The diameter of random Belyi surfaces
Résumé
We determine the asymptotic growth rate of the diameter of the random hyperbolic surfaces constructed by Brooks and Makover (J. Differential Geom. 68 (2004) 121–157). 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 n2. We show that the diameter of those random surfaces is asymptotic to 2logn in probability as n→∞.