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A perturbed cellular automaton with two phase transitions for the ergodicity

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Pith's one-line read A one-dimensional cellular automaton whose noise-perturbed version is ergodic, then not ergodic, then ergodic again as the noise rate rises.

desk verdict First two-phase-transition PCA construction with a genuinely new mechanism, but the low-noise ergodicity proof has a load-bearing gap around the synchronized-zone barrier in Proposition 5.8. read the letter →

arxiv 2507.03485 v1 pith:OGASOMMF submitted 2025-07-04 nlin.CG math.DSmath.PR

Hugo Marsan , Mathieu Sablik , Ilkka Törmä This is my paper · ORCID
classification nlin.CGmath.DSmath.PR MSC 37B1560K35
keywords probabilisticcellularautomatonergodicityphasetransitionpositiveratesconjecturenoiserobustnessMarkovadditivechaindependencecone
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

This paper constructs a one-dimensional cellular automaton $T$ whose uniform-noise perturbation $T_\epsilon$ has two phase transitions for ergodicity: for very small noise rates it is ergodic, at an intermediate rate it is not, and for rates close to one it is ergodic again. This is the first example in which the set of noise rates leading to non-ergodicity is not an initial interval, and it refutes the intuition that low noise is always the regime where information survives. The construction shows that the positive-rates conjecture is false in a stronger sense: non-ergodicity can appear in a middle window while both very low and very high noise are mixing. A sympathetic reader should care because the result changes what is known about the possible phase structure of one-dimensional probabilistic cellular automata.

What carries the argument

The workhorse is a two-layer update rule $T(x)_i=(G(y_{i-1},y_i,y_{i+1}),F(z)_i)$ with the caveat that when the $F$-layer produces $0$ or an arrow of type $0\bmod k$, the $G$-layer is replaced by the fixed symbol $0_G$. The $F$-layer itself is a clock: values increment mod $k$ each step, and an arrow particle $\nearrow_s$ moves right at speed 1 and resets the cells it passes to synchronized states, living at most $ak$ steps. Synchronized blocks longer than $2k$ cells reset all their $G$-layers to $0_G$ simultaneously every $k$ steps, so no radius-1 information can cross them; the paper treats this as an impenetrable wall. To turn this into a proof of ergodicity, the left and right borders of the dependence cone are bounded by Markov additive chains $(L_n,J_n)$ and $(R_n,K_n)$; the positive drift of $L_n$ and negative drift of $R_n$ follow from barrier-overcoming probabilities $\beta_\epsilon$ and success probabilities $\alpha_\epsilon$, with the barrier height $H=\ln(a)/s(\epsilon)$ chosen to make successive checks independent. Non-ergodicity at $\epsilon_2$ uses Theorem 2.1, a percolation-style condition on the error field: if every finite set $S$ has error probability at most $\epsilon_c^{|S|}$ conditional on the past, the perturbed system is non-ergodic.

What would settle it

Run the perturbation for the two chosen parameter regimes on a large finite ring: in the low-noise regime, initialize a lone synchronized block of length $2k+1$ and inject a single nonzero $G$-signal on one side immediately after a $0$-projection; if the signal appears on the other side before the next projection, the wall property is false. For the intermediate regime, check numerically whether the conditional error probability for a finite set $S$ at $\epsilon_2$ exceeds $\epsilon_c^{|S|}$ as Theorem 2.1 requires; a single counterexample to that bound would break the non-ergodicity leg.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: there exist integers $k,a$ and rates $0<\epsilon_1<\epsilon_2<\epsilon_3<1$ such that the uniform $\epsilon$-perturbation of a specific radius-1 cellular automaton $T$ is uniformly ergodic for all $0<\epsilon<\epsilon_1$, is not ergodic at $\epsilon_2$, and is uniformly ergodic for all $\epsilon>\epsilon_3$. The automaton is a product of two layers: an $F$-layer clock that increments modulo $k$, with right-moving arrow particles that synchronize the clocks they pass over, and a $G$-layer that is forced to a fixed symbol whenever the $F$-layer outputs $0$, so that synchronized blocks of length larger than $2k$ act as walls that stop information flow. At an intermediate rate $\epsilon_2 \sim 1/\ln\ln\ln k$, the clock layer decorrelates and the forced errors are sparse enough for the $G$-layer's built-in error correction (from the classical positive-rates counterexample, used as a black box) to preserve distant information forever, making the process non-ergodic. In the low-noise regime, arrows produced by rare errors live long and synchronize large zones, which force the dependence cone of each cell to shrink to a point almost surely; in the high-noise regime, the standard percolation argument for any perturbed cellular automaton gives ergodicity.

