Nathan Hayford: Critical Phenomena in Random Matrices: Old and New

Abstract:

Critical phenomena in the Hermitian 1-matrix model have been studied for several decades, and are by now well known and classified. The origin of the interest such phenomena in physics stems partly from the connection of the 1-matrix model to the enumeration of genus g-ribbon graphs. This connection allows one to interpret the partition function of the critical 1-matrix model as a partition function for a non-perturbative theory of 2D quantum gravity. In the 1980s, it was discovered that a generalization of this model (the 2-matrix model) carried a much wider class of critical phenomena, and allowed one to describe how certain conformal field theories change when coupled to gravity.

In this talk, I will survey some of the more classical results about critical phenomena in the 1-matrix model, and motivate why the much richer collection of phase transitions present in the 2-matrix model merits more intensive study. Most of what I will say is nothing new, but I hope to mention by the end some joint work with M. Duits and S.-Y. Lee, in which we study the first critical phenomenon in the 2-matrix model not present in the 1-matrix model, which describes the Ising model in a random background.