Correlated percolation and the correlated resistor network

TL;DR

This paper investigates percolation properties in the Ising model with extended bond ranges, providing exact results, phase diagrams, and Monte Carlo simulations to understand the critical behavior of correlated resistor networks.

Contribution

It offers new exact results for percolation thresholds with extended bonds and detailed Monte Carlo analysis of the correlated resistor network.

Findings

Percolation threshold equals the Ising critical temperature for next-nearest neighbors.

Finite size behavior shows no corrections to scaling.

Thermal exponent of conductivity is approximately 0.2000.

Abstract

We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than nearest-neighbors. We show that for a percolation range to next-nearest neighbors the percolation threshold Tp is still equal to the Ising critical temperature Tc, and present the phase diagram for this type of percolation. In addition, we present Monte Carlo calculations of the finite size behavior of the correlated resistor network defined on the Ising model. The thermal exponent t of the conductivity that follows from it is found to be t = 0.2000 +- 0.0007. We observe no corrections to scaling in its finite size behavior.