Spectral form factor in a random matrix theory

Abstract

In the theory of disordered systems the spectral form factor S(τ ), the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for τ < τ c and constant for τ > τ c . Near zero and near τ c its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function ρ(λ 1 , λ 2 ) exhibits several well-known universal properties in the large N limit. Its Fourier transform is linear as a consequence of the short distance universality of ρ(λ 1 , λ 2 ). However the cross-over near zero and τ c requires to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these cross-over oscillatory properties. The method is also extended to the time-dependent case.

Large n Limit of Gaussian Random Matrices with External Source, Part I

“…The Gaussian ensemble, V (M) = 1 2 M 2 , has been solved in the papers of Pastur [24] and Brézin-Hikami [7]- [10], by using spectral methods and a contour integration formula for the determinantal kernel. In the present work we will develop a completely different approach to the solution of the Gaussian ensemble with external source.…”

Matrix regularizing effects of Gaussian perturbations

“…the monograph [25]). There are also several results which rely on the Harish-Chandra formulae [12,4,5], and thus apply to GUE but not GOE perturbation. For the case that A N are uniformly normbounded and the perturbation is GUE it is a by-product of the study of local eigenvalue statistics by T. Shcherbina [29,30] that the Pastur law also holds in total variation distance.…”

“…Let H ϕ be as above, then the eigenvalues of H ϕ interlace those of H, therefore N ( 5) where in the last inequality we have applied the estimate (5.1) to the matrix H − E. By the invariance of the underlying Gaussian ensemble (GOE or GUE), the (N − 1)-dimensional matrix…”

“…Using the symmetry which is built into the assumptions, it suffices to establish the bound for diagonal matrices Q = diag(E 1 , E 2 , · · · ). Our goal is to prove that 5) where the sums may be restricted to E j = 0 (and the probability average is over the independent standard Gaussian variables g j ).…”

“…These consist of Hermitian matrices of the form 5) where X is an N × N matrix with independent standard real Gaussian entries in case of GOE, or independent standard complex Gaussian entries in case of GUE, and the asterisk indicates Hermitian conjugation. In both cases the probability distribution of V is of density proportional to exp − βN 4 tr V 2 with respect to the Lebesgue measure on matrices of the corresponding symmetry: real symmetric (GOE, with β = 1) or complex Hermitian (GUE, with β = 2).…”