The quasi-isometry classes of Galton--Watson trees
Abstract
We classify, up to quasi-isometry, the large-scale geometry of Galton--Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the \emph{ray class}, the \emph{full tree class} (which includes the binary tree), one \emph{chain class} (CΛ) for every possible branching semigroup Λ, and the \emph{bushy} class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely \textit{not} quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on {1,2}, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving D-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.
Related
- Infinite-mean Galton-Watson trees have zero random-interchange threshold
- Branching Random Walks on Multi-Type Galton–Watson Trees: A Multifractal Analysis
- Behaviour near explosion in continuous-time Galton-Watson trees
- Biased random walk on a locally infinite Galton-Watson tree
- Asymptotic behavior of some strongly critical decomposable multi-type Galton-Watson processes with immigration
BibTeX
@misc{athreya2026quasiisometry,
title = {{The quasi-isometry classes of Galton--Watson trees}},
author = {Jayadev S. Athreya and Sascha Troscheit},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23882},
eprint = {2609.23882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23882},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The quasi-isometry classes of Galton--Watson trees
AU - Jayadev S. Athreya
AU - Sascha Troscheit
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.23882
UR - https://arxiv.org/abs/2609.23882
AB - We classify, up to quasi-isometry, the large-scale geometry of Galton--Watson trees for every finitely supported offspring distribution. Apart from the trivial finite diameter regimes, we condition on infinite diameter. We find that the remaining classes are the \emph{ray class}, the \emph{full tree class} (which includes the binary tree), one \emph{chain class} (C_Λ) for every possible branching semigroup Λ, and the \emph{bushy} class. Two independent trees almost surely admit a root-preserving quasi-isometry when their offspring distributions belong to the same class and are almost surely \textit{not} quasi-isometric when they belong to different classes. Any two survival-conditioned supercritical realisations with finitely supported offspring laws nevertheless a.s.~admit quasi-isometric embeddings in both directions. We prove that the class of a realisation depends only on the support of the offspring distribution, not its specific distribution. For offspring distributions supported on {1,2}, we additionally obtain an explicit exponential tail bound for the probability of non-existence of a root-preserving D-quasi-isometry. The classification also implies that the corresponding random Cantor boundaries are almost surely quasisymmetrically equivalent. Conditioned on nonextinction, this applies to strongly separated fractal percolation, even when the underlying self-similar iterated function systems and retention parameters differ. We also classify two families of trees with continuous random branching times. Our proofs use new automorphism-matching theorems for random graph labellings of trees, based on contraction estimates for mismatch potentials. These matching results for Markov labellings on the binary tree may be of independent interest. The quasi-isometry classification, embeddability, and matching theorems are formally verified in Lean.
ER -