Ergodicity of the number of infinite geodesics originating from zero

TL;DR

This paper proves that the number of infinite geodesics originating from the origin in first-passage percolation is almost surely constant in higher dimensions and for general distributions, extending previous results.

Contribution

It extends the ergodicity result of the number of infinite geodesics from 2D with continuous distributions to higher dimensions and more general distributions.

Findings

Number of infinite geodesics is almost surely constant in higher dimensions.

Extension of previous 2D results to general distributions.

Supports the ergodic nature of geodesic structures in first-passage percolation.

Abstract

First-passage percolation is a random growth model which has a metric structure. An infinite geodesic is an infinite sequence whose all sub-sequences are shortest paths. One of the important quantity is the number of infinite geodesics originating from the origin. When $d=2$ and an edge distribution is continuous, it is proved to be almost surely constant [D. Ahlberg, C. Hoffman. Random coalescing geodesics in first-passage percolation]. In this paper, we will prove the same result for higher dimensions and general distributions.