Long-range correlated percolation

Abstract

This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as $x^{-a}$ for large distances $x$.. For $a<d$ the correlations are relevant if $a\nu -2<0\mathrm{}$ (where $\nu$ is the percolation correlation-length exponent), while for $a>d$ the correlations are relevant if $d\nu -2<0\mathrm{}$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\nu }_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a $q$-state Potts model with quenched disorder in the limit $q\to 1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\epsilon =6-d$ and in $\delta =4-a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\nu }_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\eta }_{\mathrm{long}}=\frac{1}{11}(\delta -\epsilon )$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.