Universal scaling limits of matrix models, and (p,q) Liouville gravity

Understand

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ~ x^{p/q}, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p,q)-minimal models kernels.

• Those (p,q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree at most p.
• For example, near a regular edge y ~ x^{1/2}, the (1,2) kernel is the Airy kernel and we recover the Airy law.
• Those kernels are associated to the (p,q) minimal model, i.e.