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

Largest projections for random walks and shortest curves in random mapping tori