  Let $\boldsymbol{X}$ be a $d$-dimensional random array on $[n]$ whose entries
take values in a finite set $\mathcal{X}$, that is, $\boldsymbol{X}=\langle
X_s:s\in \binom{[n]}{d}\rangle$ is an $\mathcal{X}$-valued stochastic process
indexed by the set $\binom{[n]}{d}$ of all $d$-element subsets of
$[n]:=\{1,\dots,n\}$. We give easily checked conditions on $\boldsymbol{X}$
that ensure, for instance, that for every function $f\colon
\mathcal{X}^{\binom{[n]}{d}}\to\mathbb{R}$ that satisfies
$\mathbb{E}[f(\boldsymbol{X})]=0$ and $\|f(\boldsymbol{X})\|_{L_p}=1$ for some
$p>1$, the random variable $f(\boldsymbol{X})$ becomes concentrated after
conditioning it on a large subarray of $\boldsymbol{X}$. These conditions cover
several classes of random arrays with not necessarily independent entries.
Applications are given in combinatorics, and examples are also presented that
show the optimality of various aspects of the results.
