  We show that the largest subsurface projection distance between a marking and
its image under the nth step of a random walk grows logarithmically in n, with
probability approaching 1 as n tends to infinity. Our setup is general and also
applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then
use this result to prove Rivin's conjecture that for a random walk $(w_n)$ on
the mapping class group, the shortest geodesic in the hyperbolic mapping torus
$M_{w_n}$ has length on the order of $1/ \log^2(n)$.
