Large $N$ Phase Transition In The Heat Kernel On The $U(N)$ Group

TL;DR

This paper studies a large N phase transition in the heat kernel on the U(N) group, revealing a relation between eigenvalue density and Young tableaux shape, and analyzing different coupling phases.

Contribution

It introduces a simple functional relation linking eigenvalue density to Young tableaux shape in the large N limit of the heat kernel on U(N).

Findings

Identifies the phase transition point in the heat kernel.

Derives a relation between eigenvalue density and boundary conditions.

Analyzes strong and weak coupling phases for specific boundary holonomies.

Abstract

The large N phase transition point is investigated in the heat kernel on the $U(N)$ group with respect to arbitrary boundary conditions. A simple functional relation is found relating the density of eigenvalues of the boundary field to the saddle point shape of the typical Young tableaux in the large $N$ limit of the character expansion of the heat kernel. Both strong coupling and weak coupling phases are investigated for some particular cases of the boundary holonomy.