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

Consider bond percolation on the square lattice Z(2) where each edge is independently open with probability p : For some positive constants p(0) is an element of( 0; 1); epsilon(1) and epsilon(2); the following holds: if p > p(0); then with probability at least 1 - epsilon(1) /n(4) there are at least epsilon(2)n/logn disjoint open left-right crossings in B-n : = 0; n 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(e(i))}(i) satisfying E (logt (e(1)))(+) < infinity; namely, lim sup(n) T-pl (0, n)/n