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Perturbations to bond retention probabilities on trees preserve whether infinite clusters exist under quantitative conditions on the factors.

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2026-07-03 18:57 UTC pith:7LDA2F5K

load-bearing objection The paper gives quantitative stability for percolation on trees under distance-dependent edge perturbations, with a limited application to the Erdős similarity conjecture.

arxiv 2607.01291 v1 pith:7LDA2F5K submitted 2026-07-01 math.PR math-phmath.CAmath.MGmath.MP

On perturbations that preserve the connectivity properties in tree percolations

Mirmukhsin Makhmudov , Ville Suomala This is my paper
classification math.PR math-phmath.CAmath.MGmath.MP
keywords bond percolationinfinite treesperturbationsinfinite clustersconnectivityErdős similarity conjectureCantor sets
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies a general bond percolation model on an infinite locally finite tree and considers replacing each edge retention probability p_e by min{1, q_|e| p_e}, where |e| is the distance from the root and {q_n} is a sequence of positive factors. It establishes that if the original model has infinite clusters, then for sequences with 0 < q_n ≤ 1 that satisfy suitable decay bounds the perturbed model retains infinite clusters. Conversely, if the original model has only finite clusters, then for sequences with q_n ≥ 1 that satisfy suitable growth bounds the perturbed model retains the absence of infinite clusters. These stability statements hold under minimal assumptions on the base percolation. The results are also used to address an instance of the Erdős similarity conjecture for Cantor sets.

Core claim

For a general bond percolation on an infinite locally finite tree, the property of admitting or not admitting infinite clusters remains unchanged when the retention probabilities are replaced by min{1, q_|e| p_e} provided the sequence {q_n} obeys quantitative conditions that keep the perturbation non-trivial yet controlled in its effect on connectivity.

What carries the argument

the distance-to-root multiplicative perturbation min{1, q_|e| p_e} applied to the original retention probabilities

Load-bearing premise

The starting point is an arbitrary bond percolation on an infinite locally finite tree.

What would settle it

An explicit tree, edge probabilities p_e, and sequence q_n obeying the paper's quantitative bound for which the perturbed model changes the infinite-cluster status of the original model.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Existence of infinite clusters is preserved by downward perturbations q_n ≤ 1 that decay sufficiently slowly.
  • Absence of infinite clusters is preserved by upward perturbations q_n ≥ 1 that grow sufficiently slowly.
  • The stability applies directly to questions of similarity between Cantor sets via the Erdős conjecture.
  • The same conclusions hold for any base percolation satisfying only the minimal assumptions stated in the setup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same perturbation technique could be tested on regular trees or other graphs with bounded degree to check whether the stability carries over.
  • Quantitative bounds on q_n might be sharpened for specific percolation models such as regular trees with constant p.
  • The approach may supply a method for constructing families of Cantor sets that are similar yet arise from perturbed percolation measures.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The paper considers general bond percolation on an infinite locally finite tree with edge retention probabilities p_e. It studies the effect of replacing p_e by min{1, q_|e| p_e} (for 0<q_n≤1) when the original model percolates, or the reverse (for q_n≥1) when it does not. Under minimal assumptions on the base model, the existence or non-existence of infinite clusters is shown to be stable for perturbation sequences {q_n} satisfying explicit summability/decay conditions. An application to the Erdős similarity conjecture for Cantor sets is discussed.

Significance. If the results hold, they establish quantitative robustness of percolation connectivity on trees under distance-from-root perturbations, extending standard branching-number or survival-probability comparisons in a controlled way. The minimal-assumption framing increases applicability, and the link to the Erdős conjecture provides a concrete bridge to geometric measure theory. The tree setting permits direct, non-asymptotic arguments that are often unavailable on general graphs.

minor comments (3)
  1. §1, paragraph after Definition 1.1: the phrase 'minimal assumptions' is used without an immediate forward reference to the precise conditions (e.g., on the branching number or on the survival probability) that are actually required; a one-sentence pointer would improve readability.
  2. Theorem 2.3 (or the main stability theorem): the summability condition on {q_n} is stated in terms of a series involving log(1/q_n), but the proof sketch does not explicitly verify that the same series controls the change in the effective branching number; adding a short displayed inequality linking the two would clarify the argument.
  3. §4 (application to Erdős conjecture): the statement of the conjecture itself is omitted; a one-line reminder of what the similarity conjecture asserts would make the application self-contained for readers outside fractal geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our work on the stability of percolation connectivity under distance-dependent perturbations on trees, as well as the link to the Erdős similarity conjecture. We appreciate the recommendation for minor revision. No major comments were listed in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes stability of percolation connectivity on infinite locally finite trees under quantitative perturbations of edge probabilities via min{1, q_|e| p_e} (or reverse). The central claims are presented as theorems derived directly from the base model under minimal assumptions on the original percolation, with explicit summability conditions on the perturbation sequence {q_n}. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or setup; the tree structure permits direct comparison of branching numbers and survival probabilities using standard percolation techniques. The derivation is self-contained against the stated model inputs without reduction to its own outputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard setup of bond percolation on trees and basic facts from probability and graph theory; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Bond percolation is defined on an infinite locally finite tree with given edge retention probabilities p_e.
    This is the starting model before any perturbation is applied.

pith-pipeline@v0.9.1-grok · 5701 in / 1195 out tokens · 32343 ms · 2026-07-03T18:57:59.204326+00:00 · methodology

0 comments
read the original abstract

We consider a general bond percolation on an infinite locally finite tree, where the edge retention probabilities $p_e$ are replaced by $\min\{1,q_{|e|}p_e\}$, where $\{q_n\}_{n\ge 1}$ is a sequence of positive perturbation factors and $|e|$ denotes the distance between the edge $e$ and the root. If the original percolation model admits infinite clusters, it is of interest to investigate under which perturbations $0<q_n\le 1$ this connectivity property is preserved. Conversely, if the original percolation does not admit infinite clusters, we are led to study the stability of such a property under perturbations satisfying $q_n\ge 1$. In both cases, under minimal assumptions on the original model, we show that the percolative behaviour is stable against certain quantitative non-trivial perturbations. We also discuss an application of our results to the Erd\H{o}s similarity conjecture for Cantor sets.

discussion (0)

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Reference graph

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