Concatenating random matchings

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 B_n(δ). For the resulting random string diagram Br_n(t), we observe a giant component if and only if n is odd, and as t → ∞ 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 Br_n(t) by random vertex-exploration processes.