Topological Expansion in the Cubic Random Matrix Model

Abstract

In this paper, we study the topological expansion in the cubic random matrix model, and we evaluate explicitly the expansion coefficients for genus 0 and 1. For genus 0 our formula coincides with the one in [6]. For higher genus, we obtain the asymptotic behavior of the coefficients in the expansion as the number of vertices of the associated graphs tends to infinity. Our study is based on the Riemann–Hilbert problem, string equations, and the Toda equation.