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.
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