  We discuss the average length l of the shortest non-contractible loop on
surfaces in the two-dimensional pure quantum gravity ensemble. The value of
$\gamma_{str}$ and the explicit form of the loop functions indicate that l
diverges at the critical point. Scaling arguments suggest that the critical
exponent of l is 1/2. We show that this value of the critical exponent is also
obtained for branched polymers where the calculation is straightforward.
