Asymptotic expansion of Toeplitz determinants of an indicator function with discrete rotational symmetry and powers of random unitary matrices

TL;DR

This paper derives a comprehensive large N asymptotic expansion for the probability related to powers of random unitary matrices, utilizing topological recursion to include oscillatory non-perturbative terms, thus advancing understanding of Toeplitz determinants with symmetric symbols.

Contribution

It provides the first explicit full asymptotic expansion of a Toeplitz determinant with a symmetric indicator function symbol, including non-perturbative oscillations, solving a previously open conjecture.

Findings

Derived a full large N asymptotic expansion for the probability involving eigenvalues of matrix powers.

Included oscillating non-perturbative terms in the asymptotic expansion.

Applied topological recursion to compute the expansion explicitly for genus g>0.

Abstract

In this short article we propose a full large $N$ asymptotic expansion of the probability that the $m^{\text{th}}$ power of a random unitary matrix of size $N$ has all its eigenvalues in a given arc-interval centered in $1$ when $N$ is large. This corresponds to the asymptotic expansion of a Toeplitz determinant whose symbol is the indicator function of several intervals having a discrete rotational symmetry. This solves and improves a conjecture left opened by the author. It also provides a rare example of the explicit computation of a full asymptotic expansion of a genus $g>0$ classical spectral curve, including the oscillating non-perturbative terms, using the topological recursion.