Abstract:We consider favorite sites, i.e., sites that achieve the maximal local time for a discrete time simple random walk. We show that the limsup of the number of favorite sites is 3 with probability one in d = Â 2. We also give sharp asymptotics of the number in higher dimensions. This talk is based on a joint work with Chenxu Hao (Peking), Xinyi Li (Peking) and Yushu Zheng (CAS).
Speaker: Wen-Tai Hsu (UMD) -Â In this talk, we discuss a diffusion process in a narrow tubular domain with reflecting boundary conditions, where the geometry serves as a singular perturbation of an underlying graph in $\mathbb{R}^2$ or $\mathbb{R}^3$. We show that, in the limit, the projected process converges weakly to a diffusion process on the graph, with gluing conditions at the vertices that depend on the relative scales of the neighborhoods. Our analysis relies on a detailed understanding of the narrow escape problem in domains with bottlenecks. In particular, we rigorously derive the asymptotic behavior of the expected escape time, establish the asymptotic exponential distribution of escape times, and obtain exit place estimates, results that may be of independent interest.
When: Wed, October 1, 2025 - 2:00pm Where: Kirwan Hall 1313
Abstract: : The directed landscape is the central object in the Kardar-Parisi-Zhang universality class, and is conjecturally the scaling limit for all models of last passage percolation, directed polymers, exclusion processes, and many other types of interface growth models arising in probability and statistical physics. It was constructed independently by Matetski-Quastel-Remenik and Dauvergne-Ortmann-Virag. In this talk, we discuss a recent result where we show that the directed landscape is a black noise. This roughly means that it is noise-sensitive and cannot be expressed as a random dynamical system driven by Gaussian white noise. In particular, it cannot be written as a stochastic PDE.
Abstract: Given an evolving string on [0,1] modeling random surface growth, one may ask: if the slopes at the two boundary points are prescribed, what does a typical realization look like? This question reduces to understanding the equilibrium state of the open KPZ equation. In recent work of Corwin–Knizel, Barraquand–Le Doussal, and Bryc–Kuznetsov–Wang–Wesołowski, this equilibrium state is described explicitly and identified as a resampled Brownian motion.
In this talk, I will present recent joint work with Alex Dunlap and Tommaso Rosati in which we provide a stochastic-analytic derivation of this equilibrium state. Our approach relies on tools such as change of measure, time reversal, Itô’s formula, and the theory of regularity structures.
Abstract: It was understood long ago that the first nonzero eigenvalue of the Laplacian on a closed hyperbolic surface cannot exceed that of the hyperbolic plane, asymptotically as the diameter goes to infinity. But until recently, it was not known whether there exist hyperbolic surfaces that attain this bound. In joint work with Magee and Puder, we show something much stronger: random covering spaces of an arbitrary hyperbolic surface have an optimal spectral gap with probability 1-o(1). I will aim to explain how this question reduces to an unusual problem of random matrix theory, whose resolution was made possible by recent advances in the theory of strong convergence.
Abstract: Alice and Bob control a random walk: alternately, each of them flips a fair coin, is supposed to report the outcome, and the random walk advances according to the report. Suppose that the random walk did not return to the origin infinitely often. We suspect that one of Alice and Bob misreported the outcomes of her or his coin. Can we identify the deviator?
More generally, several players are supposed to follow a prescribed profile of strategies (e.g., select each of Right and Left with probability 1/2). If they follow this profile, they will reach a given target (e.g., the random walk returns to the origin infinitely often). We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.
Joint work with Noga Alon (Princeton and Tel Aviv University), Benjamin Gunby (Rutgers), Xiaoyu He (Georgia Tech), and Eran Shmaya (Stony Brook).
Abstract: We consider a first passage percolation model in dimension 1 + 1 with potential given by the product of a spatial i.i.d. potential with symmetric bounded distribution and an independent i.i.d. in time sequence of signs. We assume that the density of the spatial potential near the edge of its support behaves as a power, with exponent κ > −1. We investigate the linear growth rate of the actions of optimal point-to-point lazy random walk paths as a function of the path slope and describe the structure of the resulting shape function. It has a corner at 0 and, although its restriction to positive slopes cannot be linear, we prove that it has a flat edge near 0 if κ > 0. For optimal point-to-line paths, we study their actions and locations of favorable edges that the paths tend to reach and stay at. Under an additional assumption on the time it takes for the optimal path to reach the favorable location, we prove that appropriately normalized actions converge to a limiting distribution that can be viewed as a counterpart of the Tracy–Widom law. Since the scaling exponent and the limiting distribution depend only on the parameter κ, our results provide a description of a new universality class.
Abstract: "On-scale" stretched exponential estimates for fluctuations of models within the Kardar-Parisi-Zhang universality class have played a particularly important role in mathematical work seeking to make physically motivated heuristic arguments about random growth models rigorous.
This talk will explain a simple coupling argument which recovers a moment generating function identity originally discovered by Rains using integrable probability techniques in the context of last-passage percolation. This coupling approach has proven to generalize significantly since it was first introduced about five years ago. The talk will then discuss how identities of this type can be used to derive the type of stretched exponential bounds mentioned above and when these bounds are sharp. The discussion will highlight some recent and in-progress improvements and extensions of older work.
Based on joint work with Elnur Emrah and Timo Seppäläinen.
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 the threshold and algebraic superdiffusion above it. I will discuss recent [arXiv: 2601.22142, arXiv:2404.01115] and ongoing work with Scott Armstrong and Tuomo Kuusi in which we address these problems using techniques from the theory of stochastic homogenization.
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