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ICTP Lecture Notes Series, Volume 17

(ISBN 92-95003-25-X) - *March 2004*

## School and Conference on Probability Theory

Editor: *G.F. Lawler* (Cornell Univ., USA and Duke Univ., USA)

**Introduction**

This volume includes expanded lecture notes from the School and Conference in Probability Theory held at ICTP in May, 2001. Probability theory is a very large area, too large for a single school and conference. The organizers, G. Lawler, C. Newman, and S. Varadhan chose to focus on a number of active research areas that have their roots in statistical physics. The pervasive theme in these lectures is trying to find the large time or large space behavior of models defined on discrete lattices. Usually the definition of the model is relatively simple: either assigning a particular weight to each possible configuration (equilibrium statistical mechanics) or specifying the rules under which the system evolves (nonequilibrium statistical mechanics).

Interacting particle systems is the area of probability that studies the evolution of particles (either finite or infinite in number) under random motions. The evolution of particles depends on the positions of the other particles; often one assumes that it depends only on the particles that are close to the particular particle. Thomas Liggett's lectures give an introduction to this very large area. Claudio Landim's follows up by discussing hydrodynamic limits of particle systems. The goal of this area is to describe the long time, large system size dynamics in terms of partial differential equations.

The area of random media is concerned with the properties of materials or environments that are not homogeneous. Percolation theory studies one of the simplest stated models for impurities - taking a lattice and removing some of the vertices or bonds. Luiz Renato G. Fontes and Vladas Sidoravicius give a detailed introduction to this area. Random walk in random enviroment combines two sources of randomness - a particle performing stochastic motion in which the transition probabilities depend on position and have been chosen from some probability distribution. Alain-Sol Sznitman gives a survey of recent developments in this very di cult area.

It has been conjectured for a long time that in two dimensions many models in statistical physics have limits that are in some sense conformally invariant. There has been much activity recently proving such results. Richard Kenyon's lecture focuses on the dimer model where conformal invariance can be used at the discrete level. The continuum limit of many models is now understood with the aid of the stochastic Loewner evolution (SLE), developed by Oded Schramm. Gregory Lawler describes some of these recent results, including applications to percolation and path properties of planar Brownian motion.

*G.F. Lawler*

March, 2004

**Table of Contents** (PostScript and PDF sources available)

Most (La)TeX & PS sources of Volume 17: lns017.tar.gz (5732127 bytes)

(NB: This tar-compressed file contains LaTeX files, postscript (.ps) and/or encapsulated postscript (.eps) files of each figure plus the postscript and PDF versions of each lecture note)