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

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 [4] 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 ∂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,

$$\displaystyle \mathbb {E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb {E}T(0,\partial B(n)). $$

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.