  We investigate the site percolation transition in two strongly correlated
systems in three dimensions: the massless harmonic crystal and the voter model.
In the first case we start with a Gibbs measure for the potential,
$U=\frac{J}{2} \sum_{<x,y>} (\phi(x) - \phi(y))^2$, $x,y \in \mathbb{Z}^3$, $J
> 0$ and $\phi(x) \in \mathbb{R}$, a scalar height variable, and define
occupation variables $\rho_h(x) =1,(0)$ for $\phi(x) > h (<h)$. The probability
$p$ of a site being occupied, is then a function of $h$. In the voter model we
consider the stationary measure, in which each site is either occupied or
empty, with probability $p$. In both cases the truncated pair correlation of
the occupation variables, $G(x-y)$, decays asymptotically like $|x-y|^{-1}$.
Using some novel Monte Carlo simulation methods and finite size scaling we find
accurate values of $p_c$ as well as the critical exponents for these systems.
The latter are different from that of independent percolation in $d=3$, as
expected from the work of Weinrib and Halperin [WH] for the percolation
transition of systems with $G(r) \sim r^{-a}$ [A. Weinrib and B. Halperin,
Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent
$\nu$ is very close to the predicted value of 2 supporting the conjecture by WH
that $\nu= \frac{2}{a}$ is exact.
