The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case

Abstract

We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the two-point correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii [4]; it describes the correlations of the eigenvalue density in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner [33].

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Universality of the local regime for the block band matrices with a finite number of blocks

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