  A $d$-dimensional random array on a nonempty set $I$ is a stochastic process
$\boldsymbol{X}=\langle X_s:s\in \binom{I}{d}\rangle$ indexed by the set
$\binom{I}{d}$ of all $d$-element subsets of $I$. We obtain structural
decompositions of finite, high-dimensional random arrays whose distribution is
invariant under certain symmetries.
  Our first main result is a distributional decomposition of finite,
(approximately) spreadable, high-dimensional random arrays whose entries take
values in a finite set; the two-dimensional case of this result is the finite
version of an infinitary decomposition due to Fremlin and Talagrand. Our second
main result is a physical decomposition of finite, spreadable, high-dimensional
random arrays with square-integrable entries that is the analogue of the
Hoeffding/Efron--Stein decomposition. All proofs are effective.
  We also present applications of these decompositions in the study of
concentration of functions of finite, high-dimensional random arrays.
