The Cheeger constants of random Belyi surfaces

TL;DR

This paper demonstrates that in a model of random hyperbolic surfaces constructed from ideal triangles, the Cheeger constant tends to be below a specific bound as the number of triangles increases, indicating typical geometric bottlenecks.

Contribution

It establishes a probabilistic upper bound on the Cheeger constant for large random hyperbolic surfaces in the Brooks-Makover model.

Findings

Cheeger constant is less than 3/(2π)+ε for large surfaces

Results hold for any ε>0 as the number of ideal triangles increases

Provides insight into the geometric properties of random hyperbolic surfaces

Abstract

Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this paper we show that for any $\epsilon>0$, as the number of ideal triangles goes to infinity, a generic hyperbolic surface in Brooks-Makover's model has Cheeger constant less than $\frac{3}{2\pi}+\epsilon$.