The growth rate on the volume of Mg<L(g)

Let Mg be the moduli space of hyperbolic surfaces of genus g endowed with the Weil-Petersson metric. In this paper, we introduce a function $L(g)$ of genus g and call the geodesics whose length less than $L(g)$ short geodesics. We compute the growth rate on the volume of the subset of hyperbolic surfaces with short geodesics. In particular, when g approaches infinity, if $L(g)$ also approaches infinity, then the volume of surfaces characterized by short geodesics is equal to Vg almost surely.