Mesoscopic fluctuations for the thinned Circular Unitary Ensemble

TL;DR

This paper investigates the mesoscopic fluctuation behavior of the thinned Circular Unitary Ensemble, revealing a phase transition from Gaussian to non-Gaussian fluctuations depending on scale and thinning parameters.

Contribution

It introduces a detailed analysis of the mesoscopic fluctuation regimes in the thinned CUE, identifying a universal transition phenomenon and deriving the limiting laws.

Findings

Two fluctuation regimes separated by a critical line

Gaussian CLT in one regime, non-Gaussian infinitely divisible laws on the critical line

Universality of the transition in related processes

Abstract

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.