TITLE

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

AUTHOR(S)
Auffinger, Antonio; Damron, Michael
PUB. DATE
June 2013
SOURCE
Probability Theory & Related Fields;Jun2013, Vol. 156 Issue 1/2, p193
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
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, ) and Zhang (Ann Probab 36:331-362, ). 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.
ACCESSION #
87610049

 

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