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 $p_c^{tri}=2\sin(\pi/18)$ 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 $g_{nn}=0.687\;500\;0(2)$, which confirms a conjecture by Mitra and Nienhuis in J. Stat. Mech. P10006 (2004), implying the exact value $g_{nn}=11/16$. We also determine the connectivity on a free surface as $g_n^{surf}=0.625\;000\;1(13)$ 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 $\sim L^{y_t-d}$, and the associated specific-heat-like quantities $C_n$ and $C_{nn}$ scale as $\sim L^{2y_t-d} \ln (L/L_0)$, where $d$ is the lattice dimensionality, $y_t=1/\nu$ the thermal renormalization exponent, and $L_0$ 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).