On the Spectral Form Factor for Random Matrices

TL;DR

This paper rigorously proves the universality of the spectral form factor (SFF) for a broad class of random matrices, extending previous results and confirming predictions in physics through mathematical analysis and numerical validation.

Contribution

It extends the mathematical proof of SFF universality beyond Wigner matrices to include monoparametric ensembles and larger spectral scales using multi-resolvent local laws.

Findings

Proves SFF universality up to intermediate time scales for various random matrices.

Extends universality results to monoparametric ensembles with a single random parameter.

Numerical results match theoretical predictions across the slope-dip-ramp regime.

Abstract

In the physics literature the spectral form factor (SFF), the squared Fourier transform of the empirical eigenvalue density, is the most common tool to test universality for disordered quantum systems, yet previous mathematical results have been restricted only to two exactly solvable models [Forrester 2020]. We rigorously prove the physics prediction on SFF up to an intermediate time scale for a large class of random matrices using a robust method, the multi-resolvent local laws. Beyond Wigner matrices we also consider the monoparametric ensemble and prove that universality of SFF can already be triggered by a single random parameter, extending the recently proven Wigner-Dyson universality [Cipolloni, Erd\H{o}s, Schr\"oder 2021] to some larger spectral scales. Remarkably, extensive numerics indicates that our formulas correctly predict the SFF in the entire slope-dip-ramp regime, as customarily called in physics.