Expected Covering Radius of a Translation Surface

Abstract

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 ${{\mathcal{H}}}_1(\kappa )$. The covering radius of a translation surface is the largest radius of an immersed disk. In the case of the stratum ${{\mathcal{H}}}_1(2g-2)$ 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 $\sqrt{ \frac{\log g}{g}}$, 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 ${{\mathcal{H}}}_1(\kappa )$, which is given in the appendix.