Topological expansion and boundary conditions

TL;DR

This paper presents a recursive diagrammatic method to compute the topological expansion of mixed-traces in a hermitian two matrix model, enabling enumeration of discrete surfaces with various boundary conditions and genus.

Contribution

It introduces a new recursive, diagrammatic approach to calculate the topological expansion for mixed-traces in matrix models, including boundary conditions.

Findings

Provides a simple, recursive recipe for computing surface counts

Enables enumeration of surfaces with arbitrary boundary conditions

Offers a diagrammatic representation for ease of use

Abstract

In this article, we compute the topological expansion of all possible mixed-traces in a hermitian two matrix model. In other words we give a recipe to compute the number of discrete surfaces of given genus, carrying an Ising model, and with all possible given boundary conditions. The method is recursive, and amounts to recursively cutting surfaces along interfaces. The result is best represented in a diagrammatic way, and is thus rather simple to use.