Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices H_N,N={pipe}Λ{pipe}W with elements H_jk,αβ where j,k ∈ Λ =[1,m]^d ∩ ℤ^d (they parameterize the lattice sites) and α β = 1,...., W(they parameterize the orbitals on each site). The entries H_{jk,α β} are random Gaussian variables with mean zero such that 〈 H_j1k1,α1β1,H_j2k2,α2β2〉 =δ_j1k2δ_j2k1δ_α1β2δ_β1α2J_j1k1 where J=1/W+α Δ /W, α < 1/4d. This matrices are the special case of Wegner's W-orbital models. Assuming that the number of sites {pipe}Λ{pipe} is finite, we prove universality of the local eigenvalue statistics of H_N for the energies {pipe}λ₀{pipe}< √2.