Mesoscopic colonization of a spectral band

TL;DR

This paper studies the formation of new spectral bands in large unitary matrix models, focusing on a mesoscopic regime where the number of eigenvalues in the emerging band grows slowly, revealing a transitional behavior between microscopic and macroscopic scales.

Contribution

It introduces a mesoscopic scaling regime for spectral band formation in unitary matrix models and analyzes the delicate transition using Riemann-Hilbert steepest descent methods.

Findings

Describes the mesoscopic regime where eigenvalue population grows slowly

Identifies conditions for formation of multiple new spectral bands

Develops a scaling limit controlling the mesoscopic colony

Abstract

We consider the unitary matrix model in the limit where the size of the matrices become infinite and in the critical situation when a new spectral band is about to emerge. In previous works the number of expected eigenvalues in a neighborhood of the band was fixed and finite, a situation that was termed "birth of a cut" or "first colonization". We now consider the transitional regime where this microscopic population in the new band grows without bounds but at a slower rate than the size of the matrix. The local population in the new band organizes in a "mesoscopic" regime, in between the macroscopic behavior of the full system and the previously studied microscopic one. The mesoscopic colony may form a finite number of new bands, with a maximum number dictated by the degree of criticality of the original potential. We describe the delicate scaling limit that realizes/controls the mesoscopic colony. The method we use is the steepest descent analysis of the Riemann-Hilbert problem that is satisfied by the associated orthogonal polynomials.