The authors define higher-order derivatives for first-passage percolation and express the variance of the passage time in terms of these derivatives.
Random coalescing geodesics in first-passage percolation
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
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.
fields
math.PR 2verdicts
UNVERDICTED 2representative citing papers
In FPP on hyperbolic groups, the set of exceptional directions has strictly smaller Hausdorff dimension than the boundary and exists densely with multiple disjoint geodesics when boundary dimension exceeds one.
citing papers explorer
-
Higher-order derivatives of first-passage percolation with respect to the environment
The authors define higher-order derivatives for first-passage percolation and express the variance of the passage time in terms of these derivatives.
-
Geodesic Trees and Exceptional Directions in FPP on Hyperbolic Groups
In FPP on hyperbolic groups, the set of exceptional directions has strictly smaller Hausdorff dimension than the boundary and exists densely with multiple disjoint geodesics when boundary dimension exceeds one.