Exact Solution of 1-matrix Model

TL;DR

This paper introduces a new exact method for solving 1-matrix models through genus expansion without continuum limits, enabling precise computation of free energy coefficients and validation up to genus three.

Contribution

The paper presents a novel approach to solve 1-matrix models exactly in the genus expansion, avoiding the need for physical continuum limits and confirming results up to genus three.

Findings

Exact computation of free energy coefficients in genus expansion.

Method reproduces standard results with physical tuning.

Validated results up to genus three.

Abstract

I review my new method for solving general 1-matrix models by expanding in $N^{-1}$ without taking a physical continuum limit. Using my method, each coefficient of the free energy in the genus expansion is exactly computable. One can include physical information in a function which is uniquely specified by the action of the model. My method gives completely the same result with the usual one if the physical fine tuning is done and the leading singular terms are extracted. I also calculate in the genus three case and confirm the validity of my method.