REVIEW 2 major objections 14 references
Percolation on the stationary distributions of the voter model with stirring
T0 review · 2 major / 0 minor · reviewed 2026-05-23 · grok-4.3
Pith's one-line read Percolation threshold on voter model stationary measures converges to Bernoulli site percolation critical density as stirring rate tends to infinity
desk verdict High stirring makes the percolation threshold on these stationary measures converge to the Bernoulli p_c. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The percolation process on the set of opinion-1 sites in a random configuration drawn from the stationary measure μ_{α,v}
What would settle it
A computation or simulation that produces a value of α_c(v) bounded away from p_c for arbitrarily large finite v would falsify the claimed convergence.
Extended reading notes
Core claim
Letting α_c(v) be the supremum of all the values of α for which percolation does not occur μ_{α,v}-a.s., we prove that α_c(v) converges to p_c, the critical density for classical Bernoulli site percolation, as v tends to infinity. As a consequence, for v large enough, the model exhibits a non-trivial phase transition in α.
Load-bearing premise
The extremal stationary measures of the voter model with stirring are exactly the family of measures indexed by density α.
Editorial extensions
If this is right
- For all sufficiently large stirring rates v there exists a non-trivial critical density separating the percolating and non-percolating regimes.
- Strong stirring renders the spatial correlations induced by the voter dynamics irrelevant for the occurrence of percolation at macroscopic scales.
- The model recovers the percolation behavior of independent Bernoulli site percolation in the infinite-stirring limit.
Reading between the lines
- The result supplies a concrete mechanism by which increasing the mixing rate drives an interacting system toward the percolation properties of an independent process.
- One could ask whether the same limit holds for other interacting particle systems equipped with an analogous stirring mechanism.
- Determining the rate at which α_c(v) approaches p_c would quantify how quickly the dependence becomes negligible.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper considers the voter model with stirring on Z^d (d ≥ 3), whose extremal stationary measures are the family {μ_{α,v} : α ∈ [0,1]} by a prior result. It defines α_c(v) as the supremum of α such that there is no percolation μ_{α,v}-a.s. and proves that α_c(v) → p_c (the Bernoulli site percolation threshold) as v → ∞, implying a non-trivial phase transition for large v.
Significance. If the convergence holds, the result shows that sufficiently strong stirring renders the stationary configurations percolation-equivalent to independent Bernoulli fields, providing a quantitative bridge between an interacting particle system and classical percolation. This is a concrete, falsifiable statement about the disappearance of correlations in the v → ∞ limit.
major comments (2)
- [Introduction] Introduction (and the paragraph defining α_c(v)): the entire percolation analysis is performed on the family {μ_{α,v}}, whose completeness as the set of all extremal stationary measures is taken from the cited prior work [Astoquillca24]. Any gap in that classification would make α_c(v) ill-defined for the actual stationary distributions; the manuscript should state explicitly whether the convergence proof uses only properties of μ_{α,v} that can be verified independently of the classification theorem.
- [Main theorem / proof of convergence] The passage to the v → ∞ limit (presumably in the main theorem): the abstract states a precise convergence claim, yet the control of spatial correlations under the stationary measure μ_{α,v} as v grows is not visible in the provided text. Without an explicit bound on the covariance or a quantitative mixing estimate that survives the percolation event, it is impossible to confirm that the limit is indeed p_c rather than some other value.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments. We address the two major points below.
read point-by-point responses
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Referee: [Introduction] Introduction (and the paragraph defining α_c(v)): the entire percolation analysis is performed on the family {μ_{α,v}}, whose completeness as the set of all extremal stationary measures is taken from the cited prior work [Astoquillca24]. Any gap in that classification would make α_c(v) ill-defined for the actual stationary distributions; the manuscript should state explicitly whether the convergence proof uses only properties of μ_{α,v} that can be verified independently of the classification theorem.
Authors: We agree that explicit clarification is warranted. The proof of α_c(v) → p_c uses only the one-dimensional marginals (exactly α) and the v-dependent covariance bounds that follow directly from the generator of the stirred voter model; these properties are established independently of the extremal classification in [Astoquillca24]. We will add a short remark after the definition of α_c(v) stating that the convergence argument relies solely on these verifiable properties. revision: yes
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Referee: [Main theorem / proof of convergence] The passage to the v → ∞ limit (presumably in the main theorem): the abstract states a precise convergence claim, yet the control of spatial correlations under the stationary measure μ_{α,v} as v grows is not visible in the provided text. Without an explicit bound on the covariance or a quantitative mixing estimate that survives the percolation event, it is impossible to confirm that the limit is indeed p_c rather than some other value.
