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

Kenneth S. Alexander

Published 2020-01-23Version 1

We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic properties conjectured to be true, and derive consequences from these properties. The assumptions are roughly as follows: (i) the standard deviation of the passage time on scale $r$ is of some order $\sigma_r$, with $\{\sigma_r,r>0\}$ growing approximately as a power of $r$; (ii) the tails of the passage time distributions for distance $r$ satisfy an exponential bound on scale $\sigma_r$, uniformly over $r$; and (iii) the limit shape boundary has curvature uniformly bounded away from 0 and $\infty$ (a requirement we can sometimes limit to a neighborhood of some fixed direction.) The main a.s. consequences derived are the following: (a) for one-ended geodesic rays with a given asymptotic direction $\theta$, starting in a natural halfspace $H$, for the hyperplane at distance $R$ from $H$, the density of "entry points" where some geodesic ray first crosses the hyperplane is at most $c(\log R)^K/(R\sigma_R)^{(d-1)/2}$ for some $c,K$, (b) the system has no bigeodesics, i.e. two-ended infinite geodesics, (c) given two sites $x,y$, and a third site $z$ at distance at least $\ell$ from $x$ and $y$, the probability that the geodesic from $x$ to $y$ passes through $z$ is at most $c(\log \ell)^K/(\ell\sigma_\ell)^{(d-1)/2}$ for some $c,K$, and (d) in $d=2$, the probability that the geodesic rays in a given direction from two sites have not coalesced after distance $r$ "decays like $r^{-\xi}$," where $r^\xi$ is roughly the order of transverse geodesic wandering. Our entry-point density bound compares to a natural conjecture of $c/(R\sigma_R)^{(d-1)/2}$, corresponding to a spacing of order $(R\sigma_R)^{1/2}$ between entry points, which is the conjectured scale of the transverse wandering.