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On the threshold of spread-out voter model percolation. (arXiv:1705.06244v1 [math.PR])
来源于:arXiv
In the $R$-spread out, $d$-dimensional voter model, each site $x$ of
$\mathbb{Z}^d$ has state (or 'opinion') 0 or 1 and, with rate 1, updates its
opinion by copying that of some site $y$ chosen uniformly at random among all
sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal)
stationary measures of this model is given by a family $\mu_{\alpha, R}$, where
$\alpha \in [0,1]$. Configurations sampled from this measure are polynomially
correlated fields of 0's and 1's in which the density of 1's is $\alpha$ and
the correlation weakens as $R$ becomes larger. We study these configurations
from the point of view of nearest neighbor site percolation on $\mathbb{Z}^d$,
focusing on asymptotics as $R \to \infty$. In \cite{RV15}, we have shown that,
if $R$ is large, there is a critical value $\alpha_c(R)$ such that there is
percolation if $\alpha > \alpha_c(R)$ and no percolation if $\alpha <
\alpha_c(R)$. Here we prove that, as $R \to \infty$, $\alpha_c(R)$ converges to
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