Randomly dilutedxyand resistor networks near the percolation threshold

A formulation based on that of Stephen for randomly diluted systems near the percolation threshold is analyzed in detail. By careful consideration of various limiting procedures, a treatment of xy spin models and resistor networks is given which shows that previous calculations (which indicate that these systems having continuous symmetry have the same crossover exponents as the Ising model) are in error. By studying the limit wherein the energy gap goes to zero, we exhibit the mathematical mechanism which leads to qualitatively different results for xy-like as contrasted to Ising-like systems. A distinctive feature of the results is that there is an infinite sequence of crossover exponents needed to completely describe the probability distribution for R(x,x’), the resistance between sites x and x’. Because of the difference in symmetry between the xy model and the resistor network, the former has an infinite sequence of crossover exponents in addition to those of the resistor network. The first crossover exponent ${\phi }_1$=1+ε/42 governs the scaling behavior of R(x,x’) with ‖x-x’‖≡r: [R(x,x’ ${\left)\right]}_c$∼${\mathrm{x}}^{{\phi }_1/\nu }$, where [ ${]}_c$ indicates a conditional average, subject to x and x’ being in the same cluster, ν is the correlation length exponent for percolation, and ε=6-d, where d is the spatial dimensionality.