Failure of concentration in fixed-dimensional grids

Equip $[n{]}^d$ with uniform probability measure and its nearest-neighbour graph distance divided by $d(n-1)$, for fixed positive $d$. The set $\{x:x_1\le \lceil n/2\rceil \}$ has at least half the vertices. Its radius-$\epsilon d(n-1)$ graph neighbourhood still omits a proportion tending to $1/2-\epsilon d$ when $0<\epsilon <1/(2d)$. These grids therefore do not form a Lévy family of graphs.