Generalized multicritical one-matrix models

TL;DR

This paper introduces a generalized one-matrix model that smoothly interpolates between different multicritical points by employing a heavy-tailed potential, enabling the study of polygons with arbitrarily large degrees.

Contribution

It presents a simple generalization of Kazakov's multicritical matrix model using a heavy-tailed potential, allowing continuous interpolation between multicritical points.

Findings

The generalized model features a potential with a heavy tail and a cut on the real axis.

It enables polygons of arbitrary large degrees in the combinatorial interpretation.

The model reduces to polynomial form at traditional multicritical points.

Abstract

We show that there exists a simple generalization of Kazakov's multicritical one-matrix model, which interpolates between the various multicritical points of the model. The associated multicritical potential takes the form of a power series with a heavy tail, leading to a cut of the potential and its derivative at the real axis, and reduces to a polynomial at Kazakov's multicritical points. From the combinatorial point of view the generalized model allows polygons of arbitrary large degrees (or vertices of arbitrary large degree, when considering the dual graphs), and it is the weight assigned to these large order polygons which brings about the interpolation between the multicritical points in the one-matrix model.