Douglas and Kazakov predicted about twenty years ago that the pure Euclidean Yang-Mills theory on the two-dimensional sphere with structure group U(N) exhibits a phase transition, in the limit where N tends to infinity, when the area of the sphere crosses the critical value pi^2.
In probabilistic language, this can be expressed as a phase transition for the Brownian bridge on the unitary group U(N) in the large N limit, when the length of the bridge crosses the critical value pi^2.
I will explain the nature of this phase transition and present some of the tools which can be used to study it. This is joint work in progress with Mylène Maïda.