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

Consider (independent) first-passage percolation on the edges of ℤ2. Denote the passage time of the edge e in ℤ2 by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C0} = 0 for some constant C0>0 and that E[tδ(e)]<∞ for some δ>4. Denote by b0,n the passage time from 0 to the halfplane {(x,y): x ≧ 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<C1, C2<∞ and γn such that C1(log n)1/2≦γn≦ C2(log n)1/2 and such that γn−1[b0,n−Eb0,n] and (√ 2γn)−1[T(0,nu) − ET(0,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed).