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

Tatyana Shcherbina

doi:10.1007/s10955-014-0964-4

paper

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

arXiv:1309.2120 · doi:10.1007/s10955-014-0964-4

Abstract

We consider the block band matrices, i.e. the Hermitian matrices $H_{N}$ , $N = ∣Λ∣ W$ with elements $H_{j k, α β}$ , where $j, k \in Λ = [1, m]^{d} \cap Z^{d}$ (they parameterize the lattice sites) and $α, β = 1, \dots, W$ (they parameterize the orbitals on each site). The entries $H_{j k, α β}$ are random Gaussian variables with mean zero such that $⟨ H_{j_{1} k_{1}, α_{1} β_{1}} H_{j_{2} k_{2}, α_{2} β_{2}} ⟩ = δ_{j_{1} k_{2}} δ_{j_{2} k_{1}} δ_{α_{1} β_{2}} δ_{β_{1} α_{2}} J_{j_{1} k_{1}},$ where $J = 1/ W + α Δ/ W$ , $α \leq 1/4 d$ . This matrices are the special case of Wegner's $W$ -orbital models. Assuming that the number of sites $∣Λ∣$ is finite, we prove universality of the local eigenvalue statistics of $H_{N}$ for the energies $∣ λ_{0} ∣ < 2$ .

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Superbosonization of invariant random matrix ensembles

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

Abstract

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