Random coalescing geodesics in first-passage percolation

Abstract

We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics'. Random coalescing geodesics have a range of nice asymptotic properties, such as asymptotic directions and linear Busemann functions. We show that random coalescing geodesics are (in some sense) dense in the space of geodesics. This allows us to extrapolate properties from random coalescing geodesics to obtain statements on all infinite geodesics. As an application of this theory we solve the `midpoint problem' of Benjamini, Kalai and Schramm and address a question of Furstenberg on the existence of bigeodesics.

74 pages, 15 figures

References in corpus (2)

• Geodesics Toward Corners in First Passage Percolation
• First Passage percolation on a hyperbolic graph admits bi-infinite geodesics

Cited by in corpus (9)

• Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• Non-existence of bi-infinite geodesics in the exponential corner growth model
• Geometry of geodesics through Busemann measures in directed last-passage percolation
• Optimal Exponent for Coalescence of Finite Geodesics in Exponential Last Passage Percolation
• Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights
• Stationary coalescing walks on the lattice
• Geodesics Toward Corners in First Passage Percolation
• First Passage percolation on a hyperbolic graph admits bi-infinite geodesics
• Existence and coalescence of directed infinite geodesics in the percolation cone for Durrett-Liggett class of measures