On an Airy matrix model with a logarithmic potential

Abstract

The Kontsevich–Penner model, an Airy matrix model with a logarithmic potential, may be derived from a simple Gaussian two-matrix model through a duality. In this dual version, the Fourier transforms of the n-point correlation functions can be computed in the closed form. Using Virasoro constraints, we find that in addition to the parameters tn, which appear in the Korteweg–de Vries hierarchies, one needs to introduce half-integer indices tn/2. The free energy as a function of those parameters may be obtained from these Virasoro constraints. The large N limit follows from the solution to an integral equation. This leads to explicit computations for a number of topological invariants.