Quenched scaling limit of critical percolation clusters on Galton-Watson trees

Eleanor Archer; Tanguy Lions

Quenched scaling limit of critical percolation clusters on Galton-Watson trees

T0 review · 0 major / 3 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Critical percolation clusters on fixed supercritical Galton-Watson trees converge in the GHP topology to the corresponding α-stable tree.

desk verdict They prove the quenched GHP scaling limit of the critical percolation cluster is the alpha-stable tree, matching the annealed case, via quenched tail asymptotics. read the letter →

arxiv 2501.12088 v2 submitted 2025-01-21 math.PR

Eleanor Archer , Tanguy Lions This is my paper

classification math.PR

keywords quenchedpercolationGalton-WatsontreesscalinglimitsstablecriticalGromov-Hausdorff-Prokhorovrandomwalksonclustersclustersizetails

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the scaling limit of a critical percolation cluster, when the underlying Galton-Watson tree is held fixed, is the α-stable tree in the quenched setting. This matches the known annealed limit where the tree itself is random. The result applies when the offspring distribution is supercritical with either finite variance or α-stable tails for α in (1,2). It yields the same limit object whether the tree is quenched or annealed. As direct corollaries, random walks on the clusters converge to Brownian motion on the stable tree and the cluster-size tail admits explicit quenched asymptotics that complete prior partial results.

What carries the argument

Gromov-Hausdorff-Prokhorov scaling limit of the critical percolation cluster conditioned on a fixed supercritical Galton-Watson tree.

What would settle it

A single fixed supercritical Galton-Watson tree with the allowed offspring distribution for which the GHP distance between the suitably scaled percolation cluster and the α-stable tree does not converge to zero in probability.

Watch

Extended reading notes

Core claim

We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding α-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).

Load-bearing premise

The underlying Galton-Watson tree is supercritical with either finite variance offspring distribution or α-stable tails for α in (1,2).

Editorial extensions

If this is right

Simple random walk on the percolation cluster rescales to Brownian motion on the stable tree.
The tail of the cluster size admits explicit quenched asymptotics.
The quenched scaling limit coincides with the annealed scaling limit for the same offspring distribution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The additional randomness of the tree is not needed for the scaling limit once the offspring law is fixed.
The same quenched-to-annealed coincidence may hold for critical percolation on other classes of random trees.
The result supplies a deterministic-environment version of the stable-tree limit that can be used to study percolation in fixed but irregular media.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main thing here is that the paper gets the quenched GHP scaling limit of critical percolation clusters on supercritical Galton-Watson trees to be the alpha-stable tree, the same as in the annealed setting. They do this under the usual conditions on the offspring distribution: finite variance or alpha-stable tails for alpha in (1,2). Along the way they finish off the quenched tail asymptotics for cluster size that earlier papers left partial. The argument runs through those tail bounds to get convergence of the height or contour process, then the GHP limit, plus the random walk corollary. That extension from annealed to quenched is the actual new piece, and it looks like a direct, non-circular step that builds on Michelen and Archer-Vogel without obvious gaps. The conditions on the tree are standard for stable limits, so they are not a flaw, just the setting where the result holds. No load-bearing assumptions seem invented or fitted. The whole thing is technical but internally consistent based on the outline. This is for specialists already working on scaling limits for processes on random trees or percolation in random media. A reader who knows the annealed results would pick up the quenched strengthening and the tail asymptotics without much extra background. It is narrow enough that most people outside the subfield will not need it, but the result is a clean completion of prior work. I would send it to peer review because the claim is sharp, the approach is standard for the area, and the steps appear verifiable.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that, for a supercritical Galton-Watson tree with either finite-variance offspring distribution or α-stable tails (α ∈ (1,2)), the quenched Gromov-Hausdorff-Prokhorov scaling limit of a critical percolation cluster is the corresponding α-stable tree, matching the known annealed limit. The argument proceeds by establishing quenched tail asymptotics for the cluster size (extending Michelen 2019 and Archer-Vogel 2024) and then showing convergence of the height or contour process to the stable height process; a corollary on the rescaling of simple random walk on the cluster to Brownian motion on the stable tree is also obtained.

Significance. If the central claims hold, the result is significant because it establishes robustness of the scaling limit under the quenched measure, a distinction that is often delicate in random-media settings. The work supplies the missing quenched tail asymptotics and thereby completes the program initiated in the cited earlier papers; the direct use of height-process convergence supplies a clean link to the stable-tree limit and yields the random-walk corollary without additional machinery.

minor comments (3)

[Abstract] The abstract states the main theorem but does not indicate the two-step strategy (quenched tails followed by height-process convergence); a single sentence outlining this route would improve readability for readers outside the immediate area.
[§1] Notation for the quenched probability measure P^ω and the associated scaling constants should be introduced once in §1 and then used uniformly; occasional switches between P and P^ω in the early sections can be confusing.
[Figure 1] Figure 1 (schematic of the percolation cluster) would benefit from an explicit caption stating the value of α and the offspring distribution used for the simulation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance in establishing the quenched scaling limit, and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper establishes the quenched GHP scaling limit of critical percolation clusters on supercritical Galton-Watson trees (finite variance or α-stable tails) equals the α-stable tree via quenched tail asymptotics for cluster size followed by convergence of the height/contour process. These steps are derived directly from the offspring distribution assumptions and standard branching process techniques, without reducing to fitted parameters, self-definitions, or load-bearing self-citations. The reference to completing Michelen (2019) and Archer-Vogel (2024) extends prior tail results but does not make the scaling limit tautological or equivalent to inputs by construction. The argument remains externally falsifiable and independent of the present paper's fitted values.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract-only review; the paper relies on standard domain assumptions of Galton-Watson processes and percolation theory, with no free parameters or invented entities visible.

assumptions (1)

domain assumption Standard setup and properties of supercritical Galton-Watson trees and critical percolation thereon
Invoked throughout the abstract as the setting for the quenched scaling limit.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Quenched scaling limit of critical percolation clusters on Galton-Watson trees." pith.science (2026). https://pith.science/paper/2501.12088

@misc{pith2026250112088,
  author       = {Pith},
  title        = {Pith review of: Quenched scaling limit of critical percolation clusters on Galton-Watson trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2501.12088}},
  note         = {Machine review of arXiv:2501.12088}
}

abstract

We consider quenched critical percolation on a supercritical Galton--Watson tree with either finite variance or $\alpha$-stable offspring tails for some $\alpha \in (1,2)$. We show that the GHP scaling limit of a quenched critical percolation cluster on this tree is the corresponding $\alpha$-stable tree, as is the case in the annealed setting. As a corollary we obtain that a simple random walk on the cluster also rescales to Brownian motion on the stable tree. Along the way, we also obtain quenched asymptotics for the tail of the cluster size, which completes earlier results obtained in Michelen (2019) and Archer-Vogel (2024).

Discussion (0). Sign in to comment.

Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.