on the directed polymer model. I will first introduce the question of disorder relevance for a generic physical system (with some specific examples in mind) and then explain how finding a non-trivial disordered scaling limit is related to this question. I will then turn to the directed polymer model, which describes a polymer in a heterogeneous solvent and I will present recent results on their scaling limits and their relation to the Stochastic Heat Equation.

5pm: Arka Adhikari (Stanford): Spectral Gap Estimates for Mixed p-Spin Models at High Temperature

Abstract: 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 with C. Brennecke, C. Xu, and H-T Yau.

6pm: Catherine Wolfram (MIT): Large deviations for the 3D dimer model

Abstract: A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. In this talk, I will explain how to formulate the large deviations principle in 3D, show simulations, and explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments. Time permitting, I will also describe results and problems that illustrate some of the ways that three dimensions is qualitatively different from two.

Dinner: TBA

Thursday April 27th, 2023. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Omer Angel (UBC): Permuted random walks

Abstract: I will discuss how permuting the states of a random walk after each step can effect the speed at which the random walks moves. In some cases it is possible to slow down the walker, but there are limits to what can be done, and in the case of regular trees no slow down is possible, and many settings remain open. Based on joint work with Jacob Richey, Yinon Spinka and Amir Yehudayoff.

5pm: Abdelmalek Abdesselam (Virginia): Exploring conformal probability with hierarchical models

Abstract: In the context of the AdS/CFT correspondence, in Euclidean signature, an important basic fact is the bijection between conformal transformations of the boundary and hyperbolic isometries of the bulk. An infinite regular tree with the graph distance can be seen as a quintessential bare-bones version of a hyperbolic space. It turns out there is a natural way to define analogues of conformal maps on the boundary of such a tree and, quite miraculously, these are in bijection with tree isometries. Moreover, a Euclidean QFT (a random Schwartz distribution) on this boundary is the short distance limit of a hierarchical model as considered by Dyson in his study of the long-range Ising model and by Wilson when he introduced the approximate renormalization group recursion. I will try to give a pedagogical introduction to this circle of ideas, and I will discuss a particular model where there is hope to be able to prove conformal invariance from first principles via a rigorous nonperturbative renormalization group approach.

6pm: Ioana Dumitriu (UCSD): Extreme eigenvalues in bipartite Erdos-Renyi graphs

Abstract: The asymptotic (and non-asymptotic) behavior of extreme eigenvalues is a fundamental topic in random matrix theory; such eigenvalues can be used to provide theoretical guarantees for randomized linear algebra on large data sets, with applications in machine learning, signal processing, and data science. The spectra of random bipartite graphs has a basic connection to the set of singular values of random rectangular matrices and can also connect to the spectra of more general graph and hypergraph operators; understanding their spectra is thus of particular interest. Several avenues for studying these extremal eigenvalues / singular values for non-homogeneous Erdos-Renyi graphs have been explored, most importantly through the seminal work of Bandeira, Boedihardjo, and Van Handel (2021). Though very sharp, their bounds do not extend to the near critical regime when sparsity approaches the connectivity threshold; in a new paper with Yizhe Zhu (2022), we provide a non-backtracking operator approach inspired by Benaych-Georges, Bordenave, and Knowles (2019) which, while slightly less sharp, reaches all the way to the connectivity threshold. Notably, the bound is sharp in the homogeneous case up to just above criticality. 

Dinner: La Grand Orange Cafe, 7:30pm.

Thursday March 9th, 2023. UCLA. Talks in Math Sciences Room 6627.

3:30-4pm: Tea and Coffee in the department lounge.

Special note: Jacopo Borga's talk will also be the UCLA department colloquium.

4pm: Jacopo Borga (Stanford): Meanders and Meandric Systems

Abstract: In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?”

Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit of uniform meanders and some recent results on a related model called meandric systems. A meandric system is a coupled collection of meanders. Also in this case, I will present (1) a conjecture which describes the large-scale geometry of a uniform meandric system and  (2) several rigorous results which are consistent with this conjecture.

Based on joint works with Ewain Gwynne and Xin Sun, and Ewain Gwynne and Minjae Park.

