We study first-passage percolation (FPP) on the square lattice. The model is defined using i.i.d. nonnegative random edge-weights (t e ) associated to the nearest neighbor edges of 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) = 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 the behavior of the distribution function F of t e near 0. In this paper, we consider the critical case in which T (x, y) is unbounded and prove the existence of an incipient infinite cluster (IIC) type measure, constructed by conditioning the environment on the event that the passage time from 0 to a far distance remains bounded. This IIC measure is a natural candidate for the distribution of the weights at a typical exceptional time in dynamical FPP. A major part of the analysis involves characterizing the limiting behavior of independent nonnegative random variables conditioned to have small sum. We give conditions on random variables that ensure that such limits are trivial, and several examples that exhibit nontrivial limits.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.