Planar first-passage percolation times are not tight

Abstract

We consider first-passage percolation on the two-dimensional integer lattice Z^2 with passage times that are IID exponentials of mean one. It has been conjectured, based on numerical evidence, that the variance of the time T(0,n) to reach the vertex (0,n) is of order n^{2/3}. Kesten showed that the variance of T(0,n) is at O(n). He also noted that the variance is bounded away from zero. This note improves the lower bound on the variance of T(0,n) to C log n. Simultaneously and independently, Newman and Piza have achieved the same result for {0,1}-valued passage times. Their methods extend to more general passage times, while ours work only for exponential times. On the other hand, our theorem shows that the variance comes from fluctuations of nonvanishing probability in the sense that, as n->infty, the law of T(0,n) is not tight about its median. Very recently, Newman and Piza showed that the log n may be improved to a power of n for directions in which the shape is not flat (it is not known whether the shape can be flat in any direction). As pointed out to us by Harry Kesten, in the exponential case this may also be obtained via the method given here.