Expected covering radius of a translation surface

TL;DR

This paper provides an upper bound on the expected covering radius of a translation surface across any stratum, revealing it diminishes roughly as the square root of (log g)/g, which is smaller than analogous hyperbolic diameter expectations.

Contribution

It introduces a universal upper bound on the expected covering radius for translation surfaces in any stratum, extending understanding of their geometric properties.

Findings

Expected covering radius bounded by (log g / g)^(1/2)

Bound is uniform across all strata

Expected radius is smaller than hyperbolic diameter expectations

Abstract

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the largest radius of an immersed disk. In the case of the stratum H_1(2g-2) of translation surfaces of genus g with one singularity, the covering radius is comparable to the diameter. We show that the expected covering radius of a surface is bounded above by a uniform multiple of ((log g)/g)^(1/2), independent of the stratum. This is smaller than what one would expect by analogy from the result of Mirzakhani about the expected diameter of a hyperbolic metric on a Riemann surface. To prove our result, we need an estimate for the volume of the thin part of H_1(kappa) which is given in the appendix.