A Diagrammatic Equation for Oriented Planar Graphs

TL;DR

This paper introduces a diagrammatic equation for oriented planar graphs in non-Hermitian random matrix models, derived via graph counting and saddle point analysis, revealing duality properties in the solutions.

Contribution

It presents a novel diagrammatic equation for the planar sector of non-Hermitian matrix models, connecting graph counting with saddle point methods.

Findings

Derived a fundamental diagrammatic equation for planar graphs

Solved the equation perturbatively for generic models

Identified duality properties in the perturbative solutions

Abstract

In this paper we introduce a diagrammatic equation for the planar sector of square non hermitian random matrix models strongly reminiscent of Polchinski's equation in quantum field theory. Our fundamental equation is first obtained by a graph counting argument and subsequently derived independently by a precise saddle point analysis of the corresponding random matrix integral. We solve the equation perturbatively for a generic model and conclude by exhibiting two duality properties of the perturbative solution.