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arXiv · 1309.2120

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

Tatyana Shcherbina
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Abstract

We consider the block band matrices, i.e. the Hermitian matrices $H_N$, $N=|Λ|W$ with elements $H_{jk,αβ}$, where $j,k \inΛ=[1,m]^d\cap \mathbb{Z}^d$ (they parameterize the lattice sites) and $α, β= 1,\ldots, W$ (they parameterize the orbitals on each site). The entries $H_{jk,αβ}$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,α_1β_1}H_{j_2k_2,α_2β_2}\rangle=δ_{j_1k_2}δ_{j_2k_1} δ_{α_1β_2}δ_{β_1α_2} J_{j_1k_1},$ where $J=1/W+αΔ/W$, $α\le 1/4d$. 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 $|λ_0|< \sqrt{2}$.

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BibTeXRIS

Tatyana Shcherbina. 2014-03-31. Universality of the local regime for the block band matrices with a finite number of blocks. https://doi.org/10.1007/s10955-014-0964-4

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