= Amenable boundary growth of lattice boxes
{title2=$\frac{|\partial^+B_n|}{|B_n|}=\frac{2d}{2n+1}\to0$}
For the nearest-neighbour graph on $\mathbb Z^d$, a box $B_n=[-n,n]^d\cap\mathbb Z^d$ has $(2n+1)^d$ <graph vertices> and outer <graph vertex> boundary of size $2d(2n+1)^{d-1}$. Thus $|\partial^+B_n|/|B_n|\to0$. This boundary-to-volume property is the geometric mechanism that contradicts a positive volume density of trifurcation points in the uniqueness proof. It realizes the boundary-growth property of the translation <amenable group> on the lattice.
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