5pm: Jean-Dominique Deuschel (TU Berlin): An isomorphism theorem for anharmonic fields and scaling limits

Abstract: We introduce a natural measure on bi-infinite random walk trajectories evolving in a time-dependent environment driven by the Langevin dynamics associated to a gradient Gibbs measure with convex potential. We derive an identity relating the occupation times of the Poissonian cloud induced by this measure to the square of the corresponding gradient field, which is  generically not Gaussian. In the quadratic case, we recover a well-known generalization of the second Ray-Knight theorem. We further determine the scaling limits of the various objects involved in dimension 3, which are seen to exhibit homogenization. In particular, we prove that the renormalized square of the gradient field converges under appropriate rescaling to the Wick-ordered square of a Gaussian free field on R3 with suitable diffusion matrix, thus extending a celebrated result of Naddaf and Spencer regarding the scaling limit of the field itself. 

Joint work with Pierre-Francois Rodriguez

6pm: Johannes Baumler (TU Munich): Chemical distances for long-range percolation

Abstract: Consider long-range percolation on Z^d, where there is an edge between two points x and y with probability asymptotic to β |x-y|^s independent of all other edges, for some positive parameters s and β. In this talk, we will focus on the metric properties of the long-range percolation graph. The chemical distance between two points x and y is the number of steps one needs to make in order to go from x to y. For different values of s, there are different regimes of how the chemical distance scales with the Euclidean distance. The transitions between these regimes happen at s=d and s=2d. After an overview of previous work, we will focus on the case s=2d. We will show that for s=2d, for each dimension d and for each β > 0, there exists a ϴ=ϴ(d,β) in (0,1) such that the chemical distance between x and y is of order |x-y|^ϴ. We will also discuss how the exponent ϴ depends on the parameter β. 

Dinner: Sunnin Lebanese Cafe, 1776 Westwood Blvd, Los Angeles, CA 90024. Time TBD.

Thursday 02/16/2023. USC. Talks in Kaprelian (KAP) 414.

Please note that the unusual time: Everything is shifted one hour earlier compared to the usual schedule.

2-3pm: Informal meet and greet at Cafe Annenberg.

3pm: David Renfrew (SUNY Binghamton): Eigenvalues of minors of random matrices and roots of derivatives of random polynomials

Abstract: I will describe the limiting behavior of the eigenvalues of minors of large biunitarily random matrices and the roots of derivatives of polynomials with independent, random coefficients, by giving a convolution semi-group which relates the two processes together.

4pm: Noah Halberstam (Cambridge): Infinite trees in the arboreal gas

Abstract: The arboreal gas, alternatively known as the edge weighted unrooted spanning forest model, is equivalent to Bernoulli percolation conditioned to be acyclic. Recent exciting work of Bauerschmidt, Crawford and Helmuth has established in dimensions d>2 the existence of a value for the edge weight parameter above which certain infinite volume limits contain an infinite tree almost surely, and thus that the model demonstrates a phase transition with respect to this parameter. Using probabilistic techniques, we show that in low dimensions d=3,4 the infinite tree is unique, and give strong heuristic evidence that the number of infinite trees is in fact infinite in higher dimensions. We also prove that in any dimension all such infinite trees must be one-ended almost surely. Joint work with Tom Hutchcroft.

5pm: Jorge Vargas (Caltech): Spectral stability under real random absolutely continuous perturbations

Abstract: In this talk I will discuss the following random matrix phenomenon (relevant in the design of numerical linear algebra algorithms): if one adds independent (tiny) random variables to the entries of an arbitrary deterministic matrix A, with high probability, the resulting matrix A′ will have (relatively) stable eigenvenvalues and eigenvectors.

More conretely, I will explain the key ideas behind obtaining tail bounds for the eigenvector condition number and minimum eigenvalue gap of a deterministic matrix that has been perturbed by a (small) random matrix with independent real entries, each with absolutely continuous distributions. I will also mention follow up work and open questions.

This is joint work with Jess Banks, Archit Kulkarni and Nikhil Srivastava.

Dinner: El Cholo, 1121 S. Western Ave., Los Angeles 90006

Thursday January 26th, 2023. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Barry Simon (Caltech): A tale of three coauthors: comparison of Ising models

Abstract: On Friday, Jan 14, 2022, I had a draft of a single author paper intended  for the Lieb Festschrift. Six days later, the paper had three coauthors. This talk will explain the interesting story, expose some underlying  machinery and sketch the proof of a lovely inequality on certain  finite  sums.

