Universality of the local regime for the block band matrices with a finite number of blocks

Abstract

We consider the block band matrices, i.e. the Hermitian matrices $H_N$, $N=|\Lambda|W$ with elements $H_{jk,\alpha\beta}$, where $j,k \in\Lambda=[1,m]^d\cap \mathbb{Z}^d$ (they parameterize the lattice sites) and $\alpha, \beta= 1,\ldots, W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha\beta}$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha_1\beta_1}H_{j_2k_2,\alpha_2\beta_2}\rangle=\delta_{j_1k_2}\delta_{j_2k_1} \delta_{\alpha_1\beta_2}\delta_{\beta_1\alpha_2} J_{j_1k_1},$ where $J=1/W+\alpha\Delta/W$, $\alpha\le 1/4d$. This matrices are the special case of Wegner's $W$-orbital models. Assuming that the number of sites $|\Lambda|$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda_0|< \sqrt{2}$.