Sharp phase boundaries for a lattice flux line model
type: Item
instance of: scholarly article
title: Sharp phase boundaries for a lattice flux line model
zbMATH Open document ID: 1010.82022
DOI: 10.1023/A:1018611627599
published in: Journal of Statistical Physics
publication date: 2000-08-09T00:00:00Z
review text: The authors analyse the phase diagram of a model of mutually avoiding and self-avoiding random walks on rectangles, in the limit of the rectangles increasing to \(\mathbb Z^d\). Special cases arise in \(d=2\) in connection with dimer models and other exactly solvable models, in \(d=3\) in connection with flux lines in superconductors and directed polymers. The model is as follows. A nearest neighbor self-avoiding walk \(\omega\) in \(\mathbb Z^d\) receives the weight \[ z^\omega=\prod_{i=1}^d z_{i+}^{N_{i+}(\omega)}z_{i-}^{N_{i-}(\omega)}, \] where \(z=(z_{1+},z_{1-},\dots,z_{d+}, z_{d-})\) is a vector of \(2d\) nonnegative weights, and \(N_{i\pm}(\omega)\) denotes the number of positive resp.\ negative steps in the \(i\)-direction, respectively. The main attention will be paid to the case where the weights represent flux lines in a magnetic field, i.e. \(z_{i\pm}=e^{-\beta(\varepsilon\mp h_i)}\). The grand canonical partition function in a rectangle \(\Lambda\subset\mathbb Z^d\) is defined as \(Z(z,\Lambda)=\sum_{\Omega}\prod_{\omega\in\Omega} z^\omega\), where the sum is over all sets \(\Omega\) of (an arbitrary number of) self-avoiding, mutually avoiding nearest-neighbor walks, starting and ending at the boundary of \(\Lambda\). The \(k\)-point correlation function at \(k\) points \(x_1,\dots,x_k\in\mathbb Z^d\), \(S_\Lambda(x_1,\dots,x_k)\), is defined as \(1/Z(z,\Lambda)\) times the restriction of the above sum to those configurations in which each \( x_i\) is hit by at least one of the walks. The connectivity of \(x_1,\dots,x_k\), \(\tau_\Lambda (x_1,\dots,x_k)\), is the same with the restriction to those configurations in which at least one of the walks hits all the points \(x_1,\dots,x_k\). Define the \textit{free energy} \(f(z)= \lim_{\Lambda\uparrow\mathbb Z^d}|\Lambda|^{-1}\log Z(z,\Lambda)\), and define the infinite volume limits \(S\) of \(S_\Lambda\) and \(\tau\) of \(\tau_\Lambda\). The main results are as follows. In the case of flux lines in a magne…
reviewed by: W. König
Mathematics Subject Classification ID: 82B41
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 82D55
zbMATH DE Number: 1486436
zbMATH Keywords: self-avoiding random walk
zbMATH Keywords: phase boundary
author: Christian Borgs
author: Jennifer T. Chayes
author: Christopher King
author: Neal Madras
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1486436