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

Summary. 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 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 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. Received: 6 February 1996 / In revised form: 17 July 1996