Planarity and non-separating cycles in uniform high genus quadrangulations

TL;DR

This paper investigates the global topological properties of large uniform high genus quadrangulations, revealing that local neighborhoods are planar and short non-contractible cycles exist with positive probability, extending understanding beyond local convergence.

Contribution

It demonstrates that high genus quadrangulations have planar local neighborhoods and contain short non-contractible cycles, providing new insights into their global topology.

Findings

Balls around the root are planar with high probability up to logarithmic radius.

Short non-contractible cycles exist with positive probability.

Global properties differ from local convergence results.

Abstract

We study large uniform random quadrangulations whose genus grow linearly with the number of faces, whose local convergence was recently established by Budzinski and the author arXiv:1902.00492,arXiv:2012.05813. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar whp up to logarithmic radius, and we prove that there exists short non-contractible cycles with positive probability.