Probability Archives for Fall 2021 to Spring 2022

Dynamic polymers: invariant measures and ordering by noise

When: Thu, October 21, 2021 - 11:00am
Where: 0104
Speaker: Yuri Bakhtin (NYU) -
Abstract: Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.

A Decomposition-Separation Theorem for Markov Chains and Related Problems

When: Thu, October 28, 2021 - 11:00am
Where: Kirwan Hall 3206
Speaker: Isaac Sonin (UNC Charlotte) -
Abstract: Let $(P_{n}), n=0,1,2,...$ be a sequence of finite stochastic matrices of the same size $N\times N$. How does the product $\prod_{i=k}^nP_i, 0\leq k$, behave, when $n\rightarrow \infty$, if there are no assumptions about sequence the $(P_{n})$? In probabilistic terms this question is equivalent to a question about the asymptotic behaviour of a family of nonhomogeneous Markov chains defined by a sequence $(P_{n})$ and all possible initial distributions on a finite set $S, |S|=N$ with an initial time point $k$.

The surprising answer to this question is given by a fundamental theorem, which we call the Decomposition-Separation (DS)\ Theorem. This theorem was initiated by a small paper of A. N. Kolmogorov (1936) and formulated and proved in a few stages in a series of papers: D. Blackwell (1945), H. Cohn (1971,..., 1989) and I. Sonin (1987, 1991,..., 2008).

The DS Theorem has a simple deterministic interpretation in terms of the behaviour of the simplest model of an irreversible process. Such a model can be defined, for
example, by a finite number of cups filled with a solution (e.g. tea) having
possibly different concentrations and exchanging this solution at
discrete times. This can also be considered in terms of the evolution of a
systems of particles, which connects this theorem with a broad area called Consensus Algorithms.

Some new results will also be presented, but generally the DS
Theorem leaves many open problems and probably leads to generalizations in other fields of mathematics besides probability theory.

Among other topics that will be briefly mentioned are: The Markov Chain Tree Theorem and the question: "What does it mean that two events are independent?"

Rare exit events near a repelling equilibrium

When: Wed, November 3, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Hong-Bin Chen (NYU) -
Abstract: Consider an ODE and a bounded domain in a Euclidean space. Assume, in that domain, the ODE has only one equilibrium and it is repelling. Under white noise perturbation, there is a positive probability that the dynamics emitting from the equilibrium point eventually exits the domain. We will discuss the exact asymptotics of rare exit events, e.g. exit locations and exit times, in the vanishing noise limit. It turns out that the decay rates of such rare events are polynomial powers of the noise magnitude, different from the exponential decay in the Freidlin-Wentzell large deviation theory. This talk is based on the joint work with Yuri Bakhtin.

KPZ on torus

When: Wed, November 17, 2021 - 2:00pm
Where: Physics 1410
Speaker: Yu Gu (UMD) -
Abstract: I will present a joint work with Tomasz Komorowski in which we establish the central limit theorem for the Kardar-Parisi-Zhang equation on a torus. I will explain the connection to the central limit theorem of the product of independent random matrices.

Nonlinear weak-noise stochastic heat equations in two dimensions

When: Wed, March 9, 2022 - 1:00pm
Where: Kirwan Hall 3206
Speaker: Alex Dunlap (NYU) -

Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. I will also discuss two cases in which the FBSDE can be explicitly solved: the linear stochastic heat equation, for which we recover the log-normal behavior proved by Caravenna, Sun, and Zygouras, and branching Brownian motion/super-Brownian motion, for which we obtain a solution to the Feller diffusion. This talk will be based on joint work with Yu Gu and with Cole Graham.

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

When: Wed, April 20, 2022 - 12:00pm
Where: Kirwan Hall 3206
Speaker: Elena Kosygina (CUNY) -
Abstract: Generalized Ray-Knight theorems for edge local times proved to be a very useful tool for studying the limiting behavior of some self-interacting random walks (SIRWs). Examples include some reinforced random walks, excited random walks, rotor walks with defects. I shall describe two classes of SIRWs studied by Balint Toth in mid-nineties - asymptotically free and polynomially self-repelling SIRWs - and discuss recent results (joint work with Thomas Mountford, EPFL, and Jon Peterson, Purdue University) which resolve an open question posed in Toth’s paper. We show that in the asymptotically free case the rescaled SIRWs converge to a perturbed Brownian motion (the result conjectured by Toth) while in the polynomially self-repelling case the convergence to the conjectured process fails in spite of the fact that generalized Ray-Knight theorems clearly identify the unique candidate in the class of perturbed Brownian motions. This negative result was somewhat unexpected. The question whether there is a suitable limit process in this case remains open.

Empirical measures, geodesic lengths, and a variational formula in first-passage percolation

When: Wed, April 20, 2022 - 1:00pm
Where: Kirwan Hall 3206
Speaker: Erik Bates (UW Madison) -

Abstract: We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. This induces a random ``disordered” geometry on the lattice. Our primary interest is in the empirical measures of edge-weights observed along geodesics in this geometry, say from 0 to [n\xi], where \xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

One-point distribution of the geodesic in directed last passage percolation

When: Wed, April 27, 2022 - 1:00pm
Where: Kirwan Hall 3206
Speaker: Zhipeng Liu (University of Kansas) -

Abstract: In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. On the other hand, different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently, but the precise limiting distributions of the geodesic locations remained unknown.
In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give an explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.

On large deviation principles and the Monge–Ampere equation (following Berman, Hultgren)

When: Wed, May 11, 2022 - 1:00pm
Where: Kirwan Hall 3206
Speaker: Yanir Rubinstein (UMD) -
Abstract: The talk will aim to give an exposition, accessible to both geometers and probabilists, of a fundamental
recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the
second boundary value problem for the real Monge–Ampere equation admits a probabilistic
interpretation, in terms of many particle limit of permanental point processes satisfying a
large deviation principle with a rate function given explicitly using optimal transport.
Time permitting, we will take the opportunity to present an alternative proof of the Berman-Hultgren theorem
that deals with all “temperatures” simultaneously instead of first reducing to the zero-temperature case.