5pm: Lily Reeves (Cornell): Chemical distance for 2d critical percolation

Abstract: Percolation clusters induce an intrinsic graph distance called the chemical distance. Besides its mathematical appeal, the chemical distance is connected to the study of random walks on critical percolation clusters. In this talk, I will begin with a brief survey on the chemical distance. Then, I will zoom in to the progress and challenges in the 2d critical regime. A portion of this talk is based on joint work with Philippe Sosoe.

6pm: Sourav Chatterjee (Stanford): Spin glass phase at zero temperature in the Edwards-Anderson model

Abstract: I will present the solutions to two open problems about the Edwards-Anderson model of short-range spin glasses (in all dimensions). First, I will show that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, I will prove that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region - a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. Together, these comprise the first mathematical proof of glassy behavior in a short-range spin glass model.

Dinner: La Grand Orange Cafe, 7:30pm.

Thursday December 1st, 2022. USC. Talks in Kaprelian (KAP) 414.

Please note that the unusual time: Everything is shifted one hour earlier compared to the usual schedule.

2-3pm: Informal meet and greet at Cafe Annenberg.

3pm: Vadim Gorin (Berkeley): Lozenge tilings via the dynamic loop equation

Abstract: I will present a framework for studying general discrete Markov chains with interactions between particles, assuming that these interactions are of random matrix type. The approach is based on a novel holomorphic observable for the transition probabilities. As an application we will discuss inhomogeneous (q,kappa)-distributions on lozenge tilings and uncover their rich asymptotic behavior.

4pm: Xin Sun (UPenn): Two types of integrability in Liouville quantum gravity

Abstract: There are two major resources of integrability in Liouville quantum gravity: conformal field theory and random planar maps decorated with statistical physics models. I will give a few examples of each type and explain how these two types are compatible. Recently, cutting and gluing random surfaces in LQG using SLE curves allows us to blend these two types of integrability to obtain exact results on Liouville conformal field theory, mating of trees, Schramm-Loewner evolution, and conformal loop ensemble. I will present a few results in this direction. Based on joint works with Morris Ang, Nina Holden and Guillaume Remy.

5pm: Matan Harel (Northeastern): On delocalization of planar integer-valued height functions and the Berezinskii-Kosterlitz-Thouless phase transition of two-component spin models in two dimensions

Abstract: In this talk, we will discuss the relation between two types of two-dimensional lattice models: on one hand, we will consider the spin models with an O(2)-invariant interaction, such as the XY and Villain models. On the other, we study integer-valued height function models, where the interaction depends on the discrete gradient. We show that delocalization of a height function model implies that an associated O(2)-invariant spin model has a power-law decay phase. Motivated by this observation, we also extend the recent work of Lammers to show that a certain class of integer-valued height functions delocalize for all doubly periodic graphs (in particular, on the square lattice). Together, these results give a new perspective on the Berezinksii-Kosterlitz-Thouless phase transition for two-dimensional O(2)-invariant lattice models. This is joint work with Michael Aizenman, Ron Peled, and Jacob Shapiro.

Dinner: University Club, 6:30pm

Thursday November 3rd, 2022. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House. 

Special note: There is also a UCLA math colloquium taking place 3-4pm in Math Sciences Room 6627 that may be of interest to some of our participants.

4pm: Sarai Hernandez-Torres (UNAM): Continuity of the time constant of finitary random interlacements

Abstract: Finitary random interlacements (FRI) is a Poisson point process of geometrically killed random walks on Z^d, with d ≥ 3. A parameter u > 0 modulates the intensity of the point process, while T > 0 is the expected path length. The model has gained attention because, although it lacks global monotonicity on T, FRI (u, T) exhibits a phase transition.

In the supercritical regime, FRI (u, T) has a unique infinite cluster on which we consider its chemical distance (or graph distance). This talk focuses on the associated time constant. The time constant is a normalized limit of the chemical distance between the origin and a sequence of vertices growing in a fixed direction, defining a deterministic norm. Our main result is its continuity (as a function of the parameters u and T).

This work is joint with Zhenhao Cai, Eviatar Procaccia, Ron Rosenthal and Yuan Zhang.

