Geodesics in first-passage percolation cross any pattern

TL;DR

This paper studies the behavior of geodesics in first-passage percolation, showing that they cross patterns a linear number of times with high probability, under mild conditions.

Contribution

It proves that geodesics in first-passage percolation cross any fixed pattern a linear number of times with high probability.

Findings

Number of pattern crossings is linear in geodesic length

Exponential decay of probability for deviations

Applicable under mild conditions

Abstract

In first-passage percolation, one places nonnegative i.i.d. random variables (T (e)) on the edges of Z d. A geodesic is an optimal path for the passage times T (e). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. Under mild conditions, we show that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic.