Perturbed generalized multicritical one-matrix models

TL;DR

This paper analyzes perturbations of a generalized multicritical one-matrix model with a specific potential, relating it to KdV hierarchy tau-functions and solving it via genus expansion in the double scaling limit.

Contribution

It introduces a method to study perturbations of the generalized Kazakov multicritical matrix model using KdV hierarchy tau-functions.

Findings

Partition function related to KdV tau-function.

Model solvable via genus expansion in double scaling limit.

Potential exhibits a power-law decay in coefficients.

Abstract

We study perturbations around the generalized Kazakov multicritical one-matrix model. The multicritical matrix model has a potential where the coefficients of $z^n$ only fall off as a power $1/n^{s+1}$. This implies that the potential and its derivatives have a cut along the real axis, leading to technical problems when one performs perturbations away from the generalized Kazakov model. Nevertheless it is possible to relate the perturbed partition function to the tau-function of a KdV hierarchy and solve the model by a genus expansion in the double scaling limit.