Concatenating Random Matchings

TL;DR

This paper studies the properties of concatenated random perfect matchings inspired by Brauer algebra operations, revealing phase transition phenomena and asymptotic behaviors of components and loops.

Contribution

It introduces a novel analysis of concatenated random matchings using renewal theory and vertex-exploration, providing new asymptotic and structural results.

Findings

Existence of a giant component only when n is odd

Asymptotic behavior of loops and component sizes as t approaches infinity

Local description of the giant component

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 $\mathfrak{B}_n(\delta)$. 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.