## Pacific Northwest Probability Seminar

- Date: 11/04/2023

University of Washington

The **Birnbaum Lecture in Probability** will be delivered by Robin Pemantle (University of Pennsylvania) in 2023.

Northwest Probability Seminars are mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia and the University of Oregon. There is no registration fee.

The talks will take place in Mary Gates Hall 241 (see the map).

**Schedule:**

10:00 - 11:00 **Coffee** Mary Gates Commons

11:00 - 11:50 **Stefan Steinerberger**, University of Washington

*Particle Growth Models in the Plane (DLA, DBM, ...) *

We'll discuss growth patterns in the plane. The most famous such model is DLA where new particles arrive via a Brownian motion and get stuck once they hit an existing particle: it forms the most beautiful fractal patterns (pictures will be provided). Despite this, DLA is actually fairly poorly understood and we will quickly survey the existing ideas (many of which are due to Harry Kesten). I will then present a new type of growth model that behaves similarly (many more pictures will be shown) and which can be very precisely analyzed (in certain cases). No prior knowledge is necessary, I'll explain everything from first principles.

12:00 - 12:50 **Nick Marshall**, Oregon State University

*Random High-Dimensional Binary Vectors, Kernel Methods, and Hyperdimensional Computing *

This talk explores the mathematics underlying hyperdimensional computing (HDC), a computing paradigm that employs high-dimensional binary vectors. In HDC, data is encoded by shifting and combining random high-dimensional binary vectors in various ways. We study the problem of determining the optimal distribution of random high-dimensional binary vectors for use in this construction.

1:00 - 3:00 **Lunch**, catered, Mary Gates Commons

3:00 - 3:50 **Robin Pemantle**, University of Pennsylvania

*"Birnbaum Lecture": Negative association and related properties *

Negative dependence properties of random variables have been valuable in proving limit theorems and tail bounds often substituting when independence fails. I will begin by reviewing the theory and uses of negative dependence concepts for binary random variables, beginning with origins in mathematical statistics and statistical mechanics.

Among the many concepts and definitions that have been proposed, two stand out: negative association (NA) and the strong Rayleigh property (SR). The former is a negative dependence property that is sometimes hard to prove but is very useful when it holds. The somewhat mysterious Strong Rayleigh property implies negative dependence and can in fact be a route to proving negative dependence.

The endpoint of this talk is to explore the limits of SA by looking at cases where NA holds or is expected to hold but SR does not. While this kills the hope of proving NA for these models by establishing SR, it also helps us see what allows NA to hold without SR, which I hope will motivate and enable development of new technology for proving NA.

4:00 - 4:30 **Coffee** Mary Gates Commons

4:30 - 5:20 **Lucas Teyssier**, University of British Columbia

*On the universality of fluctuations for the cover time *

How long does it take for a random walk to cover all the vertices of a graph? And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover time? We show that on vertex-transitive graphs of bounded degree, this set is decorrelated (it is close to a product measure) if and only if a simple geometric condition on the diameter of the graph holds. In this case, the cover time has universal fluctuations: properly scaled, the cover time converges to a Gumbel distribution. To prove this result we rely on recent breakthroughs in geometric group theory which give a quantitative form of Gromov's theorem on groups of polynomial growth. We also prove refined quantitative estimates showing that the hitting time of any set of vertices is (irrespective of its geometry) approximately an exponential random variable.

This talk is based on joint work with Nathanaël Berestycki and Jonathan Hermon.

6:00 No-host (likely subsidized) dinner.

Restaurant: **Mamma Melina**. There will be set menu, the first menu on this list. The cost per person will be 36 dollars (this does not include tax and gratuity) although it is likely that the conference will have funds to partly subsidize dinner. Please bring cash. There will be a vegetarian option. Wine and coffee can be ordered and paid for individually (cash only; alcohol will not be subsidized). Address: 5101 25th Ave NE, Seattle, WA 98105. See Google map. The restaurant is a 20 minute walk from Mary Gates Hall.

The Twenty Third Northwest Probability Seminar is supported by the University of Oregon, Pacific Institute for the Mathematical Sciences (PIMS), Baidu, University of Washington (Friends of Mathematics Fund), and Milliman Fund.

Omer Angel (University of British Columbia)

Chris Burdzy (University of Washington)

Zhenqing Chen (University of Washington)

David Levin (University of Oregon)

Alex Rodriguez (Oregon State University)