Passage time of the frog model has a sublinear variance

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