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 [1] 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.