On bulk singularities in the random normal matrix model

TL;DR

This paper investigates eigenvalue distributions near bulk singularities in the random normal matrix model, revealing how local properties are governed by a dominant Taylor expansion and special functions like Mittag-Leffler.

Contribution

It extends the rescaled Ward identities method to analyze eigenvalue behavior at bulk singularities, identifying the key Taylor expansion and special functions involved.

Findings

Eigenvalue distribution near bulk singularities is determined by a dominant Taylor expansion.

A special entire function, such as Mittag-Leffler, describes the local eigenvalue distribution.

The method provides a detailed microscopic description of eigenvalues at singular points.

Abstract

We extend the method of rescaled Ward identities of Ameur-Kang-Makarov to study the distribution of eigenvalues close to a bulk singularity, i.e. a point in the interior of the droplet where the density of the classical equilibrium measure vanishes. We prove results to the effect that a certain "dominant part" of the Taylor expansion determines the microscopic properties near a bulk singularity. A description of the distribution is given in terms of a special entire function, which depends on the nature of the singularity (a Mittag-Leffler function in the case of a rotationally symmetric singularity).