The Wilson loop in the Gaussian Unitary Ensemble

TL;DR

This paper presents exact finite-N calculations of the Wilson loop expectation and spectral density in the Gaussian Unitary Ensemble using supersymmetric formalism and combinatorics, providing new proofs of known formulas.

Contribution

It introduces a novel exact finite-N computation of the Wilson loop and spectral density, and offers a new combinatorial proof of the Harer-Zagier series formula.

Findings

Exact finite-N expectation of the Wilson loop

Exact spectral density formula at finite N

New combinatorial proof of Harer-Zagier series

Abstract

Using the supersymmetric formalism we compute exactly at finite $N$ the expectation of the Wilson loop in the Gaussian Unitary Ensemble and derive an exact formula for the spectral density at finite $N$. We obtain the same result by a second method relying on enumerative combinatorics and show that it leads to a novel proof of the Harer-Zagier series formula.