Percolation in the Harmonic Crystal and Voter Model in three dimensions

Abstract

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, $U=\frac{J}{2}{\sum }_{<x,y>}(\varphi (x)-\varphi (y){)}^2$, $x,y\in {\mathbb{Z}}^3$, $J>0$ and $\varphi (x)\in \mathbb{R}$, a scalar height variable, and define occupation variables ${\rho }_h(x)=1,(0)$ for $\varphi (x)>h(<h)$. 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.

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• On the threshold of spread-out voter model percolation