5pm: Tomas Berggren (MIT): Random dimer coverings of the Aztec diamond with doubly periodic edge weights

Abstract: This talk will be centered around domino tilings, or dimer coverings, of the Aztec diamond with doubly periodic edge weights. Such model exhibits a rich structure, for instance, in the limit three types of regions may appear, the frozen, rough and smooth regions (also known as solid, liquid and gas regions). We will discuss asymptotic results, both on the macroscopic and microscopic scale.

The asymptotic results rely on an expression of the correlation kernel in terms of a Wiener-Hopf factorization of a matrix valued function, which is defined in terms of the edge weights. If time permits, we will discuss how such Wiener-Hopf factorization sometimes can be obtained in a form that are suitable for asymptotic analysis. ​

6pm: Izumi Okada (Kyushu): Capacity of the range of random walk

Abstract: We study the capacity of the range of a simple random walk in three and higher dimensions. It is known that the order of the capacity of the random walk range in n dimensions is similar to that of the volume of the random walk range in n-2 dimensions. We show that this correspondence breaks down for the law of the iterated logarithm for the capacity of the random walk range in three dimensions. We also prove the law of the iterated logarithm in higher dimensions. This is joint work with Amir Dembo. 

Dinner: Shamshiri Grill, 7:30pm.

Thursday October 6th, 2022. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Tianyi Zheng (UCSD): Furstenberg entropy spectrum of stationary actions

Abstract: The study of stationary actions of groups and connections to random walks was initiated by Furstenberg, and later plays a role in the development of rigidity theory. The Furstenberg entropy is a fundamental invariant defined for a stationary system. In this talk we will discuss some aspect of the question: given a group, what is the range of the Furstenberg entropy of ergodic stationary actions of it? For the linear group SL(d,R) and its lattices, constraints on this spectrum are closely related to structure theorems due to Nevo and Zimmer; and entropy values can be realized based on random walks on random stationary graphs. 

5pm: Philip Easo (Caltech): Percolation on finite transitive graphs

Abstract: Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.

6pm: Tim Austin (UCLA): Entropy, dynamics of free groups, and probability over sparse random graphs

Abstract:   Multivariate distributions determined by an underlying graph structure are well established in probabilistic combinatorics, statistics, statistical physics, coding theory and several other disciplines.  By taking a suitable weak limit, a sequence of such graphical models can also give rise to a measure-preserving action of a free or free-product group, and this construction is perhaps the richest source of examples in the ergodic theory of those groups.

This talk will give a rough overview of this branch of ergodic theory, and then describe some recent results about one of these examples, a limit of randomly-generated low-density parity-check codes.  It provides the strongest known counterexample to several natural extensions of classical theorems about the entropy of single measure-preserving transformations.  The talk will assume only quite basic knowledge of ergodic theory.

The new results are from an ongoing joint project with Lewis Bowen, Christopher Shriver and Brandon Seward.

Dinner: La Grand Orange Cafe, 7:30pm.

Thursday June 2nd, 2022. Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Huy Tuan Pham (Stanford): On some conjectures of Talagrand on suprema of positive selector processes and empirical processes 

Abstract: Understanding suprema of stochastic (empirical) processes is an important subject in probability theory with many applications. In the Gaussian case, via generic chaining and Talagrand’s celebrated majorizing measure theorem, Talagrand showed that extreme events of suprema of Gaussian processes admit a purely geometric characterization. Moving beyond the Gaussian case is a much more challenging quest. In this talk, I will discuss recent joint work with Jinyoung Park that resolves a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on general positive empirical processes. Combining with the recent resolution of the (generalized) Bernoulli conjecture and advances in chaining, this gives the first steps towards the last missing piece in the study of suprema of general empirical processes. 

Time permitting, I will also briefly discuss how one of the ideas in our proof of Talagrand’s conjectures leads to the proof of the Kahn-Kalai conjecture, an important question in probabilistic combinatorics and random graph theory. 

5pm: Eliza O'Reilly (Caltech): Random Tessellation Features and Forests

Abstract: The Mondrian process in machine learning is a recursive partition of space with random axis-aligned cuts used to build random forests and Laplace kernel approximations.  The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT random forests achieve minimax rates for Lipschitz and C^2 functions and STIT random features approximate a large class of stationary kernels. This work opens many new questions at the intersection of stochastic geometry and machine learning. Based on joint work with Ngoc Mai Tran.

