High order expansion of matrix models and enumeration of maps

2006/08/08 by Édouard Maurel-Segala, Maurel-Segala, Edouard
Mathematics · #05C30 #15A52 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.math/0608192

openalex publication_date 2006/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Perturbation of the GUE are known in physics to be related to enumeration of graphs on surfaces. We investigate this idea and show that for a small convex perturbation, we can perform a genus expansion: the moments of the empirical measure can be developed into a series whose g-th term is a generating function of graphs on a surface of genus g.

Citations

• The degree distribution in bipartite planar maps: applications to the Ising model
• Limit laws for Random matrices and free products
• A planar diagram theory for strong interactions
• Combinatorial aspects of matrix models
• On the Distribution of the Roots of Certain Symmetric Matrices
• Invariants of algebraic curves and topological expansion

Related

• GUE Correlators and Large Genus Asymptotics
• GOE statistics on the moduli space of surfaces of large genus
• Map enumeration from a dynamical perspective
• Genus Expansions of Hermitian One-Matrix Models: Fat Graphs vs. Thin Graphs
• Strings and matrix models on genus g Riemann Surfaces