Scaling limits of random P\'olya trees

TL;DR

This paper proves that large uniform Pólya trees with degree restrictions converge to the Continuum Random Tree and provides tail bounds for their height and width.

Contribution

It establishes the convergence of Pólya trees to the Continuum Random Tree and derives optimal tail bounds for their height and width.

Findings

Convergence of Pólya trees to the Continuum Random Tree

Sub-Gaussian tail bounds for height and width

Global shape dictated by large Galton-Watson subtree

Abstract

P\'olya trees are rooted trees considered up to symmetry. We establish the convergence of large uniform random P\'olya trees with arbitrary degree restrictions to Aldous' Continuum Random Tree with respect to the Gromov-Hausdorff metric. Our proof is short and elementary, and it shows that the global shape of a random P\'olya tree is essentially dictated by a large Galton-Watson tree that it contains. We also derive sub-Gaussian tail bounds for both the height and the width, which are optimal up to constant factors in the exponent.