Corrections to scaling for percolative conduction: anomalous behavior at small L

Recently Grassberger has shown that the correction to scaling for the conductance of a bond percolation network on a square lattice is a nonmonotonic function of the linear lattice dimension with a minimum at $L = 10$, while this anomalous behavior is not present in the site percolation networks. We perform a high precision numerical study of the bond percolation random resistor networks on the square, triangular and honeycomb lattices to further examine this result. We use the arithmetic, geometric and harmonic means to obtain the conductance and find that the qualitative behavior does not change: it is not related to the shape of the conductance distribution for small system sizes. We show that the anomaly at small L is absent on the triangular and honeycomb networks. We suggest that the nonmonotonic behavior is an artifact of approximating the continuous system for which the theory is formulated by a discrete one which can be simulated on a computer. We show that by slightly changing the definition of the linear lattice size we can eliminate the minimum at small L without significantly affecting the large L limit.Comment: 3 pages, 4 figures;slightly expanded, 2 figures added. Accepted for publishing in Phys. Rev.

Order parameter for two-dimensional critical systems with boundaries

Conformal transformations can be used to obtain the order parameter for two-dimensional systems at criticality in finite geometries with fixed boundary conditions on a connected boundary. To the known examples of this class (such as the disk and the infinite strip) we contribute the case of a rectangle. We show that the order parameter profile for simply connected boundaries can be represented as a universal function (independent of the criticality model) raised to the power eta/2. The universal function can be determined from the Gaussian model or equivalently a problem in two-dimensional electrostatics. We show that fitting the order parameter profile to the theoretical form gives an accurate route to the determination of eta. We perform numerical simulations for the Ising model and percolation for comparison with these analytic predictions, and apply this approach to the study of the planar rotor model.Comment: 10 pages, 14 figures. Revisions: Removed many typos, improved presentation of most of the figure