Sharp Phase Transition for the Formation of Infinite Tubes

Shu Kanazawa , Omer Bobrowski , Primoz Skraba

Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:19 UTC · model grok-4.3

classification 🧮 math.PR

keywords percolationtubesharptubularbondclassicalcriticalityevent

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The pith

Tube percolation exhibits sharp thresholds at criticality for infinite tube formation, proven via OSSS inequality and adapted exploration algorithms.

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

Classical percolation studies when random connections on a grid let a point link to infinity. This work shifts focus to tubes made from random plaquettes that can enclose and connect a fixed loop or sphere to infinity. The authors define a tubular one-arm event measuring the chance a tube reaches far away and show its probability falls off exponentially below a critical density but stays bounded away from zero above it. They also prove that tube crossings of large boxes appear sharply at the same threshold and establish an analogue of the unique infinite cluster property despite tubes not being transitive like paths.

Core claim

the tubular one-arm event exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. [...] the existence of a box-crossing tube also exhibits a sharp threshold.

Load-bearing premise

The OSSS inequality applies to the Boolean function defined by the tubular one-arm event under the adapted exploration algorithm that respects tube topology, without hidden obstructions from non-transitivity of tube connectedness.

read the original abstract

Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization of this question, asking whether a fixed loop (or, more generally, a topological sphere) can escape to infinity via a tube formed by random plaquettes. We refer to this phenomenon as tube percolation. We first compare tube percolation with previously studied higher-dimensional percolation phenomena, including face and cycle percolation. For tubes of codimension one, we further relate the critical probability for tube percolation to those for percolation of finite clusters and shielded percolation in the dual bond percolation model. Next, we introduce a tubular analogue of the classical one-arm event, the tubular one-arm event, and prove that it exhibits a sharp threshold at criticality: below criticality, its probability decays exponentially in scale, whereas above criticality, it admits a mean-field-type lower bound. The proof relies on the O'Donnell-Saks-Schramm-Servedio (OSSS) inequality together with an exploration algorithm adapted to the topology of tubes. Finally, we study the tubular box-crossing property. Unlike ordinary path connectedness, "tube connectedness" is not transitive, and thus there is no natural notion of clusters. Nevertheless, we establish an analogue of the uniqueness of the infinite cluster from classical bond percolation. Combining this result with the sharp threshold for the tubular one-arm event, we prove that the existence of a box-crossing tube also exhibits a sharp threshold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Tube percolation gets sharp thresholds via OSSS on a non-transitive topology, but the adaptation to tubes needs explicit verification that it preserves the inequality's hypotheses.

read the letter

The paper introduces tube percolation, where a fixed loop escapes to infinity through a tube of random plaquettes, and proves sharp thresholds for the tubular one-arm event and box-crossing property. Below criticality the one-arm probability decays exponentially; above it there is a mean-field lower bound. They also relate the critical probability for codimension-one tubes to dual bond percolation quantities and show an analogue of unique infinite cluster despite the fact that tube connectedness is not transitive. This is new relative to face and cycle percolation work, and the OSSS-plus-adapted-exploration approach is a reasonable way to get sharpness without relying on standard cluster arguments. The relations to shielded percolation and finite clusters in the dual look clean and definitional rather than circular. The main soft spot is exactly the one the stress-test flags: non-transitivity means local plaquette connections do not compose in the usual way, so it is not obvious that the exploration algorithm produces revealment probabilities whose squares sum to something small enough for OSSS to deliver the exponential decay. The abstract claims the adaptation works, but without seeing the explicit check that no hidden dependencies break the hypotheses, the sharpness claim rests on an unverified step. The box-crossing result then inherits the same uncertainty. This is a technical but contained extension of percolation theory. It is aimed at people working on higher-dimensional or topological variants of percolation and random media models. The ideas are coherent on their own terms and engage the literature properly, so it deserves a serious referee even if the proofs require some tightening around the exploration step. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper introduces tube percolation as a higher-dimensional generalization of classical bond percolation, in which a fixed loop (or topological sphere) escapes to infinity through a tube of random plaquettes. For codimension-one tubes it relates the critical probability to those of finite-cluster and shielded percolation in the dual bond model. It defines a tubular one-arm event and proves it exhibits a sharp threshold at criticality (exponential decay below pc, mean-field lower bound above) via the OSSS inequality applied to a topology-adapted exploration algorithm. It further establishes an analogue of uniqueness of the infinite cluster for box-crossing tubes despite the non-transitivity of tube connectedness, and deduces a sharp threshold for the existence of box-crossing tubes.

Significance. If the central claims hold, the work supplies the first rigorous sharp-threshold results for tube percolation, extending classical percolation theory to non-transitive topological connectedness and providing a template for applying OSSS-type inequalities to higher-dimensional topological events. The explicit comparison with dual bond models and the box-crossing uniqueness result are technically substantive contributions.

major comments (2)

[Proof of tubular one-arm sharp threshold] The section establishing the sharp threshold for the tubular one-arm event (the proof that relies on OSSS plus the adapted exploration algorithm): the manuscript must explicitly verify that the non-transitivity of tube connectedness does not produce hidden dependencies or topological obstructions that invalidate the revealment-probability bounds required by OSSS. Without this check the exponential-decay claim below criticality does not follow from the stated hypotheses.
[Comparison with dual bond percolation models] The paragraph relating the tube-percolation critical probability to the dual bond model (codimension-one case): the definitions of the finite-cluster and shielded critical probabilities must be shown to be independent of the tube event itself; any implicit dependence would render the comparison circular rather than a genuine reduction.

minor comments (2)

[Notation and definitions] Notation for the tubular one-arm event and the adapted exploration algorithm should be introduced with a short diagram or pseudocode to clarify how plaquette connections are revealed while respecting tube topology.
[Box-crossing uniqueness] The statement of the box-crossing uniqueness result would benefit from an explicit comparison table with the classical infinite-cluster uniqueness theorem, highlighting where transitivity is replaced by the weaker tube-connectedness relation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper introduces no free parameters or invented entities. It rests on standard percolation axioms and the domain assumption that OSSS applies to the tubular setting.

axioms (2)

standard math Independent Bernoulli occupation of plaquettes on the lattice follows the classical percolation model.
Foundational definition of the random environment for tube formation.
domain assumption The OSSS inequality holds for the Boolean function encoding the tubular one-arm event under the adapted exploration process.
Central to the sharp-threshold proof; invoked without additional verification in the abstract.

pith-pipeline@v0.9.0 · 5578 in / 1305 out tokens · 68477 ms · 2026-05-15T03:19:24.162217+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the tubular one-arm event exhibits a sharp threshold at criticality... The proof relies on the O’Donnell–Saks–Schramm–Servedio (OSSS) inequality together with an exploration algorithm adapted to the topology of tubes... Unlike ordinary path connectedness, “tube connectedness” is not transitive
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that there exists a critical probability p_tube_c ... θ_tube(p) ≥ κ(p - p_tube_c)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.