  For First Passage Percolation in Z^d with large d, we construct a path
connecting the origin to {x_1 =1}, whose passage time has optimal order \log
d/d. Besides, an improved lower bound for the "diagonal" speed of the cluster
combined with a result by Dhar (1988) shows that the limiting shape in FPP with
exponential passage times (and thus that of Eden model) is not the euclidian
ball in dimension larger than 35.
