First passage time of the frog model has a sublinear variance

Abstract

In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.