Expected covering radius of a translation surface

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.

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