The partition function of the 2-dimensional Yang—Mills model

The partition function of the 2d Yang—Mills model is the natural mass of the Yang—Mills measure, and there is at least one reasonable way of defining it. For each oriented compact surface, it is a function on the set of all possible boundary conditions for the Yang—Mills field, which in the 2d case is finite-dimensional. This function plays for the 2d Yang—Mills field the role usually played by the transition kernel of a Markov process. The case of the sphere is unique among closed oriented surfaces in that, in the large N limit, the U(N) Yang—Mills model exhibits a third order phase transition, the Douglas—Kazakov phase transition, with respect to the total area of the sphere. This transition can be understood in terms of non-intersecting Brownian motions on a circle, as Karl Liechty and Dong Wang did, or in terms of a discrete Coulomb gas, as we did with Mylène Maïda.