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Sublinear variance in first-passage percolation for general distributions

Abstract

We prove that the variance of the passage time from the origin to a point \(x\) in first-passage percolation on \(\mathbb {Z}^d\) is sublinear in the distance to \(x\) when \(d\ge 2\), obeying the bound \(C\Vert x\Vert /\log \Vert x\Vert \), under minimal assumptions on the edge-weight distribution. The proof applies equally to absolutely continuous, discrete and singular continuous distributions and mixtures thereof, and requires only \(2+ \log \) moments. The main result extends work of Benjamini–Kalai–Schramm (Ann Prob 31, 2003) and Benaim–Rossignol (Ann Inst Henri Poincaré Prob Stat 3, 2008).

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Acknowledgments

We thank S. Sodin for helpful conversations and for pointing out the use of geometric averaging in his paper. We also thank him for careful reading of a previous draft. We are grateful to T. Seppäläinen for pointing out an error in the entropy section and A. Auffinger for finding various typos. Last, we thank an anonymous referee for comments that led to a better organized presentation.

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Correspondence to Michael Damron.

Additional information

The research of M. D. is supported by NSF Grant DMS-0901534.

The research of J. H. is supported by an NSF graduate research fellowship.

The research of P. S. is supported by an NSERC postgraduate fellowship.

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Damron, M., Hanson, J. & Sosoe, P. Sublinear variance in first-passage percolation for general distributions. Probab. Theory Relat. Fields 163, 223–258 (2015). https://doi.org/10.1007/s00440-014-0591-7

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Mathematics Subject Classification

  • Primary 60K35
  • Secondary 60E15