Conformally Invariant Processes in the Plane

Front Cover
American Mathematical Soc., 2008 - Mathematics - 242 pages
0 Reviews
Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by 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. The following topics are covered: 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 with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Some discrete processes
1
Chapter 1 Stochastic calculus
11
Chapter 2 Complex Brownian motion
43
Chapter 3 Conformal mappings
57
Chapter 4 Loewner differential equation
91
Chapter 5 Brownian measures on paths
119
Chapter 6 SchrammLoewner evolution
147
Chapter 7 More results about SLE
177
Chapter 9 Restriction measures
205
Appendix A Hausdorff dimension
217
Appendix B Hypergeometric functions
229
Appendix C Reflecting Brownian motion
233
Bibliography
237
Index
240
Index of symbols
242
Copyright

Chapter 8 Brownian intersection exponent
187

Other editions - View all

Common terms and phrases

Bibliographic information