Spectral form factor in a random matrix theory

1996/08/26 by E. Brézin, S. Hikami · 28 citations
Chemistry · Mathematics · Physics and Astronomy · #Molecular spectroscopy and chirality #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #cond-mat

paper · pdf · doi:10.1103/physreve.55.4067

36P, (+5 figures not included)

arxiv created 1996/08/26 · openalex publication_date 1997/04/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31

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(λ₁,λ₂) 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(λ₁,λ₂). 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.

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