## Summary

This is a recap of the tenth Northwest Probability Seminar was 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 the Theory Group at Microsoft Research. Usually the conference is hosted at the University of Washington, but this year the conference was hosted at Microsoft.

Supported by the **Mathematical Sciences Research Institute** (MSRI),

**Microsoft Research**, and the **Pacific Institute for the Mathematical Sciences** (PIMS).

The **Birnbaum**** Lecture in Probability** was given by **Laurent Saloff-Coste** (Cornell). The other speakers were Omer Angel (UBC), Eyal Lubetzky (MSR), Soumik Pal (UW), and Edward Waymire (OSU).

[Speaker photographs] [Past Birnbaum speakers]

The Scientific Committee for the 2008 NW Probability Seminar consisted of Martin Barlow (U British Columbia), Chris Burdzy (U Washington), Zhen-Qing Chen (U Washington), Yevgeniy Kovchegov (Oregon State U), David Levin (U Oregon), and Yuval Peres (Microsoft).

## Schedule

9:30 **Coffee and muffins**

10:30 **Laurent Saloff-Coste** (Cornell)

Behavior of Brownian motion on compact Lie groups as a function of dimension (e.g., on special orthogonal groups).

Abstract:The distribution of Brownian motion on a compact manifold converge to its equilibrium (the normalized volume measure). Quantitatively, for natural families of compact manifolds, this convergence depends on certain geometric properties. In this talk we will focus on compact Lie groups such as the family of special orthogonal groups and consider the convergence of Brownian motion as the dimension grows to infinity.

11:30 **Omer Angel** (UBC)

Colouring Voronoi

Abstract:We consider the problem of colouring the planar map given by the Voronoi tessellation corresponding to a Poisson process in R^2. We seek colouring rules that are isometry invariant and are factors of the Poisson process. We prove that 6 colours suffice.

Joint work with Itai Benjamini, Ori Gurel-Gurevich, Tom Meyerovitch, and Ron Peled.

12:15 **Open problems and microtalks**

1:00 **Lunch **(catered)

2:15 **Eyal Lubetzky** (MSR)

Cutoff phenomena for random walks on random regular graphs

Abstract:The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are conjectured to exhibit cutoff, yet establishing this fact is often extremely challenging. An important such family of chains is the random walk onG(n,d), a randomd-regular graph onnvertices. Friedman determined the constant spectral gap of this class of chains fordfixed, implying a mixing-time of O(logn). Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph iswhp(6+o(1))log_{2}n, and Peres further conjectured that for any fixeddthe simple random walk onG(n,d) has cutoffwhp.In this work we confirm the above conjectures, and establish cutoff in total-variation, its location and its optimal window, both for simple and for non-backtracking random walks on

G(n,d). Namely, for any fixedd, thesimplerandom walk onG(n,d)whphas cutoff at d/(d-2) log_{d}_{-1}nwith window order sqrt{logn}. Surprisingly, thenon-backtrackingrandom walk onG(n,d)whphas cutoff already at log_{d-1}nwithconstantwindow order. We further extend these results toG(n,d) for anyd=n^{o(1)}(beyond which the mixing time is O(1)), provide efficient algorithms for testing cutoff, as well as give explicit constructions where cutoff occurs.Joint work with Allan Sly.

3:00 **Soumik Pal** (UW)

Applications of a skew-product decomposition for the Bessel-Squared processes

Abstract:We consider different models arising in three distinct areas of probability: Watterson’s Infinitely-Many-Neutral-Alleles model from mathematical biology, the Volatility-Stabilized-Market model of Fernholz and Karatzas in mathematical finance, and the embedding of large forests of critical Galton-Watson trees in Brownian motion. We show that all these models have a structure in which a skew-product decomposition of the Bessel-Squared processes play an important role. As a conclusion we see the emergence of a limiting Poisson-Dirichlet structure for each of them. For Watterson’s model, this conclusion was proved earlier by Ethier and Kurtz via a different method. For the other models, these conclusions are new. In particular, we solve a problem posed by Fernholz and Karatzas with regards to the distribution of the ‘market weights’ functionals of the VSM model.

3:45 **Tea**

4:15 **Edward Waymire** (OSU)

Skew Brownian Motion and Applications in Fluid Dispersion

Abstract:Skew Brownian motion was introduced by Ito and Mckean in a classic 1963 paper devoted to constructions of various stochastic processes associated with Feller’s classification of one-dimensional diffusions. Its basic properties and extensions have been the subject of a number of papers in the foundations of probability theory (many by probabilists in the Pacific Northwest !) In this talk we will discuss some recent and ongoing applications that arose out of discussions with colleagues in the geosciences at OSU. In particular this has led to the derivation of apparently new formulae for joint densities involving skew Brownian motion (with drift) and local and occupation times.It is based on joint work with OSU student and colleagues Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, Brian Wood, and Jorge Ramirez (now at the University of Arizona).

5:00 **Open problem solutions, conclude **

5:45 **Dinner** (not hosted)