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The Lace Expansion and its Applications: Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004 / Edition 1

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Overview

The lace expansion is a powerful and flexible method for understanding the critical scaling of several models of interest in probability, statistical mechanics, and combinatorics, above their upper critical dimensions. These models include the self-avoiding walk, lattice trees and lattice animals, percolation, oriented percolation, and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models.

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Editorial Reviews

From the Publisher
From the reviews:

"The book consists mainly of materials for an advanced master course. … The book focusses on the modeling of mechanics, where the concept of a model shall be understood in its broadest sense, namely as a mathematical structure which describes mechanical phenomena. … A list of notations and an index are helpful to the reader. The little book with about 200 pages tackles many theories and concepts, without going much into detail." (Albrecht Bertram, Zentralblatt MATH, Vol. 1115 (17), 2007)

"This work is based on the author’s lectures on the lace expansion and its applications … . It contains the recent developments in the analysis of the lace expansion and the scaling limits of the critical objects discovered since the publication … . It also contains further applications of the lace expansion to other models and an extensive list of references. … The subject is still growing, and studying the lecture notes by Slade is a good starting point to learn the subject." (Akira Sakai, Mathematical Reviews, Issue 2007 m)

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Product Details

Meet the Author

Gordon Slade is Professor at the University of British Columbia since 1999. Before he was Lecturer at the University of Virginia from 1985 to 1986 and Professor at the McMaster University from 1986 to 1999. The Author has been awarded the UBC Killam Research Prize (Senior Science Category) in 2004 and the Prix de l'Institut Henri Poincaré—with Remco van der Hofstad—in2003. In 2003 he was Stieltjes Visiting Professor, in 1995 Coxeter-James Lecturer for the Canadian Mathematical Society. Since 2000 he is Fellow of the Royal Society of Canada.

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Table of Contents

Simple Random Walk.- The Self-Avoiding Walk.- The Lace Expansion for the Self-Avoiding Walk.- Diagrammatic Estimates for the Self-Avoiding Walk.- Convergence for the Self-Avoiding Walk.- Further Results for the Self-Avoiding Walk.- Lattice Trees.- The Lace Expansion for Lattice Trees.- Percolation.- The Expansion for Percolation.- Results for Percolation.- Oriented Percolation.- Expansions for Oriented Percolation.- The Contact Process.- Branching Random Walk.- Integrated Super-Brownian Excursion.- Super-Brownian Motion.

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