Conformally Invariant Processes in the Plane (Mathematical Surveys and Monographs) (Mathematical Surveys and Monographs)

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.

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.

Some Discrete Processes -- Ch. 1. Stochastic Calculus -- Ch. 2. Complex Brownian Motion -- Ch. 3. Conformal Mappings -- Ch. 4. Loewner Differential Equation -- Ch. 5. Brownian Measures On Paths -- Ch. 6. Schrainin-loewner Evolution -- Ch. 7. More Results About Sle -- Ch. 8. Brownian Intersection Exponent -- Ch. 9. Restriction Measures -- App. A. Hausdorff Dimension -- App. B. Hypergeometric Functions -- App. C. Reflecting Brownian Motion. Gregory F. Lawler. Includes Bibliographical References (p. 237-239) And Index. An introduction to the conformally invariant processes that appear as scaling limits. It covers topics including: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; and, the Schramm-Loewner evolution (SLE). Presents an introduction to the conformally invariant processes that appear as scaling limits. This book covers such topics as stochastic integration, and complex Brownian motion and measures derived from Brownian motion. It is suitable for those interested in random processes and their applications in theoretical physics.