Correlated percolation and the correlated resistor network

We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest neighbour case, which implies that the percolation threshold, T-p, is still equal to the Ising critical temperature T-c. In addition, we present Monte Carlo calculations of the finite size behaviour 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 behaviour.