# Probability and Statistical Mechanics

### Overview

Much of the original motivation for th study of spatially interactive stochastic systems came from stochastic models in statistical physics. An intensive area of recent research centres around the idea that complex local dynamics can lead to a small number of well-understood continuum models upon space-time rescaling. When the underlying system is at or near criticality the limit invariably seems to be closely related to super-Brownian motion. The list of such results obtained in recent years is remarkable and includes Fisher-Wright and Fleming-Viot Models in population genetics (Dawson Donnelly, Etheridge, Kurtz, March and Perkins), interacting particle systems including contact process and voter models (Bramson, Cox, Durrett, Le Gall, Perkins, and Sakai), lattice trees and animals above the critical dimension of 8 (Derbez and Slade), and percolation and oriented percolation at criticality above the critical dimensions of 6 and 4, respectively (Hara, van der Hofstad and Slade).

Other local interactions arising in models for competing species, predator-prey systems or symbiotic branching lead to more complex stochastic models which behave locally like superprocesses but with branching, migration and drift coefficients which depend on the current state of the system. Two challenging and related topics are therefore:

The development of a general theory of interactive superprocesses and in particular methods to characterize these processes and study their properties.

The use of such models in mathematical ecology and evolution.

The rescaling results of Slade and his co-authors have created some strong common interests between the statistical physics and spatial stochastic process communities. The scaling limits of low dimensional statistical physics, however, are not super-Brownian motion. It is a defining goal of statistical mechanics to identify them and to calculate their properties. At present there is excellent progress in two dimensions where the stochastic Loewner processes provide natural candidates for scaling limits (ongoing work of Lawler, Schramm and Werner). Another promising program is based on the renormalization group. The self-avoiding walk in 4 (and 4 - e ) dimensions can in principle be analyzed by these methods (on going work of Brydges, Imbrie and others).

A period of concentration in Probability and Statistical Mechanics at PIMS will start from April 2004 - August 2006.

There will be a number of short term and long term visitors, and several conferences (see `Scientific Activities '). Each summer there will be a summer school, lasting about 5 weeks, with two advanced courses on special topics in probability theory. Graduate students from Western Canadian Universities may attend these courses under the Western Dean's Agreement.

Administrative enquirers should be addressed to pims@pims.math.ca.

Scientific inquiries should be addressed to one of the coordinating committee at UBC:

Martin Barlow, David Brydges, Alexander Holroyd, Vlada Limic, Ed Perkins, Gordon Slade.

Participating departments: UBC, U Washington, Microsoft Research, U Alberta, U Saskatchewan, U Regina.

### PIMS Distinguished Chairs

- Prof. Frank Den Hollander (Leiden University and EURANDOM, The Netherlands) will be visiting UBC January 1-August 15, 2006.

The CRG had two Distinguished Chairs in 2004.

- Prof. Richard Bass (U. Connecticut) spent an academic year at UBC, arriving in Aug. 04 and leaving the following summer.
- Prof. Yaozhong Hu (University of Kansas) spent an academic year at U. of Alberta from August 1, 2004 to August 31, 2005.

### Faculty

CRG Leaders:

- David Brydges (UBC)
- Chris Burdzy (U. Washington)
- Ed Perkins (UBC)
- Byron Schmuland (U. Alberta)

UBC:

- David Brydges
- Joel Feldman
- Alexander Holroyd
- Vlada Limic
- Gordon Slade
- Martin Barlow
- Ed Perkins
- John Walsh.

U. Alberta:

U. Washington:

Microsoft Research:

- Jennifer Chayes
- Christian Borgs
- Oded Schramm
- David Wilson.

U. Regina:

University of Saskatchewan:

Other institutions: Remco van der Hofstad (Eindhoven), Don Dawson (McGill)

**PIMS Postdoctoral Fellows**

- Dr. Omer Angel, PIMS PDF at UBC, 2004-06
- Dr. Codina Cotar, PDF at UBC
- Dr. Alexander Roitershtein, PDF at UBC

**Visitors**

2006

- Jan. 2-8, 2006: Ted Cox (Syracuse University).
- Jan. 1-Aug. 18, 2006: Frank den Hollander (Leiden University and EURANDOM).
- Feb. 28-Mar. 9, 2006: Jean-Dominique Deuschel (Technische UniversitÃ¤t Berlin).
- May 21-Jun. 18, 2006: Akira Sakai (EURANDOM).
- Jun. 26-Sep. 9, 2006: Malek Abdeselam (Universite Paris 13).

2005:

- Jan. 4-Feb. 4, 2005: Nicolai Krylov (U. Minnesota) will visit UBC.
- Feb. 13-Mar. 6 2005: Alain Sznitman (ETH) will visit UBC.
- March, 2005: Bruno Remillard (HEC Montreal) will visit U. Alberta.
- June 3-July 9, 2005: Lorenzo Zambotti, (Pisa) will visit UBC.
- June and July, 2005: Dan Romik (Weizmann, Institute) will visit UBC.
- June 6 - July 1, 2005: Yuval Peres (UC Berkeley) will visit UBC.
- Two months summer 2005: Takashi Kumagai (Kyoto University) will visit UBC.
- Summer 2005: Leonid Mytnik, (Technion) will visit UBC.
- Summer 2005: Carl Mueller (U. of Rochester); Steve Evans, (Berkeley U.) will visit UBC.

2004:

- May 1 - June 8, 2004: Doug Blount (Arizona State University) will visit U. Alberta.
- May 19 - 31, 2004: Takashi Kumagai (Nagoya University) will visit UBC.
- May 10-23, 2004: Don Dawson (Carleton U.) will visit UBC.
- May 15-23, 2004: Yves Le Jan (U. Paris, Orsay) will visit UBC.
- May 15-June 15, 2004: Pierre Tarres (U. Toulouse) will visit UBC.
- May 17-23, 2004: Robert Dalang (U. Lausanne) will visit UBC.
- May 19 - June 26, 2004: Greg Lawler (Cornell U.) will visit UBC.
- June and Aug 15 - Sept 15: Erwin Bolthausen (U. Zurich) will visit UBC.
- July 2004: John Imbrie, (U. Virginia), Jon Dimock, (SUNY Buffalo), Pronob Mitter, (U. Montpellier) will visit UBC
- Aug. 2004 - Aug. 2005: Richard Bass (University of Connecticutt) will visit UBC.
- August 23 - September 5, 2004: Remco van der Hofstad (Technical University Eindhoven) will visit UBC.
- Sept. 2004: Ted Cox (Syracuse U.) will visit UBC.
- Sept. 13-21, 2004: Andreas Greven (U. Erlangen) will visit UBC.
- Oct. 11-14, 2004: Tony Guttmann (University of Melbourne) will visit UBC.
- Fall Term, 2004: Zhenqing Chen (U. of Washington) will visit UBC.
- Nov. 2004: Takashi Kumagai (Kyoto University) will visit UBC.