Scaling limits for a family of unrooted trees

2016/04/28 by Minmin Wang, Wang, Minmin
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.1604.08287

25 pages, 4 figures

openalex publication_date 2016/04/28 · arxiv created 2016/07/30 · arxiv updated 2016/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce weights on the unrooted unlabelled plane trees as follows: let μ be a probability measure on the set of nonnegative integers whose mean is no larger than 1; then the μ-weight of a plane tree t is defined as Π μ(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 μ-weights and we prove that, under the assumption that the sequence of laws μ_p, p ≥ 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.

Citations

• The coding of compact real trees by real valued functions

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