Conformally Invariant Processes in the Plane

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. Such a belief has allowed them to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. Topics include: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain whose input is a Brownian motion; application to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability.

Drawn from a graduate course at Cornell University, this textbook explores two-dimensional lattice models having continuum limits that are conformally invariant. Lawler introduces integration with respect to Brownian motion and continuous semimartingales, explains the Loewner differential equation, derives conformally invariant measures on paths from complex Brownian motion, and analyzes the Loewner differential equation driven by Brownian motion. Annotation ©2004 Book News, Inc., Portland, OR