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Differentiability at the edge of the percolation cone and related results in first-passage percolation
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  • Published: 28 March 2012

Differentiability at the edge of the percolation cone and related results in first-passage percolation

  • Antonio Auffinger1 &
  • Michael Damron2 

Probability Theory and Related Fields volume 156, pages 193–227 (2013)Cite this article

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Abstract

We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf  supp(μ) > 0 and \({\mu(\{b\})=p\geq \vec p_c}\) , the threshold for oriented percolation. We first show that for each such μ, the boundary of the limit shape for μ is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if μ is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman–Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for μ. This result confirms a prediction of Newman and Piza (Ann Probab 23:977–1005, 1995) and Zhang (Ann Probab 36:331–362, 2008). Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents χ and ξ.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Ill, 60637, USA

    Antonio Auffinger

  2. Mathematics Department, Princeton University, Fine Hall, Washington Rd., Princeton, NJ, 08544, USA

    Michael Damron

Authors
  1. Antonio Auffinger
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  2. Michael Damron
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Corresponding author

Correspondence to Antonio Auffinger.

Additional information

A. Auffinger’s research partially funded by NSF Grant DMS 0806180. M. Damron’s research funded by an NSF Postdoctoral Fellowship.

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Auffinger, A., Damron, M. Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Relat. Fields 156, 193–227 (2013). https://doi.org/10.1007/s00440-012-0425-4

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  • Received: 26 May 2011

  • Revised: 28 February 2012

  • Accepted: 03 March 2012

  • Published: 28 March 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0425-4

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Keywords

  • First-passage percolation
  • Shape fluctuations
  • Oriented percolation
  • Richardson’s growth model
  • Graph of infection

Mathematics Subject Classification

  • Primary 60K35
  • 82B43
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