A central limit theorem for ''critical'' first-passage percolation in two dimensions

Consider (independent) first-passage percolation on the edges of Z(2). Denote the passage time of the edge e in Z(2) by t(e), and assume that P{t(e) = 0} = 1/2, P{0 < t(e) < C-0} = 0 for some constant C-0 > 0 and that E[t(delta)(e)] < infinity for some delta > 4. Denote by b(0,n) the passage time from 0 to the halfplane {(x, y): x greater than or equal to n}, and by T(0,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0 < C-1, C-2 < infinity and gamma(n) such that C-1(log n)(1/2) less than or equal to gamma(n) less than or equal to C-2(log n)(1/2) and such that gamma(n)(-1)[b(0,n) - Eb(0,n)] and (root 2 gamma(n))(-1)[T(0, nu) - ET(0, nu)] converge in distribution to a standard normal variable (as n --> infinity, u fixed). A similar result holds for the site version of first-passage percolation on Z(2), when the common distribution of the passage times {t(nu)} of the vertices satisfies P{t(nu) = 0} = 1 - P{t(nu) greater than or equal to C-0} = p(c)(Z(2), site) := critical probability of site percolation on Z(2), and E[t(delta)(u)] < infinity for some delta > 4.