Hypergraph matrix models and generating functions

TL;DR

This paper introduces the hypergraph matrix model (HMM), a generalization of GUE, and derives generating functions for maps with instructions, connecting matrix models to combinatorial topology and Wright's work on graph enumeration.

Contribution

The paper develops generating functions for unicelled edge-ramified CW complexes with fixed genus, extending matrix model techniques to new combinatorial structures.

Findings

Derived generating functions for maps with instructions of fixed genus.

Connected matrix models to enumeration of CW complexes and maps.

Extended Wright's graph enumeration results to hypergraph matrix models.

Abstract

Recently we introduced the hypergraph matrix model (HMM), a Hermitian matrix model generalizing the classical Gaussian Unitary Ensemble (GUE). In this model the Gaussians of the GUE, whose moments count partitions of finite sets into pairs, are replaced by formal measures whose moments count set partitions into parts of a fixed even size 2m >= 2. Just as the expectations of the trace polynomials Tr X^{2r} in the GUE produce polynomials counting unicellular orientable maps of different genera, in the HHM these expectations give polynomials counting certain unicelled edge-ramified CW complexes with extra data that we call (orientable CW) maps with instructions. In this paper we describe generating functions for maps with instructions of fixed genus and with the number of vertices arbitrary. Our results are motivated by work of Wright. In particular Wright computed generating functions of connected graphs of fixed first Betti number as rational functions in the rooted tree function T (x), given as the solution to the functional relation x = T e^{-T}.