# Splitting of a gap in the bulk of the spectrum of random matrices

## TL;DR

This paper analyzes the probability of two gaps in the eigenvalue spectrum of Gaussian Unitary Ensemble matrices, providing detailed asymptotics for the transition between one large gap and two gaps.

## Contribution

It offers explicit uniform asymptotics for gap probabilities and characterizes the transition between single and double gaps in the spectrum.

## Key findings

- Explicit asymptotics for the transition between one and two gaps
- Asymptotic terms for the probability of two gaps in the spectrum
- Analysis of Toeplitz determinants with symbols on two arcs

## Abstract

We consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle.