Large expanders in high genus unicellular maps

TL;DR

This paper investigates large random unicellular maps with high genus, revealing that they typically contain large induced subgraphs that are expanders, thus contributing to understanding their hyperbolic geometric properties.

Contribution

It demonstrates that high genus unicellular maps almost surely contain large expander subgraphs, linking hyperbolic geometry features with combinatorial graph structures.

Findings

Maps contain large expander subgraphs with high probability

High genus maps exhibit hyperbolic geometric features

Local limits and diameter properties are consistent with hyperbolic geometry

Abstract

We study large uniform random maps with one face whose genus grows linearly with the number of edges. They can be seen as a model of discrete hyperbolic geometry. In the past, several of these hyperbolic geometric features have been discovered, such as their local limit or their logarithmic diameter. In this work, we show that with high probability such a map contains a very large induced subgraph that is an expander.