# On the threshold of spread-out voter model percolation

## TL;DR

This paper investigates the percolation properties of the spread-out voter model in high dimensions, showing that as the interaction range grows, the critical density for percolation converges to that of independent Bernoulli percolation.

## Contribution

It proves the convergence of the critical density in the spread-out voter model to the Bernoulli percolation threshold as the interaction range increases.

## Key findings

- Critical density $ o$ Bernoulli percolation threshold as $R o
$
- Established a new upper bound on joint event probabilities under $rac{lpha,R}$
- Demonstrated percolation transition behavior in high-dimensional spread-out voter models.

## Abstract

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 the critical probability for Bernoulli site percolation on $\mathbb{Z}^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu_{\alpha,R}$ which is of independent interest.