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Bigeodesics in first-passage percolation. (English) Zbl 1361.60089
First-passage percolation (FPP) is a model for the flow of a liquid through a random medium due to J. M. Hammersley and D. J. A. Welsh [in: Bernoulli-Bayes-Laplace, Anniversary Vol., Proc. Int. Res. Semin., Berkeley 1963, 61–110 (1965; Zbl 0143.40402)]. On the multi-dimensional integer lattice, every edge $$e \in E^d$$ that connects two nearest-neighbor sites of $$\mathbb{Z}^d$$ is designated a nonnegative random variable $$t_e$$, called the passage time of $$e$$. Similarly, for every path $$\gamma$$ (from some site to another site), its passage time is defined as $$T (\gamma) := \sum_{e \in \gamma} t_e$$. Then the random variable $$T (x, y)$$ is defined as the infimum of the passage times of all paths from $$x$$ to $$y$$. Such a minimizing path is called geodesic and a doubly infinite geodesic is called bigeodesic. Assume that the sequence $$(t_e)$$ is independent and identically distributed with a continuous distribution. Assume also that $$d=2$$. A well-known conjecture of Kensten asserts that there are no bigeodesics in two dimensions almost surely. Let the limit shape be a subset of $$\mathbb{R}^2$$ that contains all sites $$x$$ such that $$T (0, nx) / n \leq 1$$ as $$n \to \infty$$ almost surely. Given any deterministic direction $$\phi$$ (i.e., for a sequence $$(x_n)$$ of nearest-neighbor sites in a path, $$\|x_n\|$$ tends to infinity and the argument of $$x_n$$ tends to $$\phi$$ as $$n \to \infty$$) and assuming that the boundary of the limit shape is differentiable, the authors show that there is no bigeodesic with one end directed in $$\phi$$ almost surely.

MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
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References:
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