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We study first-passage percolation (FPP) on the square lattice. The model isdefined using i.i.d. nonnegative random edge-weights $(t_e)$ associated to thenearest neighbor edges of $\mathbb{Z}^2$. The passage time between vertices $x$and $y$, $T(x,y)$, is the minimal total weight of any lattice path from $x$ to$y$. The growth rate of $T(x,y)$ depends on the value of $F(0) =\mathbb{P}(t_e=0)$: if $F(0) < 1/2$ then $T(x,y)$ grows linearly in $|x-y|$,but if $F(0) > 1/2$ then it is stochastically bounded. In the critical case,where $F(0) = 1/2$, $T(x,y)$ can be bounded or unbounded depending on thebehavior of the distribution function $F$ of $t_e$ near 0. In this paper, weconsider the critical case in which $T(x,y)$ is unbounded and prove theexistence of an incipient infinite cluster (IIC) type measure, constructed byconditioning the environment on the event that the passage time from $0$ to afar distance remains bounded. This IIC measure is a natural candidate for thedistribution of the weights at a typical exceptional time in dynamical FPP. Amajor part of the analysis involves characterizing the limiting behavior ofindependent nonnegative random variables conditioned to have small sum. We giveconditions on random variables that ensure that such limits are trivial, andseveral examples that exhibit nontrivial limits.

Exceptional behavior in critical first-passage percolation and random sums