# Probability

## Phase Transitions for Interacting Diffusions

## Algebraic Z^d-actions

This is a written account of five Pacific Institute for the Mathematical Sciences Distinguished Chair Lectures given at the Mathematics Department, University of Victoria, BC, in November 2002. The lectures were devoted to the ergodic theory of --actions, i.e. of several commuting automorphisms of a probability space. After some introductory remarks on more general -actions the lectures focused on ‘algebraic’ -actions, their sometimes surprising properties, and their deep connections with algebra and arithmetic. Special emphasis was given to some of the very recent developments in this area, such as higher order mixing behaviour and rigidity phenomena.

## Entropy and Orbit Equivalence

## Self-Interacting Walk and Functional Integration

These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expansion, supersymmetry and the Renormalization Group. These lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a -dimensional simple cubic lattice as the number of steps grows. When the end-to-end distance has been conjectured to grow as Const. where is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the conjecture in the simplified setting known as the ``Hierarchial Lattice.''