| Popis: |
We study the critical case of first-passage percolation in two dimensions. Letting (te) 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. |
