On an Airy matrix model with a logarithmic potential

TL;DR

This paper studies the Kontsevich-Penner matrix model with a logarithmic potential, deriving correlation functions and topological invariants using duality, Virasoro constraints, and large N analysis.

Contribution

It introduces a duality-based approach to compute correlation functions and free energy in an Airy matrix model with a logarithmic potential, extending the parameter space with half-integer indices.

Findings

Closed-form Fourier transforms of n-point correlation functions

Virasoro constraints determine the free energy

Explicit calculations of topological invariants

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 closed form. Using Virasoro constraints, we find that in addition to the parameters $t_n$, which appears in the KdV hierarchies, one needs to introduce here half-integer indices $t_{n/2}$ . 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.