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

Abstract

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 δ (e)]<∞ for some δ>4. Denote by b 0,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<C 1 , C 2 <∞ and γ n such that C 1 ( log n) 1/2 ≦γ n ≦ C 2 ( log n) 1/2 and such that γ n −1 [b 0,n −Eb 0,n ] and (√ 2γ n ) −1 [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, 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(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C 0 } = p c (ℤ 2 , site ) := critical probability of site percolation on ℤ 2 , and E[t δ (u)]<∞ for some δ>4.