Almost sure Weyl law for quantized tori

TL;DR

This paper proves that the eigenvalues of randomly perturbed Toeplitz quantizations on the torus follow a Weyl law with high probability, confirming a conjecture by Christiansen and Zworski.

Contribution

It establishes a probabilistic Weyl law for eigenvalues of quantized tori under small random perturbations, confirming a prior conjecture.

Findings

Eigenvalues follow a Weyl law with high probability

Confirms a conjecture by Christiansen and Zworski

Perturbed eigenvalues distribute according to the Weyl law

Abstract

We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean $0$, variance $1$ and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by T. Christiansen and M. Zworski.