Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices
The paper considers the integrated density of states for stochastic Jacobi matrices \(h_{\omega}\). Define the operator \(h_ 0\) on \(\ell^ 2(Z^{\nu})\) by \((h_ 0u)(n)=\sum_{| \delta | =1}(u(n+\delta)-u(n))\) with \(| \delta | =\sum^{\nu}_{i=1}| \delta_ i|\). Suppose \(\{v_{\omega}(n)\}\) is a stationary ergodic random field and set \(h_{\omega}=h_ 0+v_{\omega}\). Let \(\rho_{\omega}(A)\) be the spectral resolution of \(h_{\omega}\) associated to the Borel set \(A\subset {\mathbb{R}}\) and denote by \(\rho_{\omega}(A;i,j)\) its matrix elements. The integrated density of states k(E) can be defined by \(k(\lambda)=E(\rho_{\omega}((-\infty,\lambda),0,0))\) where E denotes the expectation with respect to \(\{v_{\omega}\}\). It is proved that the function k is continuous. Moreover, it is log Hölder continuous in the sense that \(| k(\lambda)-k(\lambda ')| \leq C_ R[\log(| \lambda -\lambda '|^{-1}]^{-1}\) for \(| \lambda |<R\) and \(| \lambda -\lambda '|<\frac{1}{2}.\) The major tool for the proof is a ''Thouless-formula for a strip'' which connects the density of states of the \(h_{\omega}\) restricted to a strip (infinite in one direction) with the Lyapunov index of an associated system of differential equations. Meanwhile \textit{F. Delyon} and \textit{B. Souillard} [Remark on the continuity of the density of states of ergodic finite difference operators. ibid. 94, 289-291 (1984)] found a simpler proof for the mere continuity of k.
scientific article; zbMATH DE number 3844760
scientific article