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

Abstract

We study Bernoulli first-passage percolation (FPP) on the triangular lattice 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 et al. (Ann Probab 44:1013–1052, 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.