Disjoint crossings, positive speed and deviation estimates for first passage percolation

Consider bond percolation on the square latticeZ where each edge is independently open with probability p. For some positive constants p0 ∈ (0, 1), 1 and 2, the following holds: if p > p0, then with probability at least 1− 1 n4 there are at least 2n logn disjoint open left-right crossings in Bn := [0, n] 2 each having length at most 2n, for all n ≥ 2. Using the proof of the above, we obtain positive speed for first passage percolation with independent and identically distributed edge passage times {t(ei)}i satisfying E (log t(e1)) + < ∞; namely, lim supn Tpl(0,n) n ≤ Q a.s. for some constant Q < ∞, where Tpl(0, n) denotes the minimum passage time from the point (0, 0) to the line x = n taken over all paths contained in Bn. Finally, we also obtain corresponding deviation estimates for nonidentical passage times satisfying infiP(t(ei) = 0) > 2 3 .