Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension

Abstract: In first passage percolation (FPP) on $\mathbb{Z}^d$, i.i.d.~(bond) passage times are attached to the nearest-neighbor bonds of the lattice, and the passage time from $x$ to $y$ is the shortest sum of bond passage times among all possible paths from $x$ to $y$; the corresponding minimizing path is called a geodesic. One can also consider geodesic rays and bigeodesics, which are one-ended and two-ended infinite paths for which every finite segment is a geodesic; a $\theta$-ray is a geodesic ray with asymptotic direction $\theta$. The conjectured picture, partly verified for $d=2$ under assumptions of various strengths, is that for a given $\theta$, there is a.s.~a unique $\theta$--ray from each lattice site, and any two $\theta$--rays eventually coalesce, though there is a random null set of directions for which this fails; bigeodesics a.s.~do not exist at all. Here we establish portions of this heuristic picture in higher dimensions (where few results currently exist), at least under the assumption that certain very basic but unproven properties of FPP are valid. We establish a coalescence-like property: taking all the $\theta$--rays starting next to a given hyperplane, and looking at the set of points where they cross another hyperplane some distance $r$ ahead of the starting one, we show that the geodesics bundle together in the sense that the density of the crossing points approaches 0 (at a near-sharp rate) as $r\to\infty$. This bundling property also holds if we consider together all $\theta$--rays over a narrow range of directions $\theta$, and this fact leads to proof of the absence of bigeodesics. In $d=2$, bundling can be used to bound the probability that two $\theta$--rays do not coalesce before traveling a distance $r$.

mathematical physicsprobability

Audience: researchers in the topic

Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

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