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.

Readership

Graduate students and research mathematicians interested in random processes and their applications in theoretical physics.