Colored triangulations of arbitrary dimensions are stuffed Walsh maps

TL;DR

This paper introduces a new combinatorial representation called stuffed Walsh maps for families of edge-colored graphs that encode colored triangulations of pseudo-manifolds, extending previous models to arbitrary dimensions.

Contribution

It establishes a bijection between these graph families and stuffed Walsh maps, generalizing Walsh's hypermap representation and linking tensor models with multi-matrix models.

Findings

Bijection between edge-colored graphs and stuffed Walsh maps

Extension of the intermediate field method to arbitrary models

Complete characterization of certain maximizer maps

Abstract

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.