Short-range correlations in percolation at criticality

الأصل · EN

We derive the critical nearest-neighbor connectivity gₙ as 3/4, 3(7-9pctri)/[4(5-4pctri)], and 3(2+7pctri)/[4(5-pctri)] for bond percolation on the square, honeycomb and triangular lattice respectively, where pctri=2(π/18) is the percolation threshold for the triangular lattice; and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as gnn=0.6875000(2), which confirms a conjecture by Mitra and Nienhuis in J. Stat. Mech. P10006 (2004), implying the exact value gnn=11/16. We also determine the connectivity on a free surface as gₙsurf=0.6250001(13) and conjecture that this value is exactly equal to 5/8. In addition, we find that at criticality, the connectivities depend on the linear finite size L as Lʸᵗ⁻ᵈ, and the associated specific-heat-like quantities Cₙ and Cnn scale as L²ʸᵗ⁻ᵈ (L/L₀), where d is the lattice dimensionality, yₜ=1/ν the thermal renormalization exponent, and L₀ a non-universal constant. We provide an explanation of this logarithmic factor in the theoretical framework reported recently by Vasseur et al. in J. Stat. Mech. L07001 (2012).