Free energies and fluctuations for the Unitary Brownian Motion

TL;DR

This paper investigates the asymptotic behavior of traces of words in independent unitary Brownian motions, providing new analytic tools to study Wilson loops in the context of unitary Yang-Mills theory.

Contribution

It introduces a novel approach to analyze the Laplace transforms of traces in unitary Brownian motions and characterizes the limits via equations similar to Schwinger-Dyson's, named after Makeenko and Migdal.

Findings

Laplace transforms converge to an analytic function on a non-trivial disc.

Asymptotics of Wilson loops are studied under the unitary Yang-Mills measure.

Limiting objects are characterized by Makeenko-Migdal equations.

Abstract

We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary Yang-Mills measure on the plane. The limiting objects obtained are shown to be characterized by equations analog to Schwinger-Dyson's ones, named here after Makeenko and Migdal.