Conformally Invariant Processes in the Plane
About this Title
Gregory F. Lawler, Cornell University, Ithaca, NY
Publication: Mathematical Surveys and Monographs
Publication Year 2005: Volume 114
ISBNs: 978-0-8218-4624-7 (print); 978-1-4704-1341-5 (online)
MathSciNet review: MR2129588
MSC: Primary 60-02; Secondary 30-02, 30C35, 31A15, 60J65, 81T40, 82B27
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.
Graduate students and research mathematicians interested in random processes and their applications in theoretical physics.
Table of Contents
- 1. Stochastic calculus
- 2. Complex Brownian motion
- 3. Conformal mappings
- 4. Loewner differential equation
- 5. Brownian measures on paths
- 6. Schramm-Loewner evolution
- 7. More results about $SLE$
- 8. Brownian intersection exponent
- 9. Restriction measures