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...
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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
Some discrete processes1Ch. 1Stochastic calculus11Ch. 2Complex Brownian motion43Ch. 3Conformal mappings57Ch. 4Loewner differential equation91Ch. 5Brownian measures on paths119Ch. 6Schrainin-Loewner evolution147Ch. 7More results about SLE177Ch. 8Brownian intersection exponent187Ch. 9Restriction measures205App. AHausdorff dimension217App. BHypergeometric functions229App. CReflecting Brownian motion233
