Random trees with superexponential branching weights

TL;DR

This paper investigates the behavior of rooted planar random trees with weights growing faster than exponentially, revealing convergence to a tree with a single infinite-degree vertex and providing detailed analysis for specific factorial-based weights.

Contribution

It introduces a new class of random trees with superexponential weights and characterizes their asymptotic structure and convergence properties.

Findings

Measures concentrate on a single infinite-degree vertex as size grows

Explicit results for weights of the form $w_n=((n-1)!)^eta$

Refined analysis of the approach to the infinite volume limit

Abstract

We study rooted planar random trees with a probability distribution which is proportional to a product of weight factors $w_n$ associated to the vertices of the tree and depending only on their individual degrees $n$. We focus on the case when $w_n$ grows faster than exponentially with $n$. In this case the measures on trees of finite size $N$ converge weakly as $N$ tends to infinity to a measure which is concentrated on a single tree with one vertex of infinite degree. For explicit weight factors of the form $w_n=((n-1)!)^\alpha$ with $\alpha >0$ we obtain more refined results about the approach to the infinite volume limit.