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

Abstract

We show existence, uniqueness, and directedness properties for infinite geodesics in the FPP model. After giving the fundamental definitions, we describe results by Newman and collaborators giving existence and uniqueness of directed geodesics under an unproven curvature assumption. We then give two proofs of the existence of at least two infinite geodesics under no unproven assumptions. In the final two sections, we give proofs of directedness statements for infinite geodesics using more recent methods which give information even under no unproven assumptions and prove a generalized uniqueness statement for infinite geodesics.

Busemann functions, geodesics, and the competition interface for directed last-passage percolation

“…Details can be found in Appendix A of [27] (which is an extended version of [28]). See also the proof of Theorem 2.3 in [34]. First observe that if x u,ξ,1 0,8 and x v,ξ,1 0,8 ever intersect, then from there on they follow the same evolution (smallest B ξ increment and increment e 1 in case of a tie).…”