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<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.
Article PDF
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Author information
Authors and Affiliations
Additional information
Received: 6 February 1996 / In revised form: 17 July 1996
Rights and permissions
About this article
Cite this article
Kesten, H., Zhang, Y. A central limit theorem for “critical” first-passage percolation in two dimensions. Probab Theory Relat Fields 107, 137–160 (1997). https://doi.org/10.1007/s004400050080
Issue Date:
DOI: https://doi.org/10.1007/s004400050080