Percolation transitions in two dimensions

TL;DR

This study numerically analyzes percolation thresholds on various two-dimensional lattices, confirming theoretical predictions about scaling corrections and revealing lattice orientation effects on correction amplitudes.

Contribution

It provides new numerical estimates of percolation thresholds for several 2D lattices and investigates the nature of scaling corrections, including logarithmic factors, in these models.

Findings

Corrections to scaling follow the second temperature dimension $X_{t2}=4$ as predicted.

Evidence of additional logarithmic correction terms in some cases.

Lattice orientation influences the amplitude of power-law correction terms.

Abstract

We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension $X_{t2}=4$ predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice such a logarithmic term appears to be small or absent. The amplitude of the power-law correction associated with $X_{t2}=4$ is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.