# Northwest Probability Seminar

### Supported by the Pacific Institute for Mathematical Sciences

The event in pictures.

Northwest Probability Seminars are one-day mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia, the University of Oregon, and the Theory Group at the Microsoft Research. There is no registration fee. Participants are requested to contact Chris Burdzy (burdzy@math.washington.edu ) in advance so that adequate facilities may be arranged for.

The talks will take place in MEB 238 and MEB 242 (Mechanical Engineering Building). See the map of north-central campus. The Mechanical Engineering Building is marked as MEB in red.

More campus maps are available at the UW Web site.

Parking on UW campus is free on Saturdays after 12:00 (noon). More information is available at a parking Web site provided by UW.

### Tentative schedule

• 12:00 Informal lunch in the Mathematics Department Lounge (Padelford Hall)
• 1:00 Martin T. Barlow, University of British Columbia
• Random walks on supercritical percolation clusters
Consider bond percolation on ${\bf Z}^d$. It is well known that for $p>p_c$ there exists (except for a set of $\omega$ with probability zero) a unique infinite cluster $C(\omega)$, which has positive density. I will discuss the behaviour of simple random walk on $C(\omega)$. The problem divides into two parts. The first is to use fairly well known properties of supercritical percolation to obtain volume growth and Poincare inequalities for most' balls in $C(\omega)$. The second is to apply heat kernel' methods, which have mainly been developed for very regular graphs, to this situation, where there are small local irregularities.
• 2:00 Scott Sheffield, Theory Group, Microsoft Research
• Crystal facets and the amoeba
Why do crystals have facets? Why do the facets always rational slopes? What causes particular facets in an equilibrium crystal (e.g., certain surfactant and condensed helium crystals) to disappear and reappear when parameters (e.g., temperature) are changed? In the real world, these questions are difficult to answer quantitatively. But for a certain class of "random surface" models (which arise as "height functions" of random perfect matchings of a weighted, bipartite doubly periodic graph G, embedded in the plane), we can precisely describe the facets that arise in the "thermodynamic limit" using an algebraic geometric construction called the "amoeba." We show that the slopes of these facets always lie in the dual of the lattice of translation symmetries of G; depending on the edge weights, some, none, or all of these possible facets may be present. An ergodic Gibbs measure mu on the space of perfect matchings of G is said to be a rough phase if, when two matchings chosen independently from mu are superimposed, there are almost surely infinitely many cycles that surround the origin. If the origin is almost surely contained in only finitely many cycles, then mu is smooth. We will see that crystal facets correspond to smooth phases. The criteria for the existence of smooth phases (and hence crystal facets) are surprisingly simple and can be stated in terms of the perfect matchings of a single period of G. One consequence of our analysis is that sampling a perfect matching from any rough phase is equivalent to constructing a spanning tree of a so-called "T-graph" using Wilson's algorithm and a zero-drift loop-erased random walk. The results are joint work with Andrei Okounkov and Richard Kenyon.
• 3:00 Coffee break
• 3:30 Hao Wang, University of Oregon
• A class of interacting superprocesses
This talk will present some of my recent research progress and joint work with D. Dawson and Z. Li on a class of interacting measure-valued diffusion processes which include Super-Brownian motion as a special case. The talk is intended to cover a class of interacting branching particle systems with location dependent branching, the state classification of the limiting superprocesses, singular interacting branching particle systems, coalescence property, degenerate stochastic partial differential equation for a purely-atomic measure valued process and the strong uniqueness of the solution of this degenerated SPDE.
• 5:30 No host dinner at Aqua Verde Paddle Club restaurant (1303 N.E. Boat St., Seattle, WA 98105). See the map.