On The Douglas-Kazakov Phase Transition

TL;DR

This paper rigorously proves the Douglas-Kazakov phase transition in two-dimensional gauge theories, analyzing the asymptotic eigenvalue behavior of the unitary Brownian bridge as N approaches infinity.

Contribution

It provides a rigorous mathematical proof of the phase transition and offers insights into the eigenvalue asymptotics using Fourier analysis on the unitary group.

Findings

Confirmed the occurrence of the phase transition in the large N limit.

Connected the phase transition to eigenvalue distribution changes.

Applied Fourier analysis to study the asymptotic behavior.

Abstract

We give a rigorous proof of the fact that a phase transition discovered by Douglas and Kazakov in 1993 in the context of two-dimensional gauge theories occurs. This phase transition can be formulated in terms of the Brownian bridge on the unitary group U(N) when N tends to infinity. We explain how it can be understood by considering the asymptotic behaviour of the eigenvalues of the unitary Brownian bridge, and how it can be technically approached by means of Fourier analysis on the unitary group. Moreover, we advertise some more or less classical methods for solving certain minimisation problems which play a fundamental role in the study of the phase transition.