The Annals of Probability

Singularity points for first passage percolation

J. E. Yukich and Yu Zhang

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Let 0<a<b<∞ be fixed scalars. Assign independently to each edge in the lattice ℤ2 the value a with probability p or the value b with probability 1−p. For all u,v∈ℤ2, let T(u,v) denote the first passage time between u and v. We show that there are points x∈ℝ2 such that the “time constant” in the direction of x, namely, lim n→∞n−1Ep[T(0,nx)], is not a three times differentiable function of p.

Article information

Ann. Probab., Volume 34, Number 2 (2006), 577-592.

First available in Project Euclid: 9 May 2006

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

First passage percolation shape theory the right-hand edge nondifferentiability of time constants


Yukich, J. E.; Zhang, Yu. Singularity points for first passage percolation. Ann. Probab. 34 (2006), no. 2, 577--592. doi:10.1214/009117905000000819.

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