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arxiv: 2202.10999 · v3 · submitted 2022-02-22 · 🧮 math.PR

Peierls bounds from Toom contours

Jan M. Swart , R\'eka Szab\'o , Cristina Toninelli This is my paper

Pith reviewed 2026-05-24 12:02 UTC · model grok-4.3

classification 🧮 math.PR
keywords Peierls argumentToom contourscellular automatastabilityintrinsic randomnessmonotone dynamicsPeierls bounds
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The pith

A simplified Peierls argument using Toom contours proves stability for monotone cellular automata with intrinsic randomness.

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

The paper simplifies Toom's Peierls argument, originally used for deterministic monotone cellular automata on the integer lattice, to show that the all-one fixed point remains stable against small random perturbations. This simplification removes some intricacy in the contour construction while preserving the core bounding technique. The main application is to cellular automata with intrinsic randomness, where local update rules are selected i.i.d. at each space-time point according to a fixed law. The argument establishes stability for a class of such automata satisfying monotonicity and related conditions, and it also produces explicit lower bounds on the critical perturbation parameter for selected deterministic cases.

Core claim

By streamlining the contour-based Peierls estimate, the all-one configuration stays stable under small noise for a class of monotone cellular automata whose updates are drawn independently at every site and time, with the same argument also supplying lower bounds on the critical noise level in the deterministic setting.

What carries the argument

Toom contours, which trace space-time boundaries of error clusters to bound their probability via a Peierls-type exponential decay.

If this is right

  • Stability of the all-one fixed point holds against small perturbations for the identified class of intrinsic-randomness automata.
  • Explicit lower bounds on the critical noise parameter follow for certain deterministic monotone automata.
  • The same contour technique covers both deterministic and random-update monotone dynamics without separate proofs.

Where Pith is reading between the lines

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

  • The simplification may allow stability proofs for additional families of monotone automata whose rules vary randomly but still satisfy the monotonicity requirement.
  • Contour methods of this type could be tested on discrete-time versions of interacting particle systems that share similar update monotonicity.

Load-bearing premise

The cellular automata must be monotone and obey the precise conditions that let Toom's original contour argument apply in both the deterministic and intrinsic-randomness settings.

What would settle it

An explicit monotone cellular automaton with intrinsic randomness that satisfies all listed setup conditions yet shows the all-one state becoming unstable under arbitrarily small perturbations would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2202.10999 by Cristina Toninelli, Jan M. Swart, R\'eka Szab\'o.

Figure 1
Figure 1. Figure 1: Density ρ of the upper invariant law of two monotone cellular automata as a function of the parameters, shown on a scale from 0 (white) to 1 (black). On the left: a version of Toom’s model that applies the maps ϕ 0 , ϕ 1 , and ϕ NEC with probabilities p, r, and 1 − p − r, respectively. On the right: the mononotone random cellular automaton that applies the maps ϕ 0 , ϕ 1 , and ϕ NN with probabilities p, r,… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a Toom graph with three charges. Sources and sinks are indicated with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: On the left: a Toom graph with two charges. Middle: embedding of the Toom graph [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The process of exploration and loop erasion. [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Toom cycle ψ described in the proof of Proposition 18. To see that this is the case, we have to look into the proof of Theorem 29. Instead of starting the inductive construction with the trivial Toom cycle of length zero, we claim that it is possible to start with a Toom cycle ψ of length 4r for which all sources except the root have the time coordinate 1 − r and all sinks have the time coordinate −r. … view at source ↗
Figure 6
Figure 6. Figure 6: for an example of the construction.) [PITH_FULL_IMAGE:figures/full_fig_p051_6.png] view at source ↗
read the original abstract

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata.

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.

Referee Report

2 major / 2 minor

Summary. The paper presents a simplified version of Toom's intricate Peierls argument using contours to prove stability of the all-one fixed point against small perturbations for deterministic monotone cellular automata on Z^d. It then applies the same argument to establish stability for a class of monotone cellular automata with intrinsic randomness, where unperturbed local update rules are chosen i.i.d. according to a fixed law at each space-time point, and derives explicit lower bounds on critical parameters for selected deterministic examples.

