Asymptotics of shortest filling closed multi-geodesics

In this paper, we investigate the asymptotics of shortest filling closed multi-geodesics of closed hyperbolic surfaces as systole $\to 0$ or as genus $\to \mathrm{\infty }$. We first show that for a closed hyperbolic surface $X_g$ of genus $g$, the length of a shortest filling closed multi-geodesic of $X_g$ is uniformly comparable to $$(g+\sum _{\mathit{\text{closed}}\mathit{\text{geodesic}}\gamma \subset X_g,\mathrm{\ell }(\gamma )<1}\log \left(\frac{1}{\mathrm{\ell }(\gamma )}\right))\mathrm{.}$$ As an application, we show that as $g\to \mathrm{\infty }$, a Weil-Petersson random hyperbolic surface has a shortest closed multi-geodesic of length uniformly comparable to $g$. We also show that this is true for a random hyperbolic surface in the Brooks-Makover model.