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

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.