The diameter of random Belyi surfaces

TL;DR

This paper analyzes the asymptotic behavior of the diameter of random hyperbolic surfaces constructed via a specific gluing process, showing it grows logarithmically with the number of triangles involved.

Contribution

It provides the first precise asymptotic estimate for the diameter of Brooks and Makover's random hyperbolic surfaces as the number of triangles increases.

Findings

Diameter asymptotically proportional to 2 log n

Diameter grows logarithmically with the number of triangles

Results hold in probability as n approaches infinity

Abstract

We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover. This model consists of a uniform gluing of $2n$ hyperbolic ideal triangles along their sides followed by a compactification to get a random hyperbolic surface of genus roughly $n/2$. We show that the diameter of those random surfaces is asymptotic to $2 \log n$ in probability as $n \to \infty$.