Load-bearing premise

The low-noise proof assumes that a contiguous block of more than $2k$ cells whose clock layer is synchronized is an impenetrable wall that no radius-1 information can cross, and this wall property is argued in a paragraph rather than isolated as a formal lemma.

Editorial extensions

If this is right

  • If Theorem 3.1 is correct, the set of noise rates for which a positive-rate one-dimensional PCA is ergodic can be a non-monotone subset of $[0,1]$, not just an interval above a critical value.
  • The low-noise regime is not automatically the regime where information is best preserved: here it is the regime where synchronized clock blocks scramble all transmitted information and force ergodicity.
  • Because every perturbed cellular automaton is ergodic at noise rates close to 1, the new content is that non-ergodicity can occupy a middle window strictly between two ergodic regimes.

Reading between the lines

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

  • A direct numerical test: for the $k,a$ of the construction, the survival time of arrows in the $F$-layer should grow as $\epsilon$ shrinks, and the crossing of the dependence-cone borders should occur at times consistent with the drift inequalities; measuring these in finite simulations would test the mechanism without waiting for a full proof.
  • If the wall property is the weakest point, then a small modification of the $F$-layer (for example, allowing occasional errors in a synchronized block) might destroy low-noise ergodicity, which would imply the phenomenon is sensitive to the exact synchrony rather than a general principle.
  • The open question posed by the authors — which subsets of $[0,1]$ can be realized as the ergodicity set of a perturbed CA — is sharpened by this example; iterating the clock-layer idea might produce three or more phase transitions, though the paper notes the construction is hard to adapt because the base automaton's invariant measures are not understood.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper constructs a one-dimensional probabilistic cellular automaton T_ε by coupling Gács's non-ergodic CA G with an auxiliary 'clock' layer F of counters modulo k and right-moving arrow particles. The main result (Theorem 3.1) asserts that there are parameters k,a and rates 0<ε1<ε2<ε3<1 such that T_ε is uniformly ergodic for ε∈(0,ε1), non-ergodic at ε=ε2, and uniformly ergodic for ε>ε3. The non-ergodicity regime is obtained by showing that at ε2 the errors injected into the G-layer by the F-layer plus noise satisfy the conditional probability condition of Gács's theorem (Theorem 2.1). The low-noise ergodicity regime is approached via coupling-from-the-past and dependence cones; the boundaries of the dependence cone are controlled by Markov additive chains whose positive (resp. negative) drift is computed in Sections 5.4–5.10. The high-noise regime follows from a known percolation argument. The conclusion would be the first example of two phase transitions for ergodicity in a one-dimensional PCA.

Significance. If the proof is completed, this is a substantial result: it answers a natural open question, shows that the ergodicity region in noise parameter can be disconnected, and provides a promising template for constructing multi-phase-transition PCAs by coupling a reliable automaton with a controlling layer. The paper also has clear strengths: precise definitions, a coherent overall architecture, and explicit asymptotic computations of the Markov additive chain drifts. The use of Gács's automaton as a black box is methodologically appealing. However, the significance is conditional on closing the gap concerning the synchronized-zone barrier described below.

