Convergence of limit shapes for 2D near-critical first-passage percolation

Abstract

We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the classical "shape theorem", and let $L(p)$ be the correlation length. We show that as $p\uparrow p_c$, the rescaled limit shape $L(p)^{-1}\mathcal {B}(p)$ converges to a Euclidean disk. This improves a result of Chayes et al. [J. Stat. Phys. 45 (1986) 933--951]. The proof relies on the scaling limit of near-critical percolation established by Garban et al. [J. Eur. Math. Soc. 20 (2018) 1195--1268], and uses the construction of the collection of continuum clusters in the scaling limit introduced by Camia et al. [Springer Proceedings in Mathematics \& Statistics, 299 (2019) 44--89].