Percolation thresholds in Ising magnets and conducting mixtures

Abstract

The effect of the percolation threshold on conduction in mixtures and on the magnetic ordering temperature in alloys of high- and low- ${T}_{C}$ material is described using a real-space rescaling approximation (introduced by Kadanoff, following an idea of Migdal) which is applicable to systems of arbitrary spatial dimensionality $d$. We calculate exponents describing the regimes just above and just below ${p}_{c}$ for the case of bond dilution in resistor networks or Ising ferromagnets. The results are accurate for $d=2$ in those cases where comparison is possible and qualitatively correct for $d=3$, but are invalid in the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Ising spin glasses do not show order in 2 dimensions, in this approximation, but may do so in 3 dimensions.