Matrix Model Calculations beyond the Spherical Limit

TL;DR

This paper introduces an improved iterative method for calculating higher genus contributions in hermitian matrix models, providing explicit results up to genus four and establishing equivalences with other models in the double scaling limit.

Contribution

It develops a new iterative scheme for higher genus calculations and proves the equivalence of hermitian, complex, and Kontsevich models in the double scaling limit.

Findings

Explicit higher genus results up to genus four.

Proof of equivalence between hermitian and complex matrix models in the double scaling limit.

Demonstration of equivalence to the Kontsevich model in the double scaling limit.

Abstract

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.