Cubic lattice (${\mathbb{Z}}^d$)

The cubic lattice in dimension $d$ has graph vertex set ${\mathbb{Z}}^d$ and an edge between $x,y$ exactly when ${\sum }_i|x_i-y_i|=1$. For $d=2$ it is the square lattice. Every graph vertex has $2d$ graph neighbours, so it is a locally finite graph.

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 $|{\partial }^{+}{B}_n|/|B_n|\to 0$. 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.