Concentration of discrete cubes at diameter scale ($d_n(x,y)=d_{Q_n}(x,y)/n$)

Equip the Boolean hypercube with uniform probability measure and graph distance divided by its graph diameter. These spaces form a Lévy family of graphs. The vertex-isoperimetric inequality in the discrete cube reduces the complement of a positive-density set's radius-$\epsilon n$ graph neighbourhood to a binomial distribution tail. For fixed density $\delta >0$, Chebyshev's inequality puts a lower density quantile at least $n/2-C_{\delta }\sqrt{n}-O(1)$. After enlarging by $\epsilon n$, the omitted mass is at most $n/[4(\epsilon n-C_{\delta }\sqrt{n}-O(1){)}^2]$, which tends to zero.