6pm: Steven Heilman (USC): Convex Cylinders and the Symmetric Gaussian Isoperimetric Problem

Abstract:  The symmetric Gaussian isoperimetric problem of Barthe (2001) asks for the symmetric Euclidean set of fixed Gaussian volume with smallest Gaussian surface area.  We show that, if such a set is a cylinder, then this set or its complement must be convex.  Moreover, except for one case for the sign of the Gaussian mean curvature of the optimal set in R^n, the boundary of the optimal set must be r S^k × R^(n-k) for some (n-1)^(1/2) ≤ r ≤ (n+1)^(1/2)  and for some 1 ≤ k ≤ n-1 (if k=0 we do not constrain r).   It has been known since the 1970s that a Euclidean set with fixed Gaussian volume and smallest Gaussian surface area is a half space, and this implies e.g. Gaussian concentration of measure.  The added symmetry assumption of Barthe's problem introduces extra difficulties.

Dinner: La Grand Orange Cafe, 7:30pm.

Thursday May 5th, 2022. UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Nathanael Berestycki (Vienna): Near-critical dimers and massive SLE. 

Abstract: A programme initiated by Makarov and Smirnov is to describe near-critical scaling limits of planar statistical mechanics models in terms of massive SLE and/or Gaussian free field. We will consider in this talk the dimer model on the square and hexagonal lattices with doubly periodic weights. Although in the near-critical regime the Kasteleyn matrix is related to a massive Laplacian, Chhita proved that on the whole plane square lattice, the corresponding height function has a scaling limit which surprisingly is not even Gaussian. 

In joint work with Levi Haunschmid (TU Vienna) we obtain some new results about this model in three different directions: (a) we establish a rigorous connection with the massive SLE$_2$ constructed by Makarov and Smirnov; (b) we show that the convergence takes place in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. Our techniques rely on Temperley's bijection and the "imaginary geometry" approach developed in earlier work with Benoit Laslier and Gourab Ray.

5pm: Emmanuel Michta (UBC): Plateaux estimates and their implications for percolation and self-avoiding walks in high-dimensions

Abstract: For torus percolation in high-dimensions the two-point function has a whole critical window (depending on the size of the torus) where it behaves more or less like the critical two-point function in Z^d. This fact can be summarized in a quantitative estimate called the plateau for the torus two-point function. A similar estimate conjecturally holds for the self-avoiding walk two-point function on a torus and partial results have been recently obtained in this direction. In this talk we will focus on plateaux estimates for those two models as well as on the various implications they have : ranging from the torus triangle condition in percolation to the  asymptotic number of (weakly) self-avoiding walks on the torus. This is based on joint work with Tom Hutchcroft, Gordon Slade and Jiwoon Park.

6pm: Marianna Russkikh (MIT): Dimers and embeddings

Abstract: The dimer model is a model from statistical mechanics corresponding to random perfect matchings on graphs. Circle patterns are a class of embeddings of planar graphs such that every face admits a circumcircle. We introduce a concept of ‘t-embeddings’ (or a circle pattern) of a dimer planar graph, and discuss algebro-geometric properties of these embeddings. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model.

Dinner: Sunnin Lebanese Cafe, 7:30pm.

Thursday April 7th, 2022, Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Antoine Jego (MSRI): Multiplicative chaos of the Brownian loop soup.

Abstract: 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.

5pm: Yilin Wang (MSRI/MIT): SLE, energy duality, and foliations by Weil-Petersson quasicircles

Abstract: The Loewner energy for Jordan curves first arises from the small-parameter large deviations of Schramm-Loewner evolution (SLE). It is finite if and only if the curve is a Weil-Petersson quasicircle, an interesting class of Jordan curves appearing in Teichmuller theory, geometric function theory, and string theory with currently more than 20 equivalent definitions. In this talk, I will show that the large-parameter large deviations of SLE gives rise to a new Loewner-Kufarev energy, which is dual to the Loewner energy via foliations by Weil-Petersson quasicircles and exhibits remarkable features and symmetries. Based on joint works with Morris Ang and Minjae Park (MIT) and with Fredrik Viklund (KTH).

