On the Speed of Convergence in First-Passage Percolation

We consider the standard first-passage percolation problem on Zd:{t(e):ean edge ofZd} is an i.i.d. family of random variables with common distribution F,a0,n:=inf{.k1t(e1):(e1,.,ek) a path on Zd from 0 to n.1}, where .1 is the first coordinate vector. We show that .2(a0,n).C1n and that P{|a0,n.Ea0,n|.x.n}.C2exp(.C3x) for x.C4n and for some constants 0<Ci<.. It is known that .:=lim(1/n)Ea0,n exists. We show also that C5n.1.Ea0,n.n..C6n5/6(logn)1/3. There are corresponding statements for the roughness of the boundary of the set ~B(t)={.:. a vertex of Zd that can be reached from the origin by a path (e1,.,ek) with .t(ei).t}.