## Pacific Northwest Probability Seminar

The Twentieth Northwest Probability Seminar
October 20, 2018
Supported by the Pacific Institute for the Mathematical Sciences (PIMS) and Microsoft Research.

The Birnbaum Lecture in Probability will be delivered by Jeremy Quastel (University of Toronto) in 2018.

The 20th Northwest Probability Seminar, a one-day mini-conference organized by the University of Washington, the Oregon State University, the University of British Columbia, the University of Oregon, and Microsoft Research, will be held on October 20, 2018. The conference will be hosted at Microsoft, supported by Microsoft Research and the Pacific Institute for the Mathematical Sciences (PIMS).

There is no registration fee. Breakfast, lunch, and coffee will be free.

The talks will take place in Building 99 at Microsoft. Parking at Microsoft is free.

## Schedule

 10:00 — 11:00 Coffee and muffins 11:00 — 11:40 Moumanti Podder (University of Washington) First Order Logic on Galton-Watson Trees 11:50 — 12:30 Yinon Spinka (University of British Columbia) Finitary codings for random fields on $\mathbb{Z}^d$ 12:30 — 1:40 Lunch (catered) 1:10 — 2:15 Probability demos and open problems (overlaps with lunch) 2:20 — 3:10 Jeremy Quastel (University of Toronto; Birnbaum Lecture) The KPZ fixed point 3:20 — 4:00 Alex Zhai (Stanford University) Cutoff for product replacement on finite groups 4:00 — 4:30 Tea and snacks 4:30 — 5:10 Radu Dascaliuc (Oregon State University) Stochastic explosions in branching processes and non-uniqueness for nonlinear PDE 6:00 — no host dinner at Haiku sushi & seafood buffet, downtown Redmond

## Titles and abstracts

### Radu Dascaliuc (Oregon State University)

Title: Stochastic explosions in branching processes and non-uniqueness for nonlinear PDE

Abstract: We will discuss stochastic processes, Le Jan-Sznitman cascades, that can be associated with certain nonlinear PDE and how explosion of these cascades can be exploited to prove non-uniques for the associated Cauchy problems. In particular, we are interested in using these techniques to explore uniqueness problems for the 3D Navier-Stokes equations.

This research is a collaboration with E. Thomann and E. Waymire.

### Moumanti Podder (University of Washington)

Title: First Order Logic on Galton-Watson Trees

Abstract: This talk will focus on the rooted Galton-Watson (GW) tree, and we shall limit ourselves to Poisson$(\lambda)$ offspring distributions. We discuss the first order (FO) language, derived from mathematical logic, on rooted trees. FO sentences describe finite structures inside the tree. We analyze the probabilities of FO properties under the GW measure, and obtain these probabilities as fixed points of contracting distributional maps. Moreover, we show that the probabilities of FO sentences, conditioned on survival of the GW tree, are expressible as nice functions of $\lambda$ and $p_{\lambda}$, the survival probability.

Time permitting, we shall briefly touch on some of the recently concluded work on second order logic on random rooted trees.

Joint work with Joel Spencer.

### Jeremy Quastel (University of Toronto)

Title: The KPZ fixed point

Abstract: The (1d) KPZ universality class contains random growth models, directed random polymers, stochastic Hamilton-Jacobi equations (e.g. the eponymous Kardar-Parisi-Zhang equation). It is characterized by unusual scale of fluctuations, some of which appeared earlier in random matrix theory, and which depend on the initial data. The explanation is that on large scales everything approaches a special scaling invariant Markov process, the KPZ fixed point. It is obtained by solving one model in the class, TASEP, and passing to the limit. Both TASEP and the KPZ fixed point turn out to have a novel structure: "stochastic integrable systems".

Joint work with Konstantin Matetski and Daniel Remenik.

### Yinon Spinka (University of British Columbia)

Title: Finitary codings for random fields on $\mathbb{Z}^d$

Abstract: Let $X$ be a translation-invariant random field on $\mathbb{Z}^d$ (i.e., a stationary $\mathbb{Z}^d$-process). We say that $X$ can be coded by an i.i.d. process if there is a deterministic and translation-invariant way to construct a realization of $X$ from i.i.d. variables associated to the sites of $\mathbb{Z}^d$. That is, if there is an i.i.d. process $Y$ and a measurable map $F$ from the underlying space of $Y$ to that of $X$, which commutes with translations of $\mathbb{Z}^d$ and satisfies that $F(Y)=X$ in distribution. Such a coding is called finitary if in order to determine the value of $X$ at a given site, one only needs to look at a finite (but random) region of $Y$.

It is known that a phase transition (existence of multiple Gibbs states) is an obstruction for the existence of such a finitary coding. We on the other hand discuss conditions which guarantee that $X$ can be finitarily coded by an i.i.d. process. One such condition is weak spatial mixing for Markov random fields. Another condition is monotonicity and uniqueness of the Gibbs measure. Applications are given to numerous models, including the the Potts model, the random-cluster model, proper colorings and the hard-core model.

Based on work with Matan Harel.

### Alex Zhai (Stanford University)

Title: Cutoff for product replacement on finite groups

Abstract: Let $G$ be a finite group, and consider the following \emph{product replacement walk} on the set of generating $n$-tuples of elements of $G$: randomly pick two of the $n$ elements, say $g$ and $h$, and replace $g$ with either $gh$ or $gh^{-1}$, with equal probability. This walk has been studied in the context of computational group theory for its mixing properties. It can also be seen as part of a more general class of Markov chains that includes random walks on the group of invertible matrices $SL_n(\mathbb{F}_q)$ and the East model in interacting particle systems. In this talk, based on joint work with Yuval Peres and Ryokichi Tanaka, we show that as $n \rightarrow \infty$ (with $G$ fixed), the total-variation mixing time of the product replacement walk has a cutoff at time $\frac{3}{2} n \log n$ with window of order $n$. This generalizes a recent result of Ben-Hamou and Peres, who established the result for $G = \mathbb{Z}/2$. For general groups, the previous best bound due to Diaconis and Saloff-Coste was $O(n^2 \log n)$, who also conjectured the bound $O(n \log n)$.

## Dinner info

No host dinner (6:00 onwards) at Haiku sushi & seafood buffet, downtown Redmond