Abstract
We derive the critical nearest-neighbor connectivity as , , and for bond percolation on the square, honeycomb, and triangular lattice, respectively, where 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 , which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value . We also determine the connectivity on a free surface as and conjecture that this value is exactly equal to . In addition, we find that at criticality, the connectivities depend on the linear finite size as , and the associated specific-heat-like quantities and scale as , where is the lattice dimensionality, the thermal renormalization exponent, and 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].
- Received 24 July 2014
DOI:https://doi.org/10.1103/PhysRevE.90.042106
©2014 American Physical Society