This book presents the contents of five courses delivered by five different lecturers on various aspects of statistical mechanics at the joint Institute for Advanced Study – Park City Mathematics Institute summer school in 2007. Each course consisted of about six hours worth of lectures and was aimed at graduate students with strong backgrounds in analysis and probability. The goal was to describe recent developments and concepts in statistical mechanics. Although statistical mechanics originated with, and remains strongly motivated by, questions in physics, these lectures have a clearly mathematical orientation. No background in physics is assumed. However, for at least some of the material, knowledge of martingales, stochastic differential equations and conformal mappings is expected.
A basic theme of the lectures was the “scaling limits” of two-dimensional statistical mechanical models on increasingly fine lattices approaching the continuum limit of R2. The simplest example of this is Brownian motion in the plane as the scaling limit of random walk on a two-dimensional grid. The continuum models are expected to be universal, in some sense, because they should arise as scaling limits of many different models on two-dimensional lattices. The scaling limit also has additional symmetries — Brownian motion, for example, is rotationally invariant. Another significant element in statistical mechanics is the existence of a parameter — temperature, for example — that must be tuned to a critical value for the scaling limit to have these symmetries. The “critical temperature” usually marks a transition from order to disorder.
Some statistical mechanical systems (such as the Ising model and percolation models) are, at least in a broad sense, conformally invariant as well as scale and rotation invariant. Because the group of conformal transformations is so large, it offers a useful tool for developing new insights into two-dimensional critical phenomena. When, in 1999, Oded Schramm offered an elegant description of two-dimensional conformally invariant statistical mechanical systems in terms of what is now called Schramm-Loewner Evolution, it inspired a new burst of activity in the field.
This book is clearly intended for specialists in the field or those intending to become specialists. The most accessible section of the book may be the last set of lectures in the book, by Wendelin Werner, partly because he focuses more narrowly on two-dimensional critical percolation. His lectures are also more self-contained. Alice Guionnet contributes a set of lectures that demonstrate how classical ideas and tools from statistical mechanics can be used to study random matrices. The lectures by David Brydges present the renormalization group, a critical tool of modern physics. Although the basic idea of renormalization is not difficult, its application to statistical mechanics is complex and challenging. Greg Lawler devotes his lectures to the development of Schramm-Loewner Evolution equations, beginning with background on classical discrete models of statistical mechanics such as self-avoiding walk. Richard Kenyon describes planar dimer models in his lectures. This subject, developed in terms of graph theory, is based on a natural generalization of random walk on the integers. Its primary goal is to provide a statistical mechanical model of random two-dimensional interfaces in R3.
The writing throughout this book is consistently good, but there are few concessions to any reader new to the field.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.