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

2021/04/02 by Yao, Chang-Long
#60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2104.01211

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∈(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].

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