A rigorous replica trick approach to Anderson localization in one dimension
type: Item
instance of: scholarly article
title: A rigorous replica trick approach to Anderson localization in one dimension
zbMATH Open document ID: 0614.60098
DOI: 10.1007/BF01463399
author: Abel Klein
author: J. Fernando Perez
author: Fabio Martinelli
published in: Communications in Mathematical Physics
publication date: 1986-01-01T00:00:00Z
review text: Let \(H=-\Delta +V\) on \(\ell^ 2({\mathbb{Z}})\), where V(x), \(x\in {\mathbb{Z}}\), are i.i.d. r.v.'s, and let \(G_ L(x,y;E+i\eta)=<x| (H_ L- (E+i\eta))^{-1}| y>\), where \(H_ L\) denotes the operator H restricted to \(\{\)-L,\(-L+1,...,L\}\) with Dirichlet boundary conditions. We use a supersymmetric replica trick to prove that \[ {\mathbb{E}}(| G_ L(0,x;E+i\eta)|^ 2)\leq K\eta^{-2} \exp \{-m| \log \eta |^{-\sigma}| x| \} \] for some \(m>0\), \(\sigma >0\), \(K<\infty\), uniformly in L and E. This estimate, together with the usual necessary estimate on the density of states, implies zero conductivity and gives exponential localization by the method of \textit{J. Fröhlich}, \textit{F. Martinelli}, \textit{E. Scoppola}, and \textit{T. Spencer} [ibid. 101, 21-46 (1985; Zbl 0573.60096)].
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60J70
zbMATH DE Number: 3994716
zbMATH Keywords: replica trick
zbMATH Keywords: field theoretic methods
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: exponential localization
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01463399
OpenAlex ID: W2067586198
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description: scientific article; zbMATH DE number 3994716