Spectral properties of the discrete random displacement model (Q2275588)

The authors study a discrete version of the random displacement model, which is a random Schrödinger operator on \(l^2(\mathbb Z^d)\). Precisely, the operator consists of the discrete (Neumann) Laplacian and a random potential. The random potential is constructed by dividing \(\mathbb Z^d\) into boxes of size \(M=M_1\times \dots \times M_d\). Then, for a function \(q:B\subset \mathbb Z^d\to\mathbb R\) whose support is in a box \(B\) of size smaller than \(M\), the potential in each box is given by a random translate of \(q\) (so that the support of the resulting potential remains in the box). The translates are i.i.d. for each box. If \(k\) indexes the boxes of size \(M\) and \(\omega=(\omega_k)\) are the i.i.d. translations, the random operator can be written as \[ h_{\omega}= -\Delta_{\text{discrete}}+\sum_{k\in\mathbb Z^d}q(\cdot +kM+\omega_k). \] Under mild assumptions on the potential function \(q\), the authors find a configuration of translates \(\omega^*\) that is spectrally minimizing/maximizing in the sense that \(\min \sigma(h_{\omega^*})=\inf_{\omega} (\min \sigma(h_\omega))\) (and analogously for the maximum). It follows from standard ergodic arguments that the spectrum \(\sigma(h_\omega)\) is the same closed set, say \(\Sigma\), for almost all \(\omega\) under the probability measure of the translates. The authors characterize \(\inf \Sigma\) and \(\sup \Sigma\) in terms of the lower and upper edge of \(\sigma(h_{\omega^*})\). Finally, they study the case \(d=1\), with \(M=2\), where the potential reduces to single-site potentials whose location is chosen randomly between the two possible sites in each box (Bernoulli random displacements). For this example, they are able to obtain asymptotics for the integrated density of states near the spectral edges, obtaining a singularity of the type \(1/\log^2\).