Thermodynamic Limit for Large Random Trees

TL;DR

This paper investigates the asymptotic behavior of Gibbs distributions on large random plane trees, deriving explicit limits for local neighborhoods and introducing an infinite tree model with Markov properties.

Contribution

It provides a rigorous analysis of the thermodynamic limit for Gibbs distributions on large random trees, including explicit limiting distributions and properties of the infinite tree model.

Findings

Convergence of local distributions to a limit as tree size grows

Explicit computation of the limiting distribution

Introduction of an infinite random tree with Markov properties

Abstract

We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation.