# Probability Seminar

# Fall 2018

**Thursdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.
**We usually end for questions at 3:15 PM.**

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## Friday, August 10, 10am, B239 Van Vleck András Mészáros, Central European University, Budapest

Title: **The distribution of sandpile groups of random regular graphs**

Abstract: We study the distribution of the sandpile group of random [math]d[/math]-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the [math]p[/math]-Sylow subgroup of the sandpile group is a given [math]p[/math]-group [math]P[/math], is proportional to [math]|\operatorname{Aut}(P)|^{-1}[/math]. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.

Our results extends a recent theorem of Huang saying that the adjacency matrices of random [math]d[/math]-regular directed graphs are invertible with high probability to the undirected case.

## September 20, Hao Shen, UW-Madison

Title: **Stochastic quantization of Yang-Mills**

Abstract: "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise. In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].

## September 27, Timo Seppäläinen

Title:**Random walk in random environment and the Kardar-Parisi-Zhang class**

Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).