  Short geodesics are important in the study of the geometry and the spectra of
Riemann surfaces. Bers' theorem gives a global bound on the length of the first
$3g-3$ geodesics. We use the construction of Brooks and Makover of random
Riemann surfaces to investigate the distribution of short ($< \log (g)$)
geodesics on a random Riemann surfaces. We calculate the expected value of the
shortest geodesic, and show that if one orders prime non-intersecting geodesics
by length $\gamma_1\le \gamma_2\le ... \le \gamma_i ,...$, then for fixed $k$,
if one allows the genus to go to infinity, the length of $\gamma_{k}$ is
independent of the genus.
