## 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) ??? 3:20 — 4:00 Alex Zhai (Stanford University) ??? 4:00 — 4:30 Tea and snacks 4:30 — 5:10 Radu Dascaliuc (Oregon State University) ??? 6:00 — no host dinner at Haiku sushi & seafood buffet, downtown Redmond

## Titles and abstracts

### 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.

### 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.

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## Dinner info

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