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

2018/04/16 by Hanson, Jack
#60K35 #60K37 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.05720

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.

Related

• Multiple geodesics with the same direction
• Existence and coalescence of directed infinite geodesics in the percolation cone for Durrett-Liggett class of measures
• Random coalescing geodesics in first-passage percolation
• Absence of backward infinite paths for first-passage percolation in arbitrary dimension
• Ergodicity of the number of infinite geodesics originating from zero