TY - JOUR

T1 - Differentiability at the edge of the percolation cone and related results in first-passage percolation

AU - Auffinger, Antonio

AU - Damron, Michael

N1 - Funding Information:
A. Auffinger’s research partially funded by NSF Grant DMS 0806180. M. Damron’s research funded by an NSF Postdoctoral Fellowship.

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - First-passage percolation

KW - Graph of infection

KW - Oriented percolation

KW - Richardson's growth model

KW - Shape fluctuations

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U2 - 10.1007/s00440-012-0425-4

DO - 10.1007/s00440-012-0425-4

M3 - Article

AN - SCOPUS:84878112215

SN - 0178-8051

VL - 156

SP - 193

EP - 227

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -