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We investigate the site percolation transition in two strongly correlatedsystems 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 defineoccupation 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 weconsider the stationary measure, in which each site is either occupied orempty, with probability $p$. In both cases the truncated pair correlation ofthe occupation variables, $G(x-y)$, decays asymptotically like $|x-y|^{-1}$.Using some novel Monte Carlo simulation methods and finite size scaling we findaccurate values of $p_c$ as well as the critical exponents for these systems.The latter are different from that of independent percolation in $d=3$, asexpected from the work of Weinrib and Halperin [WH] for the percolationtransition 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 WHthat $\nu= \frac{2}{a}$ is exact.Comment: 8 figures. new version significantly different from the old one,  includes new results, figures et

Percolation in the harmonic crystal and voter model in three dimensions