Short-range correlations in percolation at criticality.

Abstract

We derive the critical nearest-neighbor connectivity gn as 3/4, 3(7-9pc(tri))/4(5-4pc(tri)), and 3(2+7pc(tri))/4(5-pc(tri)) for bond percolation on the square, honeycomb, and triangular lattice, respectively, where pc(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 gnn=0.6875000(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 gn(surf)=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(yt-d), and the associated specific-heat-like quantities Cn and Cnn scale as ∼L(2yt-d)ln(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].