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

Abstract

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. the (p,q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1-matrix model, for which q=2.

pdflatex, 44 pages, 2 figures

References in corpus (6)

• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Algebraic methods in random matrices and enumerative geometry
• On Correlation Numbers in 2D Minimal Gravity and Matrix Models
• Combinatorial aspects of matrix models
• Determinantal formulae and loop equations
• From Liouville Theory to the Quantum Geometry of Riemann Surfaces

Cited by in corpus (7)

• JT gravity as a matrix integral
• Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
• Torus Amplitudes in Minimal Liouville Gravity and Matrix Models
• From Noncommutative Geometry to Random Matrix Theory
• Tracy-Widom GUE law and symplectic invariants
• Double scaling limits of Dirac ensembles and Liouville quantum gravity
• Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions