Universal distribution of random matrix eigenvalues near the "birth of a cut" transition

TL;DR

This paper investigates the universal behavior of eigenvalue distributions in random matrices at a unique transition point where a new eigenvalue support component emerges, characterized by non-analytical features and logarithmic behaviors.

Contribution

It introduces a new class of eigenvalue transition with non-rational singularities, expanding understanding beyond conformal minimal models.

Findings

Eigenvalue density exhibits logarithmic and non-analytical behaviors at the transition.

No critical exponent exists; the power of N varies in a sawtooth pattern.

The transition differs from previously studied rational singularity cases.

Abstract

We study the eigenvalue distribution of a random matrix, at a transition where a new connected component of the eigenvalue density support appears away from other connected components. Unlike previously studied critical points, which correspond to rational singularities \rho(x)\sim x^{p/q} classified by conformal minimal models and integrable hierarchies, this transition shows logarithmic and non-analytical behaviours. There is no critical exponent, instead, the power of N changes in a saw teeth behaviour.