Local limits of bipartite maps with prescribed face degrees in high genus

TL;DR

This paper investigates the local limits of high genus bipartite maps with fixed face degrees, demonstrating convergence to infinite maps with hyperbolic properties and establishing universality across various models.

Contribution

It introduces the q-IBPMs as universal local limits for high genus bipartite maps with prescribed face degrees, extending previous work on triangulations.

Findings

Convergence to q-IBPMs with hyperbolic behavior

Universality of local limits across different models

Applicable when the expected degree of the root face is finite

Abstract

We study the local limits of uniform high genus bipartite maps with prescribed face degrees. We prove the convergence towards a family of infinite maps of the plane, the q-IBPMs, which exhibit both a spatial Markov property and a hyperbolic behaviour. Therefore, we observe a similar local behaviour for a wide class of models of random high genus maps, which can be seen as a result of universality. Our results cover all the regimes where the expected degree of the root face remains finite in the limit. This follows a work by the same authors on high genus triangulations arXiv:1902.00492.