major comments (3)
  1. [Section 5.3 (Proposition 5.8) and Section 5.9.2 (Proposition 5.11)] The proof of Proposition 5.8 asserts that a synchronized F-layer zone of width greater than 2k is an impenetrable barrier because 'no arrows on the F-layer cross this region.' This is not a consequence of the events G_i^t, B_i^t, O_i^t defined in Section 5.2.3: those events are formulated entirely in terms of the noise variables E_i^t being ⊥ in the green cells and the yellow triangle, and they do not constrain the F-layer configuration inherited from time -t. The F-rule gives priority to a left arrow: if z_{i-1}=↗_s then cell i becomes ↗_{s+1} regardless of its previous value. Hence an arrow initially to the left of the zone (or created by noise outside the checked region) can move into the zone without triggering any E≠⊥. When the synchronized cells next project 0_G, the cell occupied by this arrow need not project 0_G, so the wall has a moving hole and a radius-1 dependence path could cross. Since Proposition 5.8 is the only bridge from the Markov additive chain drift (inequality (5.4)) to l_t→+∞, the low-noise ergodicity leg of Theorem 3.1 does not follow. The same gap affects the right-border version (Section 5.9.2, inequality (5.8)). The authors should either prove the barrier property from the F dynamics for all configurations compatible with the noise, or enlarge the event G_i^t to exclude crossing arrows and recompute α_ε, β_ε, and the drift inequalities.
  2. [Section 2.2 (Theorem 2.1)] Theorem 2.1 is stated as 'a direct consequence' of the main result of [4], but no theorem or lemma number from [4] is given, and the condition 'for all finite S and all events H in the past, P(error in each cell of S | H) ≤ ε_c^{|S|}' is a strong conditional-uniform error condition. Because Section 4.4 concludes non-ergodicity of T_{ε2} from this theorem, the authors need to provide a precise reference (theorem number) and a derivation, or prove the statement. As it stands, a reader cannot verify that Gács's theorem applies to the error process produced by the F-layer plus uniform noise.
  3. [Section 4, opening paragraph] The sentence 'if T_{ε2} is non-ergodic on this layer, then it is non-ergodic globally' is asserted without proof. The proof that follows bounds the probability of errors in the G-coordinate, not the full-symbol errors of T_{ε2}; since Theorem 2.1 is stated for Gács's CA on alphabet G, the logical step from the G-coordinate bound to non-ergodicity of the joint system T_{ε2} on A=G×F needs to be supplied. One possible route is to argue that a G-coordinate error implies a full-symbol error and to apply a version of Gács's theorem to the joint system, but this is not written out. Please clarify this implication.
minor comments (5)
  1. [Section 1] The sentence 'In all known examples of CA robust to noise, there is critical value ϵc such that the perturbed cellular automaton is ergodic for any ϵ < ϵc' appears to contradict the surrounding discussion; it should presumably read 'non-ergodic for any ϵ < ϵc'.
  2. [Section 5.2.3] In the definitions of G_i^t and B_i^t, the condition 'n-j ≥ t-i' in the yellow triangle is not translation-invariant and is likely a typo; since these events should be invariant under spacetime translations, please replace it with the intended condition (for example n ≥ j).
  3. [Section 5.9.2] The sentence 'We claim that the proof of Proposition 5.8 still stands for the next proposition' should be replaced by an actual proof, especially given the issue raised in the first major comment.
  4. [Title and Abstract] The title and abstract contain a typo: 'PER TURBED' should be 'PERTURBED'.
  5. [Section 2.2] The text contains the typo 'ergordic' where 'ergodic' is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two phase transitions are supported by independent probabilistic estimates and an external Gács black box.

full rationale

The derivation chain of Theorem 3.1 does not reduce any of its conclusions to the assumptions by construction. The non-ergodicity leg (Section 4) verifies the conditional error-probability condition of Theorem 2.1, which is taken from Gács's external published work [4]; the parameters ϵ(k)=1/ln ln ln k and d(k)=ln ln k are chosen from independent noise estimates, and no fitted value is later renamed as a prediction. The low-noise ergodicity leg (Section 5) uses the coupling-from-the-past criterion of Proposition 5.2, cited from [10]; this is an external, published sufficient condition for uniform ergodicity and is not the target statement of this paper. The Markov additive chain drift inequalities (5.4) and (5.8) are derived from the independently defined probabilities p_good, p_bad, αϵ and βϵ, which are computed from the noise model rather than fitted to force the conclusion. The synchronized-zone barrier argument in Proposition 5.8 is asserted as a geometric consequence of the F-layer rule and the projection of 0_G; even if it is under-proved and represents a correctness or rigor concern, it is not a circular step: it does not define the target property in terms of itself, and no equation is made true by definition. The high-noise ergodicity statement is explicitly attributed to [10, Proposition 3.6] and is standard for any perturbed CA. The only self-citation is [10], which includes one author of the present paper, but its use is as a general external criterion, not as an unverified premise that is logically equivalent to the main result. Therefore no load-bearing circularity is present; the appropriate finding is a non-finding.

