Journal of Statistical Physics

, Volume 155, Issue 3, pp 466–499 | Cite as

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

  • Tatyana Shcherbina


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 \(J=1/W+\alpha \Delta /W\), \(\alpha < 1/4d\). 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 \(|\lambda _0|< \sqrt{2}\).


Random matrices Wegner model Band matrices Universality 



Supported by NSF grant DMS 1128155. This research also was partially supported by RF Government grant 11.G34.31.0026 and by JSC “Gazprom Neft”.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Chebyshev LaboratorySt. Petersburg State UniversitySaint Petersburg Russia
  2. 2.Institute for Advanced StudyPrincetonUSA

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