Conformally invariant processes in the plane / Gregory F. Lawler

Auteur: Lawler, Gregory F. (1955-) - AuteurType de document: MonographieCollection: Mathematical surveys and monographs ; 114Langue: anglaisPays: Etats UnisÉditeur: Providence : American Mathematical Society, 2005Description: 1 vol. (XI-242 p.) : ill. ; 26 cm ISBN: 0821836773 ; rel. Résumé: 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. (source : AMS).Bibliographie: Bibliogr. p. 237-239. Sujets MSC: 30C35 Functions of a complex variable -- Geometric function theory -- General theory of conformal mappings
31A15 Potential theory -- Two-dimensional theory -- Potentials and capacity, harmonic measure, extremal length
60H30 Probability theory and stochastic processes -- Stochastic analysis -- Applications of stochastic analysis (to PDE, etc.)
60J65 Probability theory and stochastic processes -- Markov processes -- Brownian motion
81T40 Quantum theory -- Quantum field theory; related classical field theories -- Two-dimensional field theories, conformal field theories, etc
82B27 Statistical mechanics, structure of matter -- Equilibrium statistical mechanics -- Critical phenomena
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Bibliogr. p. 237-239

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. (source : AMS)

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