  Compared to the classical binomial random (hyper)graph model, the study of
random regular hypergraphs is made more challenging due to correlations between
the occurrence of different edges. We develop an edge-switching technique for
hypergraphs which allows us to show that these correlations are limited for a
large range of densities. This extends some previous results of Kim, Sudakov
and Vu for graphs. From our results we deduce several corollaries on subgraph
counts in random $d$-regular hypergraphs. We also prove a conjecture of Dudek,
Frieze, Ruci\'nski and \v{S}ileikis on the threshold for the existence of an
$\ell$-overlapping Hamilton cycle in a random $d$-regular $r$-graph.
  Moreover, we apply our results to prove bounds on the query complexity of
testing subgraph-freeness. The problem of testing subgraph-freeness in the
general graphs model was first studied by Alon, Kaufman, Krivelevich and Ron,
who obtained several bounds on the query complexity of testing
triangle-freeness. We extend some of these previous results beyond the triangle
setting and to the hypergraph setting.
