Non-Optimality of Invaded Geodesics in 2d Critical First-Passage Percolation

TL;DR

This paper demonstrates that in 2D critical first-passage percolation, the invasion geodesics are not optimal, showing a strict inequality between the expected passage times of the invasion cluster and the original model, indicating different structures.

Contribution

It proves that the invasion geodesics are not optimal by establishing a strict inequality between their expected passage times and those of the original model in 2D critical first-passage percolation.

Findings

Expected invasion passage time exceeds the original by a positive constant factor.

The time constants for the invasion model and the original model differ.

Geodesic structures in invasion and original models are fundamentally different.

Abstract

We study the critical case of first-passage percolation in two dimensions. Letting $(t_e)$ be i.i.d. nonnegative weights assigned to the edges of $\mathbb{Z}^2$ with $\mathbb{P}(t_e=0)=1/2$, consider the induced pseudometric (passage time) $T(x,y)$ for vertices $x,y$. It was shown in [2] that the growth of the sequence $\mathbb{E}T(0,\partial B(n))$ (where $B(n) = [-n,n]^2$) has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. In this paper, we show that this constant factor cannot be taken to be 1. That is, there exists $c>0$ such that for all $n$, \[ \mathbb{E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb{E}T(0,\partial B(n)). \] This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structure.