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

Antonio Auffinger*, Michael Damron

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations

Abstract

We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf supp(μ) > 0 and μ({b})=p≥ pc, 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.

Original languageEnglish (US)
Pages (from-to)193-227
Number of pages35
JournalProbability Theory and Related Fields
Volume156
Issue number1-2
DOIs
StatePublished - Jun 2013

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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