Long Range Correlated Percolation

Abstract

In this note we study the field theory of dynamic isotropic percolation (DIP) with quenched randomness that has long range correlations decaying as $r^{-a}$. We argue that the quasi static limit of this field theory describes the critical point of long range correlated percolation. We perform a one loop double RG expansion in $\epsilon=6-d$, d the spacial dimension, and $\delta = 4-a$ and calculate both the static exponents and the dynamic exponent corresponding to the long range stable fixed point. The results for the static exponents as well as the region of stability for this long range fixed point agree with the results from a previous work on the subject that used a different representation of the problem \cite{aweinrib}. For the special case $\delta = \epsilon$ we perform a two loop calculation. We confirm that the scaling relation $\nu = \frac{2}{a}$, $\nu$ is the correlation length critical exponent, is satisfied to two loop order. Simulation results for the spreading exponent in $d=3$ differ significantly from the value we obtain after Pade-Borel resummation was performed on the $\epsilon$ expansion result. This is in sharp contrast with the result of a two loop $\epsilon$ expansion for the spreading exponent for DIP where there is a very good agreement with results from simulations for $d \geq 2$.