Assumptions & free parameters 5 free parameters · 5 assumptions · 1 invented entities

The construction is modular: Gács's CA supplies reliable computation from outside the paper, and the new layer is a finite clock mod k plus finite-lifetime arrows. All parameters (k, a, epsilon2 = 1/ln(ln(ln(k))), d(k) = ln(ln(k)), H = ln(a)/s(epsilon)) are proof-scaling choices selected in Sections 4.3.6 and 5 so that explicit inequalities hold; none encode the conclusion. The only invented ontological objects are the arrows and the synchronized zones they create.

free parameters (5)
  • k
    Modulus of the F-layer clock. Chosen large enough (and even) to satisfy the Section 4.3.6 error-rate bounds and Claim 4.1 conditions; a proof-scaling parameter, not a fit.
  • a
    Arrow lifetime multiplier. Chosen large enough to satisfy the drift inequalities (5.6), (5.9) and the expansion of C_{a,k}; a proof-scaling parameter.
  • epsilon2 = 1/ln(ln(ln(k)))
    The intermediate noise rate where non-ergodicity is proved. Chosen to satisfy the five error-rate conditions in Section 4.3.6 (including 1/epsilon = o(k), d = o(ln k), epsilon >= 2|G|(1-(4/k)^{4/k})); no empirical fit.
  • d(k) = ln(ln(k))
    Cell-classification threshold separating S3 from S4 in Section 4.3. Chosen to satisfy d = o(ln k), d <= k/4, and 1/d = o(epsilon); a proof-scaling parameter.
  • H = ln(a)/s(epsilon)
    Barrier size in the Markov additive chain analysis. Chosen so that (p_other)^H tends to 1/a as epsilon goes to 0, enabling the beta_epsilon lower bound; a proof device, not a dynamical quantity.
assumptions (5)
  • domain assumption Gács's theorem [4] formulated as Theorem 2.1: a PCA whose error process satisfies P(error on every cell of S | H) <= epsilon_c^{|S|} for all finite S and past events H is non-ergodic.
    Used as a black box for the non-ergodicity leg (Section 4.4 conclusion). The exact correspondence between this percolation-style condition and the statements of [4] is asserted rather than mapped to a specific theorem in [4].
  • domain assumption Coupling-from-the-past criterion [10, Prop 3.3]: if P(x maps to Psi^t(x; U^{-t},...,U^0)_0 is constant) tends to 1, then T_epsilon is uniformly ergodic.
    Load-bearing for the low-noise ergodicity leg (Section 5.1). [10] shares an author with this paper but is an established published result used as stated.
  • standard math Law of large numbers for Markov additive chains (Asmussen [1], ch. XI): if the mean drift kappa' > 0 then L_n tends to +infinity almost surely.
    Converts the drift inequalities (5.4) and (5.8) into almost-sure divergence of the border chains in Section 5.2.2.
  • ad hoc to paper Synchronized-zone barrier property: a synchronized block longer than 2k on the F layer blocks all radius-1 information flow on the G layer.
    The core physical mechanism of the low-noise leg, argued in the proof of Prop 5.8 and asserted for the right border in Section 5.9.2; no standalone formal lemma is given.
  • domain assumption Gács CA G has radius 1 with finite alphabet; the noise model is uniform over A with an independent error field.
    Radius 1 is used for the dependence-cone speed bound (Lemma 5.5); the independent error field formalization appears in Section 4.1.
invented entities (1)
  • Arrow particles (rising to the right) with types 0 <= s < ak on the F layer
    purpose: Particles moving right at speed 1 that reset the mod-k clocks of cells they pass and die after ak steps; they create the synchronized zones that control where the G layer is forced to 0_G and, at low noise, act as information barriers.
    Internal to the construction; no falsifiable handle outside the paper (Section 3, Figures 3.1-3.2).

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Pith. "Pith review of A perturbed cellular automaton with two phase transitions for the ergodicity." pith.science (2026). https://pith.science/paper/OGASOMMF

@misc{pith2026250703485,
  author       = {Pith},
  title        = {Pith review of: A perturbed cellular automaton with two phase transitions for the ergodicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGASOMMF}},
  note         = {Machine review of arXiv:2507.03485}
}
read the original abstract

The positive rates conjecture states that a one-dimensional probabilistic cellular automaton (PCA) with strictly positive transition rates must be ergodic. The conjecture has been refuted by G\'acs, whose counterexample is a cellular automaton that is non-ergodic under uniform random noise with sufficiently small rate. For all known counterexamples, non-ergodicity has been proved under small enough rates. Conversely, all cellular automata are ergodic with sufficiently high-rate noise. No other types of phase transitions of ergodicity are known, and the behavior of known counterexamples under intermediate noise rates is unknown. We present an example of a cellular automaton with two phase transitions. Using G\'acs's result as a black box, we construct a cellular automaton that is ergodic under small noise rates, non-ergodic for slightly higher rates, and again ergodic for rates close to 1.

