For i.i.d. weights with P(X>t)~Ct^{-2}, the last passage time is shown to be at least c n(log n)^{3/4}/log log n with high probability, and a finite-second-moment distribution is exhibited whose last passage time grows superlinearly.
Directed polymers in a random environment with heavy tails
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Last passage percolation in hierarchical environments
For i.i.d. weights with P(X>t)~Ct^{-2}, the last passage time is shown to be at least c n(log n)^{3/4}/log log n with high probability, and a finite-second-moment distribution is exhibited whose last passage time grows superlinearly.