Geodesic rays and exponents in ergodic planar first passage percolation

TL;DR

This paper investigates ergodic planar first passage percolation, establishing the asymptotic shape as the $ ext{L}^1$ ball, characterizing the number of infinite geodesics, and analyzing variance and wandering exponents, revealing deviations from expected relationships.

Contribution

It proves the asymptotic shape is the $ ext{L}^1$ ball, counts infinite geodesics, and determines variance and wandering exponents, highlighting differences from independent models.

Findings

Asymptotic shape is the $ ext{L}^1$ ball.

Exactly four infinite geodesics start at the origin.

Variance and wandering exponents do not follow the $oldsymbol{ ext{chi}=2 ext{ extbeta}-1}$ relation.

Abstract

We study first passage percolation on the plane for a family of invariant, ergodic measures on $\mathbb{Z}^2$. We prove that for all of these models the asymptotic shape is the $\ell$-$1$ ball and that there are exactly four infinite geodesics starting at the origin a.s. In addition we determine the exponents for the variance and wandering of finite geodesics. We show that the variance and wandering exponents do not satisfy the relationship of $\chi=2\xi-1$ which is expected for independent first passage percolation.