A rigorous replica trick approach to Anderson localization in one dimension
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)].
scientific article; zbMATH DE number 3994716
scientific article