Thermal Phase Transition of the Dilute s -State Potts and n -Vector Models at the Percolation Threshold

Abstract

A general theory is given for the quenched dilute $s$-state Potts and $n$-vector models in any dimension $d$. It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any $s$ and $d$. For the $n$-vector model ($n>1$), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.