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

Abstract

Consider bond percolation on the square lattice \(\mathbb{Z}^2\) where each edge is independently open with probability \(p.\) For some positive constants \(p_0 \in (0,1), \epsilon_1\) and \(\epsilon_2,\) the following holds: if \(p > p_0,\) then with probability at least \(1-\frac{\epsilon_1}{n^{4}}\) there are at least \(\frac{\epsilon_2 n}{\log{n}}\) disjoint open left-right crossings in \(B_n := [0,n]^2\) each having length at most \(2n,\) for all \(n \geq 2.\) 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 \(\mathbb{E}\left(\log{t(e_1)}\right)^+<\infty;\) namely, \(\limsup_n \frac{T_{pl}(0,n)}{n} \leq Q\) a.s. for some  constant \(Q < \infty,\) where \(T_{pl}(0,n)\) denotes the minimum passage time from the point \((0,0)\) to the line \(x=n\) taken over all paths contained in \(B_n.\) Finally, we also obtain deviation corresponding estimates for nonidentical passage times satisfying \(\inf_i\mathbb{P}(t(e_i) = 0) > \frac{1}{2}.\)