This volume, with contributions by leading experts in the field, is a collection of lecture notes of the six minicourses given at the IAS/Park City Summer Mathematics Institute. It introduces advanced graduates and researchers in probability theory to several of the currently active research areas in the field. Each course is self-contained with references and contains basic materials and recent results. Topics include interacting particle systems, percolation theory, analysis on path and loop spaces, and mathematical finance.

The volume gives a balanced overview of the current status of probability theory. An extensive bibliography for further study and research is included. This unique collection presents several important areas of current research and a valuable survey reflecting the diversity of the field.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Readership

Advanced graduate students and researchers in probability theory; physicists and researchers in finance and economics.

Table of Contents

• Introduction

• Introduction
• The voter model
• Coalescing random walks
• Voter model with mutation
• The block construction
• Long range limits
• Rapid stirring limits
• Bibliography

• Preface
• The basics of percolation
• Rescaling and finite-size scaling in percolation
• Critical exponent inequalities
• Two fundamental questions
• Finite-size scaling and the incipient infinite cluster
• The BK(R) inequality
• The Potts model and the random cluster model
• Bibliography

• Introduction
• The simple exclusion model
• Proof of Theorem 1.4
• Local ergodicity
• Two-block estimate
• Relative entropy
• The Green-Kubo formula and asymmetric simple exclusion processes
• Some open problems
• Bibliography

• Introduction
• Gaussian measures on a Hilbert space
• Rolling on
• About \(\mathcal{W}_M\)
• A few facts, and something else
• Bibliography

• Introduction
• Euclidean Brownian motion
• Gradient operator
• Ornstein-Uhlenbeck operator
• Brownian motion on manifolds
• Gradient formulas
• Integration by parts
• Logarithmic Sobolev inequalities
• Bibliographical comments
• Bibliography

• Introduction
• An introduction to option pricing and the mathematical theory of risk
• Bibliography