Short-range correlations in percolation at criticality

We derive the critical nearest-neighbor connectivity g_n as 3/4, 3(7-9p_c^tri)/4(5-4p_c^tri), and 3(2+7p_c^tri)/4(5-p_c^tri) for bond percolation on the square, honeycomb, and triangular lattice, respectively, where p_c^tri=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 g_nn=0.6875000(2), which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006, 10.1088/1742-5468/2004/10/P10006], implying the exact value g_nn=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 ∼L^y_t-d, and the associated specific-heat-like quantities C_n and C_nn scale as ∼L^2yt-dln(L /L0), where d is the lattice dimensionality, y_t=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, 10.1088/1742-5468/2012/07/L07001].