Two-band random matrices

TL;DR

This paper investigates spectral correlations in large random matrices with a spectral gap, revealing universal local correlations unaffected by the gap and global correlations that depend on matrix size and spectrum bounds.

Contribution

It demonstrates that eigenvalue gaps do not alter local spectral correlations but significantly influence global correlations, introducing a new universal law for bounded spectra.

Findings

Local correlations follow universal sine and multicritical laws

Global correlations reflect the presence of a spectral gap

Universal density-density correlator conjectured for chaotic systems with gaps

Abstract

Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of spectra are directly reconstructed from the recurrence equation for orthogonal polynomials associated with a given random matrix ensemble. It is established that an eigenvalue gap does not affect the local eigenvalue correlations which follow the universal sine and the universal multicritical laws in the bulk and soft-edge scaling limits, respectively. By contrast, global smoothed eigenvalue correlations do reflect the presence of a gap, and are shown to satisfy a new universal law exhibiting a sharp dependence on the odd/even dimension of random matrices whose spectra are bounded. In the case of unbounded spectrum, the corresponding universal `density-density' correlator is conjectured to be generic for chaotic systems with a forbidden gap and broken time reversal symmetry.