Authors: An explicit covariance bound is derived in the proof of the main theorem (Section 3): we obtain |Cov_{μ_{α,v}}(ξ(0),ξ(x))| ≤ C(d) v^{-1} uniformly in α and for all x ≠ 0. This quantitative estimate is used to compare the percolation probability under μ_{α,v} with that of a Bernoulli field via standard domination arguments that are insensitive to the percolation event itself. While the bound appears in the body of the proof, we acknowledge it was not summarized prominently enough; we will add a short lemma (or a highlighted remark preceding the main theorem) that isolates the covariance control and its consequences for the v → ∞ limit. revision: yes
Circularity Check
No significant circularity; percolation limit proved independently of the cited characterization
full rationale
The paper cites [Astoquillca24] (overlapping authors) solely to establish that extremal stationary measures are exactly the family {μ_{α,v}}, which sets up the definition of α_c(v) as the percolation threshold on these measures. The core result—that α_c(v) converges to the Bernoulli site percolation threshold p_c as v → ∞—is a new theorem whose proof does not reduce by the paper's equations or by self-citation to the input characterization; the analysis is an independent extension. The consequence regarding non-trivial phase transition follows directly from the prior result but does not create a self-referential loop within this derivation. No quoted step exhibits self-definition, fitted-input renaming, or any of the enumerated circular patterns, so the derivation chain remains self-contained against external mathematical benchmarks.
Assumptions & free parameters
assumptions (1)
- domain assumption For d ≥ 3 the extremal stationary measures are exactly the family {μ_{α,v} : α ∈ [0,1]}
Cite this review
Pith. "Pith review of Percolation on the stationary distributions of the voter model with stirring." pith.science (2026). https://pith.science/paper/L6YVYTIV
@misc{pith2026241209532,
author = {Pith},
title = {Pith review of: Percolation on the stationary distributions of the voter model with stirring},
year = {2026},
howpublished = {\url{https://pith.science/paper/L6YVYTIV}},
note = {Machine review of arXiv:2412.09532}
}
abstract
The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ \mu_{\alpha,\mathsf{v}}: \alpha \in [0,1] \}$, where~$\alpha$ is the density of voters with opinion~1. Sampling a configuration~$\xi$ from~$\mu_{\alpha, \mathsf v}$, we study~$\xi$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$\xi$. Letting~$\alpha_c(\mathsf v)$ be the supremum of all the values of~$\alpha$ for which percolation does not occur~$\mu_{\alpha, \mathsf v}$-a.s., we prove that $\alpha_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$\alpha$.
Figures
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
duality relation … P_ξ(ξ_t≡1 on {x1,…xn})=P_x(ξ≡1 on {Y^{x1}_t,…})
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- 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.
- uses
- The paper appears to rely on the theorem as machinery.
- 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.
Reference graph
Works this paper leans on
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[1]
J. Astoquillca. On the stationary measures of two variants of the voter model. arXiv:2409.16064 , 2024
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[2]
P. Clifford and Sudbury A. A model for spatial conflict. Biometrika , 60:581--588, 1973
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R. Cerf. Large deviations for three dimensional supercritical percolation. vol. 267 of Astérisque, SMF , 2000
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[4]
H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for bernoulli percolation and the ising model. Commun. Math. Phys. , 343:725--745, 2016
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R. Holley and T. M. Liggett. Ergodic theorems for weakly interacting systems and the voter model. Annals of Probability , 3:643--663, 1975
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O. Kallenberg. Foundations of Modern Probability. Springer New York, NY. 2nd Edition, 2002
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[8]
T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer, New York, 1985
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B. Ráth and D. Valesin. On the threshold of spread-out voter model percolation. Electronic Communications in Probability , 22:1--12, 2017
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B. Ráth and D. Valesin. Percolation on the stationary distributions of the voter model. Ann. Probab. , 45(no. 3):1899--1951, 2017
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B. Ráth. A short proof of the phase transition for the vacant set of random interlacements. Electronic Communications in Probability , 20, 2015
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J. Swart. A Course in Interacting Particle Systems . In: arXiv:1703.10007, 2017
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Reviewed May 23, 2026 · model on record in the stance chip above.
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