Failure of concentration in fixed-dimensional grids
= 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> $\{x:x_1\leq\lceil n/2\rceil\}$ has at least half the vertices. Its radius-$\varepsilon d(n-1)$ <graph neighbourhood> still omits a proportion tending to $1/2-\varepsilon d$ when $0<\varepsilon<1/(2d)$. These grids therefore do not form a <Lévy family of graphs>.