Asymptotics for 2D critical and near-critical first-passage percolation

Abstract

We study Bernoulli first-passage percolation (FPP) on the triangular lattice $\mathbb{T}$ in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. Denote by ${\mathcal{C}}_{\infty }$ the infinite cluster with 0-time sites when $p>p_c$, where $p_c=1/2$ is the critical probability. Denote by $T(0,{\mathcal{C}}_{\infty })$ the passage time from the origin 0 to ${\mathcal{C}}_{\infty }$. First we obtain explicit limit theorem for $T(0,{\mathcal{C}}_{\infty })$ as $p\searrow p_c$. The proof relies on the limit theorem in the critical case, the critical exponent for correlation length and Kesten's scaling relations. Next, for the usual point-to-point passage time $a_{0,n}$ in the critical case, we construct subsequences of sites with different growth rate along the axis. The main tool involves the large deviation estimates on the nesting of CLE${}_6$ loops derived by Miller, Watson and Wilson (2016). Finally, we apply the limit theorem for critical Bernoulli FPP to a random graph called cluster graph, obtaining explicit strong law of large numbers for graph distance.

35 pages, 3 figures

References in corpus (1)

• Conformal radii for conformal loop ensembles

Cited by in corpus (1)

• Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman