Probability and Statistical Mechanics

2004 2006

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(link is external)David Brydges(link is external)Alexander Holroyd(link is external)Vlada Limic(link is external)Ed Perkins(link is external)Gordon Slade(link is external).

Participating departments: UBC(link is external)U Washington(link is external)Microsoft Research(link is external)U Alberta(link is external)U Saskatchewan(link is external)U Regina(link is external).

PIMS Distinguished Chairs

 

The CRG had two Distinguished Chairs in 2004.

Faculty

CRG Leaders:

UBC:

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 (link is external)(Eindhoven), Don Dawson (link is external)(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

2005:

2004: