Percolation in the harmonic crystal and voter model in three dimensions

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=(J∕2)∑⟨x,y⟩[φ(x)-φ(y)]², x,y∊Z³, $J>0$, and φ(x)∊R, a scalar height variable, and define occupation variables ρₕ(x)=1 (0) for φ(x)>h $(<h)$. The probability p of a site being occupied is then a function of h. In the voter model we consider a 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 as x-y^-1. Using some Monte Carlo simulation methods and finite-size scaling we find accurate values of pc 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)~r^-a [Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent ν is very close to the predicted value of 2, supporting the conjecture by WH that ν=2/a is exact.