Concatenating Random Matchings

Abstract

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 ${\mathfrak{B}}_n(\hat{I}´)$. 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.

31 pages, 3 figures