The unitary invariant ensembles of random matrix theory (collectively referred to as the `1-matrix model’) are one of the central objects of study in the theory of random matrices. Upon tuning the parameters of this model, one can realize certain `higher-order' eigenvalue correlations at the endpoints of the density of eigenvalues. In the 1-matrix model, these higher-order correlation kernels can be written in terms of special solutions of the Painlevé I and II hierarchies. We refer to such special tunings as critical phenomena.

The 2-matrix model is an extension of the 1-matrix model, which came to be well-studied in part because it admits a much richer class of critical phenomena. In this talk, I will survey some of the conjectures regarding these critical phenomena from the physics literature, and discuss some of the physical implications of these results. In particular, I will discuss some forthcoming work (joint with Maurice Duits and Seung-Yeop Lee) regarding a special critical phenomenon in the 2-matrix model, which has implications in the theory of the Ising minimal model coupled to topological gravity.