Random Matrices, Vicious Walkers and Yang-Mills Gauge Theory

I will discuss three apparently unrelated subjects: (i) Wishart random matrices that appear in statistics and data analysis; (ii) Vicious Random Walkers model in statistical mechanics introduced by de Gennes and Fisher; (iii) pure Yang-Mills gauge theory in two dimensions on a sphere. The goal of this talk is to establish a connection between these three subjects. I'll show that they all share the same third-order phase transition as a suitable parameter is tuned to its critical value. In case of Wishart matrices, this transition can now be seen experimentally in a coupled laser system. On the theoretical side, we will see how the celebrated Tracy-Widom distribution of the largest eigenvalue of a random matrix also shows up in the vicious walker problem as well as in the double scaling limit of the two-dimensional Yang- Mills gauge theory. Upon exploiting this gauge theory connection, one obtains some rather beautiful new results in the Vicious Walker problem.