ABSTRACTS
Spring 2024
Jan 24 @ Wharton Statistics and Data Science Colloquium : Lingfu Zhang (UC Berkeley)
Probabilistic perspective toward KPZ class models
A striking phenomenon in probability theory is universality, where different probabilistic models produce the same large-scale or long-time limit. One example is the Kardar-Parisi-Zhang (KPZ) universality class, encompassing a wide range of natural models such as growth processes modeling bacterial colonies, eigenvalues of random matrices, and traffic flow models originating from mRNA translation. Historically, these KPZ class models have mostly been studied via algebraic methods. In this talk, I will introduce a general strategy that adopts a more probabilistic perspective. This approach has enabled us to successfully resolve many open problems. I will present a selection of these results, including (1) local statistics of Last Passage Percolation – a pivotal model in the KPZ class, (2) the slow bond problem in a related traffic flow model, and (3) the cutoff phenomenon in Metropolis biased card shuffling. No prior knowledge of the topic will be assumed.
Jan 30 @ Temple: Mokshay Madiman (University of Delaware)
Log-concavity in 1-d Coulomb gas ensembles
The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log-concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, and Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. We then obtain the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is the Poissonized version of a conjecture of Chen. This is ongoing joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.
Feb 6 @ Penn: Zhengjiang Lin (NYU Courant)
Asymptotic topological statistics of Gaussian random zero sets
We will briefly discuss some asymptotic topological statistics of Gaussian random zero sets, which include a random distribution on knots as a special case. We will also discuss some results on zero sets of random Laplacian eigenfunctions, which are related to Courant’s nodal domain theorem and Milnor-Thom’s theorem on Betti numbers of real algebraic varieties.
Feb 13 @ Penn: Yuxin Zhou (UChicago)
The spherical mixed p-spin glass at zero temperature
In this talk I will discuss the spherical mixed p-spin glass model at zero temperature. I will present some recent results that classify the possible structure of the functional ordered parameter. For spherical p+s spin glasses, we classify all possibilities for the Parisi measure as a function of the model. Moreover, for the spherical spin models with n components, the Parisi measure at zero temperature is at most n-RSB or n-FRSB. Some of these results are jointly with Antonio Auffinger.
Feb 20 @ Temple: Brian Rider (Temple)
A matrix model for conditioned Stochastic Airy
There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure. What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator conditioned on having no eigenvalues below a fixed level. I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).
Feb 27 @ Temple: Fraydoun Rezakhanlou (UC Berkeley)
Periodic orbits of stochastic Hamiltonian ODEs
According to Conley-Zehnder's theorem, any periodic Hamiltonian ODE in R^2n has at least 2n+1 geometrically distinct periodic orbits. For a stochastically stationary Hamiltonian ODE, the set of periodic orbits yields a translation invariant random process. In this talk, I will discuss an ergodic theorem for the density of periodic orbits, and formulate some open questions which are the stochastic variants of Conley-Zehnder's theorem.
Mar 12 @ Penn: Sayan Das (UChicago)
Weak Universality in Random Walks in Random Environments
We consider one dimensional simple random walks whose all one step transition probabilities are iid [0,1]-valued mean 1/2 random variables. In this talk, we will explain how under a certain moderate deviation scaling the quenched density of the walk converges weakly to Stochastic Heat Equation with multiplicative noise. Our result captures universality in the sense that it holds for all non-trivial laws for random environments. Time permitting, we will discuss briefly how our proof techniques depart from the existing techniques in literature. Based on a joint work with Hindy Drillick and Shalin Parekh.
Mar 26 @ Penn: Oanh Nguyen (Brown)
Contact process on large networks
The contact process serves as a model for the spread of epidemics on networks, with three popular variations: the Susceptible-Infected-Recovered-Susceptible (SIRS), SIR, and SIS. Our focus lies in understanding the temporal evolution of these processes, especially regarding survival time and its associated phase transitions. I will provide a brief overview of related literature, recent progress, and open problems.
Apr 2 @ Temple: Paul Jung (Fordham University)
A generalization of hierarchical exchangeability on trees to Directed Acyclic Graphs
We discuss a class of partially exchangeable random arrays which generalizes the notion of hierarchical exchangeability introduced in Austin and Panchenko (2014). We say that our partially exchangeable arrays are DAG-exchangeable since their partially exchangeable structure is governed by a collection of Directed Acyclic Graphs (DAG). More specifically, such a random array is indexed by N^|V| for some DAG, G = (V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem which generalizes the Aldous-Hoover and Austin-Panchenko representation theorems.
Apr 9 @ Penn: Zhongyang Li (UConn)
Scaling limits in dimers and tableaux
We investigate limit shapes and height fluctuations in statistical mechanical models, such as dimers and lecture hall tableaux, through the asymptotics of symmetric polynomials. Confirming a conjecture by Corteel, Keating, and Nicoletti, we show that the rescaled height functions' slopes in the scaling limit of lecture hall tableaux adhere to a complex Burgers equation.
Apr 16 @ Temple: Yujin Kim (NYU Courant)
The shape of the front of multidimensional branching Brownian motion
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.
Apr 23 @ Penn: Dana Randall (Georgia Tech)
Programmable Matter and Emergent Computation
Programmable matter explores how collections of computationally limited agents acting locally and asynchronously can achieve some useful coordinated behavior. We take a stochastic approach using techniques from randomized algorithms and statistical physics to develop distributed algorithms for emergent collective behaviors that give guarantees and are robust to failures.
This talk will also be streamed over Zoom.
Feb 20 @ Temple: Atilla Yilmaz (Temple)
Homogenization of nonconvex Hamilton-Jacobi equations in stationary ergodic media
I will start with a self-contained introduction to the homogenization of inviscid (first-order) and viscous (second-order) Hamilton-Jacobi (HJ) equations in stationary ergodic media in any dimension. After a brief account of the now-classical works that are concerned with periodic media or convex Hamiltonians, I will return to the general setting and outline the results obtained in the last decade that: (i) established homogenization for inviscid HJ equations in one dimension; and (ii) provided counterexamples to homogenization in the inviscid and viscous cases in dimensions two and higher. Finally, I will present my recent joint work with E. Kosygina in which we prove homogenization for viscous HJ equations in one dimension, and also describe how the solution of this problem qualitatively differs from that of its inviscid counterpart.