Scaling limits for a family of unrooted trees

TL;DR

This paper investigates the scaling limits of weighted unrooted plane trees with fixed diameter, demonstrating convergence to unrooted Levy trees under certain probabilistic conditions on the weights.

Contribution

It introduces a new weighted model for unrooted plane trees and establishes their convergence to unrooted Levy trees in the scaling limit.

Findings

Weighted unrooted plane trees converge to unrooted Levy trees

Scaling limits depend on the domain of attraction of the weight distribution

Results extend the understanding of tree scaling limits in probabilistic combinatorics

Abstract

We introduce weights on the unrooted unlabelled plane trees as follows: let $\mu$ be a probability measure on the set of nonnegative integers whose mean is no larger than $1$; then the $\mu$-weight of a plane tree $t$ is defined as $\Pi \, \mu (degree (v) -1)$, where the product is over the set of vertices $v$ of $t$. We study the random plane tree with a fixed diameter $p$ sampled according to probabilities proportional to these $\mu$-weights and we prove that, under the assumption that the sequence of laws $\mu_p$, $p\! \geq \! 1$, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of such random plane trees are random compact real trees called the unrooted Levy trees, which have been introduced in Duquense & Wang.