We consider the $d$-dimensional voter model for $d \geq 3$. Our interest is the large scale limit of the equilibrium state of the voter model, where we prove the $d = 3$ results of  for $d \geq 4$, which turn out to be of a different nature than for $d = 3$. For this purpose we use the historical process.We establish some surprising facts about the Green’s function of random walks in dimension $d \geq 4$,which lead to the different features in $d = 3$ versus $d \geq 4$. Secondly, we prove an analogous result for the voter model on the hierarchical group.
"Renormalization of the Voter Model in Equilibrium." Ann. Probab. 29 (3) 1262 - 1302, July 2001. https://doi.org/10.1214/aop/1015345603