Probability Seminar

Fall 2021

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

ZOOM LINK. Valid only for online seminars.

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September 16, 2021, in person: Hanbaek Lyu (UW-Madison)

Scaling limit of soliton statistics of a multicolor box-ball system

The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.

September 30, 2021, in person: Marianna Russkikh (MIT)

Lozenge tilings and the Gaussian free field on a cylinder

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.

October 7, 2021, ZOOM: Barbara Dembin (ETH Zurich)

The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$

We consider the standard model of i.i.d. first passage percolation on $\mathbb Z^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty))>p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We prove that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$. This is a joint work with Raphaël Cerf.

December 9, 2021, ZOOM: Sunil Chhita (Durham)

GOE Fluctuations for the maximum of the top path in ASMs

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling. This talk is based on joint work with Arvind Ayyer and Kurt Johansson.

Past Seminars