Random Matrices, Vicious Walkers and Yang-Mills Gauge Theory

Abstract

I will discuss three apparently unrelated subjects: 

• Wishart random matrices that appear in statistics and data analysis
• Vicious Random Walkers model in statistical mechanics introduced by de Gennes and Fisher
• 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.

The Colloquium will be followed by coffee at 2pm in the Tutorial Room 3212, just along from the Lecture Theatres