Short-range correlations in percolation at criticality

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=2sin(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=0.6870.28em0ex5000.28em0ex0(2), which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value gₙₙ=11/16. We also determine the connectivity on a free surface as gₙˢᵘʳᶠ=0.6250.28em0ex0000.28em0ex1(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 ~L^yₜ-d, and the associated specific-heat-like quantities Cₙ and Cₙₙ scale as ~L^2yₜ-dln(L/L₀), where d is the lattice dimensionality, yₜ=1/ν the thermal renormalization exponent, and L₀ a nonuniversal constant. We provide an explanation of this logarithmic factor within the theoretical framework reported recently by Vasseur et al. [J. Stat. Mech. (2012) L07001].