6pm: Mitia Duerinckx (ULB/CNRS/UCLA): Eigenvalue fluctuations for random elliptic operators in homogenization regime

Abstract: This talk is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is derived for eigenvalues towards those of the homogenized problem, as well as a quantitative two-scale expansion result for eigenspaces. Next, a quantitative central limit theorem is presented for fluctuations of isolated eigenvalues: more precisely, in the spirit of previous works in collaboration with Gloria and Otto, we derive a pathwise characterization of eigenvalue fluctuations.

Dinner: La Grand Orange Cafe, 7:30pm.

Thursday March 10th, 2022, UCLA. Talks in Math Sciences Room 6627.

3-4pm: Informal meet & greet at the Kerckhoff Coffee House.

4pm: Diederik van Engelenburg (Vienna): An elementary proof of phase transition in the planar XY model. 

Abstract: We derive, with elementary methods, a power-law bound on the two-point function of the planar XY model at low temperatures and therefore show the model undergoes a Berezinskii-Kosterlitz-Thouless phase transition. This was famously first rigorously proved by Fröhlich and Spencer in the eighties. Our argument relies on a new loop representation of spin correlations and a recent result by Lammers on the delocalisation of integer-valued height functions. The main contribution is a switching lemma for the loop representation that can also be used to prove some classical correlation inequalities. Joint work with Marcin Lis.

5pm: Giulio Tiozzo (Toronto): Singularity of harmonic measure for random walks on cocompact Fuchsian groups

Abstract: A recurring question in the theory of random walks on groups of isometries of hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift, also known as the “fundamental” inequality. For certain finitely-supported random walks on cocompact Fuchsian groups, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1. Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups. Joint with P. Kosenko.

6pm: Eviatar Procaccia (Technion): The chemical distance in random interlacements in the low-intensity regime

Abstract: Random interlacements is a Poissonian soup of doubly-infinite random walk trajectories on Z^d. A parameter u > 0 controls the intensity of the Poisson point process. In a natural way, the model defines percolation on the edges of Z^d with long-range correlations. We consider the time constant associated to the chemical distance in random interlacements at low intensity u > 0. It is conjectured that the time constant times u^{1/2} converges to the Euclidean norm. We prove a sharp upper bound (of order u^{-1/2}) and an almost sharp lower bound (of order u^{-1/2+\epsilon}) for the time constant as the intensity decays to zero. Joint work with Sarai Hernandez-Torres and Ron Rosenthal.

Dinner: Sunnin Lebanese Cafe, 7:30pm.

Thursday February 10th, 2022, Caltech. Talks in 310 Linde Hall.

3-4pm: Informal meet & greet at the Red Door Cafe.

4pm: Ellen Powell (Durham): Characterising the Gaussian free field

Abstract: I will discuss recent approaches to characterising the Gaussian free field in the plane, and in higher dimensions. The talk will be based on joint work with Juhan Aru, Nathanael Berestycki and Gourab Ray.

5pm: Swee Hong Chan (UCLA): Combinatorial atlas for log-concave inequalities

Abstract: The study of log-concave inequalities for combinatorial objects have seen much progress in recent years. One such progress is the solution to the strongest form of Mason's conjecture (independently by Anari et. al. and Brándën-Huh). In the case of graphs, this says that the sequence f_k of the number of forests of the graph with k edges, form an ultra log-concave sequence. In this talk, we discuss an improved version of all these results, proved by using a new tool called the combinatorial atlas method. This is a joint work with Igor Pak. This talk is aimed at a general audience.

6pm: Reza Gheissari (Berkeley): Entropic Repulsion of 3D Ising Interfaces

Abstract: We will discuss recent progress on understanding the entropic repulsion phenomenon for low-temperature 3D Ising interfaces. More precisely, if one considers the low-temperature Ising model in a box of side-length n in three dimensions, with Dobrushin's boundary conditions (plus below the xy-plane and minus above), then the interface separating the predominantly plus region from the predominantly minus region is localized about height zero (independently of n). Suppose one conditions on this interface staying entirely above a hard barrier at height -k_n: when k_n is large enough, the barrier effect is not significant and the interface remains localized, but when k_n is zero, say, it causes the entire interface to lift off and have diverging average height. We will describe recent work joint with Eyal Lubetzky identifying the critical k_n* delineating this localization/delocalization transition. 

Dinner: La Grand Orange Cafe, 7:30pm.