The Multicritical Kontsevich-Penner Model

TL;DR

This paper introduces a hermitian matrix model with an external field and Penner-like interaction, providing explicit solutions and analyzing its critical behavior, revealing connections to $c=1$ theories and multi-critical points with conformal operators.

Contribution

The paper presents an explicit leading-order solution for a new matrix model with external field and Penner interaction, exploring its critical regimes and conformal properties.

Findings

Explicit leading order solution in $N$ for the model.

Identification of $c=1$ type critical behavior.

Existence of multiple multi-critical points with conformal operators.

Abstract

We consider the hermitian matrix model with an external field entering the quadratic term $\tr(\Lambda X\Lambda X)$ and Penner--like interaction term $\alpha N(\log(1+X)-X)$. An explicit solution in the leading order in $N$ is presented. The critical behaviour is given by the second derivative of the free energy in $\alpha$ which appears to be a pure logarithm, that is a feature of $c=1$ theories. Various critical regimes are possible, some of them corresponds to critical points of the usual Penner model, but there exists an infinite set of multi-critical points which differ by values of scaling dimensions of proper conformal operators. Their correlators with the puncture operator are given in genus zero by Legendre polynomials whose argument is determined by an analog of the string equation.