Large Representation Recurrences in Large N Random Unitary Matrix Models

TL;DR

This paper investigates the behavior of characters in large N random unitary matrices for large symmetric tensor representations, revealing eigenvalue localization, limitations of semiclassical methods, and approximate periodicity in expectation values.

Contribution

It demonstrates that semiclassical techniques can be adapted for large representations and uncovers a novel periodicity in character expectation values at large N.

Findings

Eigenvalues localize on an extremum of the effective potential.

Standard semiclassical methods are limited for large k/N.

Expectation values exhibit approximate periodicity as a function of k/N.

Abstract

In a random unitary matrix model at large N, we study the properties of the expectation value of the character of the unitary matrix in the rank k symmetric tensor representation. We address the problem of whether the standard semiclassical technique for solving the model in the large N limit can be applied when the representation is very large, with k of order N. We find that the eigenvalues do indeed localize on an extremum of the effective potential; however, for finite but sufficiently large k/N, it is not possible to replace the discrete eigenvalue density with a continuous one. Nonetheless, the expectation value of the character has a well-defined large N limit, and when the discreteness of the eigenvalues is properly accounted for, it shows an intriguing approximate periodicity as a function of k/N.