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 $\mathfrak{B}_n(\delta)$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2023
- DOI:
- 10.48550/arXiv.2306.11596
- arXiv:
- arXiv:2306.11596
- Bibcode:
- 2023arXiv230611596B
- Keywords:
-
- Mathematics - Probability;
- Mathematics - Combinatorics;
- 60C05 (Primary);
- 05C30 (Secondary)
- E-Print:
- 30 pages, 3 figures