Geodesics Toward Corners in First Passage Percolation

Abstract

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson [4], Ahlberg and Hoffman [1], and others. However the main results do not cover geodesics in the direction of corners of the limit shape B, where two facets meet. We construct an example with the following properties: (i) the limiting shape is an octagon, (ii) semi-infinite geodesics exist only in the four axis directions, and (iii) in each axis direction there are multiple such geodesics. Consequently, the set of points of ∂B which are in the direction of some geodesic does not have all of B as its convex hull.2010 Mathematics Subject Classification. Primary: 60K35, 82B43.

Geodesic rays and exponents in ergodic planar first passage percolation

“…When B is a polygon, little is known about existence of geodesics rays in the direction of the corners of B. Recently, Alexander and Berger [AB18] exhibit a model for which the limiting shape is an octagon and all (possibly infinitely many) geodesic rays are directed along the coordinate axis. Our first result shows that our model has exactly four geodesic rays a.s.. To the best of our knowledge, this is the first known FPP model for which |T 0 | is finite.…”

Last passage percolation in an exponential environment with discontinuous rates

“…[21] showed that any compact set with lattice symmetries can be obtained as a limit shape of a stationary FPP process. For example, an octagon has been obtained as a limit shape in [2], where also geodesics in the direction of corners (points of non-differentiability) were studied. It is more difficult to obtain polygonal shapes and having points of non differentiability in an LPP setting when the environment is independent and has a continuous distribution, but with the speed functions we are considering here it is possible.…”

“…By hypothesis (1), assume ε is small enough so that the first time h ε touches the top boundary [0, a] × {b}, is precisely at some point x ε > a − δ. Consider a parametrisation for h, (h (1) (s), h (2)…”

“…When the environment τ i,j ∼ Exp (1), the last passage model is one of the exactly solvable models of the KPZ class (see [10] for a survey). The strong law of large numbers in the exponential model is explicitly computed in [33] lim n→∞ G nx , ny n = γ(x, y) = ( √ x + √ y) 2 , P-a.s. (1.3) In this article we derive the limiting constant for a sequence of scaled last passage times on the lattice. The passage times themselves are coupled through a common realization of exponential random variables.…”

Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation

“…In recent work by Brito and Hoffman [3], an ergodic weight distribution exhibiting this kind of counterintuitive behavior is constructed: in their example, there exist exactly four infinite geodesics, and each wanders across an entire quadrant of the lattice. See also the recent work by Alexander and Berger [1], where an ergodic distribution is constructed having an octogon as its limit shape, but having geodesics directed only toward the axis directions of the lattice.…”