Many mathematical models of statistical physics in two dimensions are either known or
conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions …

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In
spite of this simple definition, many of the most basic questions about this model are difficult …

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and
prove some rigorous theorems on the regularity properties and possible pathologies of the …

Simple random walks-or equivalently, sums of independent random vari ables-have long
been a standard topic of probability theory and mathemat ical physics. In the 1950s, non …

We make a high-precision Monte Carlo study of two-and three-dimensional self-avoiding
walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby …

We introduce a fast implementation of the pivot algorithm for self-avoiding walks, which we
use to obtain large samples of walks on the cubic lattice of up to 33× 10 6 steps …

This book focuses on percolation on high-dimensional lattices. We give a general
introduction to percolation, stating the main results and defining the central objects. We …

We prove that nearest-neighbor percolation in dimensions d≧11 displays mean-field
behavior by proving that the infrared bound holds, in turn implying the finiteness of the …

The self-avoiding walk, and lattice spin systems such as the φ 4 model, are models of
interest both in mathematics and in physics. Many of their important mathematical problems …

Polymer chains that interact with themselves and/or with their environment are fascinating
objects, displaying a range of interesting physical and chemical phenomena. The focus in …