Figures

Figures reproduced from arXiv: 2507.03485 by the authors.

Figure 3.1
Figure 3.1. Some steps for k = 10, a = 2. Time goes up and only the F layer is shown. The pale colors represent the different synchronized zones created by the arrows. In green, an error creates a ↗6. In red, the cells where a 0 is projected onto the G layer. Finally, let us define a 1-dimensional cellular automaton T on A of radius 1 as the following: for any configuration x = (xi)i∈Z, with xi = (yi , zi) ∈ A = G × F, we have … view at source ↗
Figure 3.2
Figure 3.2. Illustrations of ϵ-perturbed orbits of F, for k = 5 and several values of a and ϵ. In each, we represent the subtraction of the value in a given cell by the time, mod k: the blocks of same color have synchronized values. A smaller ϵ increases the lifetime of the synchronized blocks, while a larger a decreases it (there is more arrows) but increases their size (the lifetime of the arrows increases). In the low noise … view at source ↗
Figure 4.1
Figure 4.1. Decomposition of a given S into each Si . In grey, the known initial configuration. In green, the visible errors on the G layer [PITH_FULL_IMAGE:figures/full_fig_p008_4_1.png] view at source ↗
Figures from the paper (11 more)
Figure 5.1
Figure 5.1. Figure 5.1: Illustration of lt. The stars represents the errors, where the values on these cells depend only on the noise. In blue, the information flow from the cells in the configuration at t = −t1 that can influence the values on the right semi-configuration at t = 0. In red,…
Figure 5.2
Figure 5.2. Figure 5.2: Left: the decision process. Right: its illustration. The green cells are at position (i + m, t + m) with 0 < m < 2k. The size of the yellow triangle depends on the nature s of the arrow ↗s seen at (i, t). One more formally defines the events as follows [PITH_FULL_IM…
Figure 5.3
Figure 5.3. Figure 5.3: Suppose Ln = i and tn = t. In green, the column where the first arrow is looked for: it appears after M −1 “other” cases (no errors on the 2k cells in diagonal, in blue). As the arrow is of type s and there is no error on its path (yellow cells), it’s a success: the …
Figure 5.4
Figure 5.4. Figure 5.4: A success is obtained when the arrow defining the synchronized zone the considered cell is [PITH_FULL_IMAGE:figures/full_fig_p020_5_4.png]
Figure 5.4
Figure 5.4. Figure 5.4: First steps of (Ln, Jn) and Lt (in blue). The red-and-blue cells are at times tn in position Ln. The initial state is (L0, J0) = (0, 1). The first steps represented are: (1) The initial cell is not in a good synchronized zone: we encounter a bad case before H checks,…
Figure 5
Figure 5. Figure 5: illustrates the right shift of the left border of the dependence cone after a success. [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 5.5
Figure 5.5. Figure 5.5: Illustration of a success in Ln (not to scale). The colored area is synchro￾nized and blocks all information flow on the G-layer. The colors correspond roughly to [PITH_FULL_IMAGE:figures/full_fig_p023_5_5.png]
Figure 5.6
Figure 5.6. Figure 5.6: Left: the decision process. Right: its illustration. The green cells are in position (i + m, t + m) with 0 < m < 2k. The yellow zone is of height 2k + 1. We can then compute the respective probabilities of these events: P [PITH_FULL_IMAGE:figures/full_fig_p027_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Suppose that Rn = i and t ′ n = t. In green the column where we search for an arrow: here after M − 1 “other” cases (no errors in the 2k cells in diagonal, in dark blue) and no errors in the light blue area. As the arrow is of age l and there is no errors on its path…
Figure 5.8
Figure 5.8. Figure 5.8: The first steps of (Rn, Kn) and Rt (in blue, defined analogously to Lt). The red-and-blue cells are at position Rn at time t ′ n . The initial state is (R0, K0) = (0, 1). The first steps are: (1) The initial cell is not in a good synchronized zone: we encounter a bad…
Figure 5.9
Figure 5.9. Figure 5.9: Illustration of a success in Rn (not to scale). The interpretation is analogous to [PITH_FULL_IMAGE:figures/full_fig_p031_5_9.png]

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