  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.