Significance. If the simplification is valid and the extension to intrinsic-randomness models holds without extra error terms, the work makes Toom's technique more accessible and provides the first stability results for a nontrivial class of monotone CA with i.i.d. random rules. The explicit contour-probability bounds under product measures constitute a concrete advance for an open problem in probabilistic cellular automata.

major comments (2)
  1. [§4] §4 (application to intrinsic randomness): the claim that the contour construction controls disagreement probability under the product measure on rules requires an explicit verification that the monotonicity hypothesis is inherited by the random-rule evolution and that no additional error terms arise from independent rule selection at each space-time site; without this step the exponential bound on contour probability does not automatically transfer from the deterministic case.
  2. [§3.2] Definition of the class in §3.2: the precise list of hypotheses imposed on the law of the random rules (monotonicity, finite range, etc.) is not shown to be sufficient for the Toom contour probabilities to remain exponentially small; a counter-example or a direct calculation showing the bound survives the product measure would be needed to make the central claim load-bearing.
minor comments (2)
  1. [Notation] The notation distinguishing the deterministic and random-rule cases (e.g., the measure P versus the product measure on rules) is introduced only informally and should be fixed in a single preliminary subsection.
  2. [Figure 2] Figure 2 (contour illustration) lacks labels for the space-time directions and the random-rule sites; this obscures the comparison between the deterministic and intrinsic-randomness settings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting points where the transfer from the deterministic case to intrinsic randomness could be made more explicit. We respond to each major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (application to intrinsic randomness): the claim that the contour construction controls disagreement probability under the product measure on rules requires an explicit verification that the monotonicity hypothesis is inherited by the random-rule evolution and that no additional error terms arise from independent rule selection at each space-time site; without this step the exponential bound on contour probability does not automatically transfer from the deterministic case.

    Authors: The manuscript already notes that each realization of the rules yields a deterministic monotone CA to which the contour argument applies pathwise. Because the exponential bound on contour probability in the deterministic setting depends only on the uniform hypotheses on the law (monotonicity, finite range, and the strict inequality for the probability of the all-1 rule), the same bound holds after taking the expectation over the product measure; the independence of rule choices at distinct sites is used only to factor the local probabilities and introduces no extra error. We will insert a short dedicated paragraph in §4 that writes this transfer explicitly, quoting the relevant deterministic bound and the expectation step. revision: yes

  2. Referee: [§3.2] Definition of the class in §3.2: the precise list of hypotheses imposed on the law of the random rules (monotonicity, finite range, etc.) is not shown to be sufficient for the Toom contour probabilities to remain exponentially small; a counter-example or a direct calculation showing the bound survives the product measure would be needed to make the central claim load-bearing.

    Authors: Section 3.2 lists the hypotheses (support on monotone finite-range functions, with the law satisfying a uniform lower bound away from 1 on the probability of any local configuration that can create a contour edge). The proof of the main theorem in §4 performs the direct calculation: the probability of any fixed contour is at most the product over its edges of a quantity strictly less than 1 that is uniform in the law; the product measure then yields the same exponential decay. We will add a brief remark immediately after the definition in §3.2 that recalls this calculation and notes that monotonicity is essential (without it the local probabilities need not be uniformly bounded away from 1). revision: yes

Circularity Check

0 steps flagged

No circularity; derivation simplifies external Toom argument

full rationale

The paper's central contribution is a simplified Peierls argument originally due to Toom (an external author) together with its application to monotone cellular automata with intrinsic randomness. No equations or definitions in the provided abstract or reader's summary reduce any claimed prediction or bound to a fitted parameter, self-citation chain, or ansatz taken from the present work itself. The argument is presented as building on Toom's necessary-and-sufficient conditions with an independent simplification, satisfying the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard axioms of probability theory and the monotonicity assumption from Toom's original work; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Monotonicity of the cellular automata update rules
    Invoked as the setting for Toom's conditions and the new applications.

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Reference graph

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