The Lace Expansion and its Applications

Abstract

ii The Lace Expansion and its ApplicationsThe Lace Expansion and its Applications iii Preface Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. Although these models have been studied by mathematicians for about half a century, exciting new developments continue to occur and the subject is flourishing. Much progress has been made, but it remains a major challenge for mathematical physics and probability theory to obtain a complete and mathematically rigorous understanding of the scaling theory of these models at criticality.These lecture notes concern the lace expansion, which is a powerful tool for the analysis of the critical scaling of several models above their upper critical dimensions, namely:• the self-avoiding walk on Z d for d > 4, • lattice trees and lattice animals onResults include proofs of existence of critical exponents, with mean-field values, and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion. There are two distinct goals for these notes. The first goal is to provide a written accompaniment to my lectures at the Saint-Flour summer school in 2004, and at the Pacific Institute for the Mathematical Sciences -University of British Columbia summer school in 2005. The notes contain an introduction to the lace expansion and several of its applications, with sufficient background and depth to prepare a newcomer to do research using the lace expansion. Basic graduate level probability theory will be used, but no previous knowledge of the lace expansion or super-Brownian motion is assumed. The second goal is to provide a survey of the field, so that an interested reader can follow up by consulting the original literature. In pursuit of the second goal, these notes include more material than can be covered during a summer school course.Following a brief initial section concerning random walk, the notes can be divided into four parts, whose contents are summarized as follows.Part I, which concerns the self-avoiding walk, consists of Sections 2-6. A complete and self-contained proof is given of the convergence of the lace expansion for the nearest-neighbour model in dimensions d 4, and for the spread-out model of self-avoiding walks which take steps of length at most L, with L 1, in dimensions d > 4. The convergence proof presented here seems simpler than all previous lace expansion convergence proofs. As a consequence of convergence, it is shown that the critical exponent γ for the generating function of the number of n-step self-avoiding walks exists and is equal to 1. A survey is then given of the many extensions of this result that have been obtained using the lace expansion. Part II, which concerns lattice trees and lattice animals, consists of Sections 7-8. It is shown how a minor modification of the expansion for the self-avoiding walk can be applied to give expansions for lattice trees and lattice animals, and an indicatio...

Current Trends in Random Walks on Random Lattices

“…Besides, there is a variety of excellent books and survey articles on random walks or directly related to random walks by Bingham [95,96], Bladt and Nielsen [97], Blanchard and Volchenkov [21], Brémaud [22], Fayolle, Iasnogorodski, and Malyshev [15], Foss, Korshunov, and Zachary [98], Fujie and Zhang [23], Gut [99], Hildebrand [16], Iksanov [100], Lawler [101], Redner [79], Shi [25], Slade [102], Takács [2], Telcs [103], and Wijesundera, Halgamuge, and Nanayakkara [104].…”

Borel-type bounds for the self-avoiding walk connective constant

“…The series expansion for μ was placed on a much firmer footing by Hara and Slade using the lace expansion [7]. The lace expansion is a powerful technique for exploring the properties of the self-avoiding walk in dimensions d > 4; we refer the reader to [8] for a recent introduction. Hara and Slade showed that the connective constant μ has an asymptotic expansion in integer powers of 1/(2d) to all orders, with all the coefficients taking integer values.…”