On the shape of random P\'olya structures

TL;DR

This paper refines the understanding of the structure of random Pólya trees, showing that attached forests are of size Θ(log n), providing combinatorial interpretations, and extending results to other Pólya structures.

Contribution

It improves existing results by precisely characterizing forest sizes, offers a combinatorial interpretation of weights, and extends findings to broader Pólya structures.

Findings

Attached forests are of size Θ(log n) with high probability.

Derived the limit probability for the size of attached forests.

Extended structural results to other Pólya structures.

Abstract

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random P\'olya trees: a uniform random P\'olya tree of size $n$ consists of a conditioned critical Galton-Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n(v)$, that is attached to a node $v$ in $C_n$, is maximally of size $\vert F_n(v)\vert=O(\log n)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\vert F_n(v)\vert$, namely $\vert F_n(v)\vert=\Theta(\log n)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given P\'olya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n(v)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other P\'olya structures.