## Pacific Northwest Probability Seminar

The Nineteenth Northwest Probability Seminar
November 4, 2017
Supported by the Pacific Institute for the Mathematical Sciences (PIMS) and Microsoft Research.

The Birnbaum Lecture in Probability will be delivered by Michel Ledoux (University of Toulouse) in 2017.

The 19th 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 November 4, 2017. 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 Gourab Ray (University of Victoria) A characterization theorem for the Gaussian free field 11:50 — 12:30 Yuval Peres (Microsoft Research) Gravitational allocation to uniform points on the sphere 12:30 — 1:40 Lunch (catered) 1:10 — 2:15 Probability demos and open problems (overlaps with lunch) 2:20 — 3:10 Michel Ledoux (University of Toulouse; Birnbaum Lecture) Optimal matching of Gaussian samples 3:20 — 4:00 Yevgeniy Kovchegov (Oregon State University) Random self-similar trees: dynamical pruning and its applications to inviscid Burgers equations 4:00 — 4:30 Tea and snacks 4:30 — 5:10 Persi Diaconis (Stanford University) An Analysis of Spatial Mixing 6:00 — no host dinner at Haiku sushi & seafood buffet, downtown Redmond

## Titles and abstracts

### Persi Diaconis (Stanford University)

Title: An Analysis of Spatial Mixing

Abstract: In joint work with Soumik Pal, we study natural mixing processes where cards (or dominoes or mahjong tiles) are 'smushed' around on a table with two hands. How long should mixing continue. If things are not well mixed, what patterns remain? We study this in practice (!): experiments indicate that about 30 seconds of smushing suffice to mix 52 cards. We also study it in theory introducing a variety of models which permit analysis. Part of the analysis passes to a reflecting, jump- diffusion limit and uses this and a novel 'shadow coupling' to give reasonably precise bounds on the mixing time.

### Yevgeniy Kovchegov (Oregon State University)

Title: Random self-similar trees: dynamical pruning and its applications to inviscid Burgers equations

Abstract: We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree $T$, starting from the leaves, according to a pruning function defined on subtrees within $T$. The generalized pruning encompasses a number of previously studied discrete and continuous pruning operations, including the tree erasure and Horton pruning. For example, a finite critical binary Galton-Watson tree with exponential edge lengths is invariant with respect to the generalized dynamical pruning for arbitrary admissible pruning function. We will discuss an application in which we examine a one dimensional inviscid Burgers equation with a piece-wise linear initial potential with unit slopes. The Burgers dynamics in this case is equivalent to a generalized pruning of the level set tree of the potential, with the pruning function equal to the total tree length. We give a complete description of the Burgers dynamics for the Harris path of a critical binary Galton-Watson tree with i.i.d. exponential edge lengths.
This work was done in collaboration with Ilya Zaliapin (University of Nevada Reno) and Maxim Arnold (University of Texas at Dallas).

### Michel Ledoux (University of Toulouse)

Title: Optimal matching of Gaussian samples

Abstract: Optimal matching problems are random variational problems widely investigated in the mathematics and physics literature. We discuss here the optimal matching problem of an empirical measure on a sample of iid random variables to the common law in Kantorovich-Wasserstein distances, which is a classical topic in probability and statistics. Two-dimensional matching of uniform samples gave rise to deep results investigated from various view points (combinatorial, generic chaining). We study here the case of Gaussian samples, first in dimension one on the basis of explicit representations of Kantorovich metrics and a sharp analysis of more general log-concave distributions in terms of their isoperimetric profile (joint work with S. Bobkov), and then in dimension two (and higher) following the PDE and transportation approach recently put forward by L. Ambrosia, F. Stra and D. Trevisan.

### Yuval Peres (Microsoft Research)

Title: Gravitational allocation to uniform points on the sphere

Abstract: Given $n$ uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition - with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf.) Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of $n$ uniform blue points to $n$ uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984).
Joint work with Nina Holden and Alex Zhai.

### Gourab Ray (University of Victoria)

Title: A characterization theorem for the Gaussian free field

Abstract: We prove that any random distribution satisfying conformal invariance and a form of domain Markov property and having a finite moment condition must be the Gaussian free field. We also present some open problems regarding what happens beyond the Gaussian free field. Ongoing joint work with Nathanael Berestycki and Ellen Powell.

## Dinner info

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