Differentiability at the edge of the percolation cone and related results in first-passage percolation

TL;DR

This paper investigates the geometric and probabilistic properties of first-passage percolation in two dimensions, establishing differentiability of the limit shape boundary, non-polygonality, and fluctuation bounds for passage times, with implications for growth models.

Contribution

It proves the differentiability of the limit shape boundary at flat edges, shows the non-polygonal nature of the shape, and extends fluctuation lower bounds to measures with percolation thresholds.

Findings

Limit shape boundary is differentiable at flat edges.

Limit shape is non-polygonal for measures with p ≥ p_c.

Variance of passage time has a lower bound of order log n outside the percolation cone.

Abstract

We study first-passage percolation in two dimensions, using measures mu on passage times with b:=inf supp(mu) >0 and mu({b})=p \geq p_c, the threshold for oriented percolation. We first show that for each such mu, the boundary of the limit shape for mu is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if mu is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman-Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for mu. This result confirms a prediction of Newman-Piza and Zhang. Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents chi and xi.