  We consider the concatenation of $t$ uniformly random perfect matchings on
$2n$ vertices, where the operation of concatenation is inspired by the
multiplication of generators of the Brauer algebra $\mathfrak{B}_n(\delta)$.
For the resulting random string diagram $\mathsf{Br}_n(t)$, we observe a giant
component if and only if $n$ is odd, and as $t\to\infty$ we obtain asymptotic
results concerning the number of loops, the size of the giant component, and
the number of loops of a given shape. Moreover, we give a local description of
the giant component. These results mainly rely on the use of renewal theory and
the coding of connected components of $\mathsf{Br}_n(t)$ by random
vertex-exploration processes.
