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Conformally Invariant Processes in the Plane
Gregory F. Lawler, Cornell University, Ithaca, NY
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Mathematical Surveys and Monographs
2005; 242 pp; softcover
Volume: 114
ISBN-10: 0-8218-4624-8
ISBN-13: 978-0-8218-4624-7
List Price: US$62
Member Price: US$49.60
Order Code: SURV/114.S
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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.

Readership

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

Reviews

"This nice book celebrates the fruitful marriage of Brownian motion and complex analysis."

-- Zentralblatt MATH

"This book gives a nice and systematic introduction to the contiuous time conformally invariant processes in the plane, assuming only knowledge of first year graduate real analysis, complex analysis and probability theory ... This books is very well written, and can also be used as a graudate textbook for a topic course on SLE."

-- Mathematical Reviews

Table of Contents

  • Some discrete processes
  • Stochastic calculus
  • Complex Brownian motion
  • Conformal mappings
  • Loewner differential equation
  • Brownian measures on paths
  • Schramm-Loewner evolution
  • More results about SLE
  • Brownian intersection exponent
  • Restriction measures
  • Hausdorff dimension
  • Hypergeometric functions
  • Reflecting Brownian motion
  • Bibliography
  • Index
  • Index of symbols
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