Trajectories in random minimal transposition factorizations

TL;DR

This paper investigates the local structure of trajectories in random minimal factorizations of an n-cycle, revealing convergence to a labeled infinite Galton-Watson tree through a novel encoding method.

Contribution

It introduces a new local convergence theorem for trajectories in random minimal transposition factorizations of cycles, using an innovative tree encoding approach.

Findings

Trajectories converge locally to a labeled infinite Galton-Watson tree.

The encoding involves an edge and vertex-labeled tree that captures the factorization structure.

The convergence is established via a local exploration algorithm.

Abstract

We study random typical minimal factorizations of the $n$-cycle, which are factorizations of $(1, \ldots,n)$ as a product of $n-1$ transpositions, chosen uniformly at random. Our main result is, roughly speaking, a local convergence theorem for the trajectories of finitely many points in the factorization. The main tool is an encoding of the factorization by an edge and vertex-labelled tree, which is shown to converge to Kesten's infinite Bienaym\'e-Galton-Watson tree with Poisson offspring distribution, uniform i.i.d. edge labels and vertex labels obtained by a local exploration algorithm.