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A central limit theorem for “critical” first-passage percolation in two dimensions
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  • Published: February 1997

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

  • Harry Kesten1 &
  • Yu Zhang2 

Probability Theory and Related Fields volume 107, pages 137–160 (1997)Cite this article

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  • 25 Citations

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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.

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Authors and Affiliations

  1. Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14853, USA 2 (e-mail: kesten@math.cornell.edu) , , , , , , US

    Harry Kesten

  2. Department of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA (e-mail: yzhang@vision.uccs.edu) , , , , , , US

    Yu Zhang

Authors
  1. Harry Kesten
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  2. Yu Zhang
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Received: 6 February 1996 / In revised form: 17 July 1996

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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

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  • Issue Date: February 1997

  • DOI: https://doi.org/10.1007/s004400050080

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  • Mathematics Subject Classification (1991): 60K35
  • 60F05
  • 82B43
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