Self-avoiding walk in 5 or more dimensions
type: Item
instance of: scholarly article
title: Self-avoiding walk in 5 or more dimensions
zbMATH Open document ID: 0575.60099
DOI: 10.1007/BF01206182
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: We consider a random walk on a simple cubic lattice of five or more dimensions. The probability for an N step walk, w is proportional to \(\exp (-\beta \sum n^ 2_ x(w))\), where the sum is over all sites x in the lattice, \(n_ x(w)\) is the number of times w visits x and \(\beta\geq 0\) is a parameter. This is the Domb-Joyce model. We prove for small \(\beta\) results equivalent to a central limit theorem for the endpoint of the walk. In particular for \(\beta >0\) sufficiently small E(exp(ikw(st)/\(\sqrt{t}))\to \exp (-const.sk^ 2)\) as \(t\to \infty\) uniformly on compacts in k and s, where w(st) is the endpoint of a walk with (integer part of) st steps. The method of proof combines some ideas from expansions in statistical mechanics with an induction on time-scales reminiscent of the renormalization group philosophy.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3919520
zbMATH Keywords: high temperature expansion
zbMATH Keywords: cubic lattice
zbMATH Keywords: Domb-Joyce model
zbMATH Keywords: central limit theorem
zbMATH Keywords: statistical mechanics
zbMATH Keywords: renormalization group
author: David Brydges
author: Thomas Spencer
MaRDI profile type: Publication
cites work: Geometric analysis of \(\phi^ 4\) fields and Ising models. I, II
cites work: A self-avoiding random walk
full work available at URL: https://doi.org/10.1007/bf01206182
OpenAlex ID: W2082111184
description: scientific article; zbMATH DE number 3919520