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Finitary codings and stochastic domination for Poisson representable processes

T0 review · 6 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes that exponential moments of the sizes of activated finite sets are the threshold for when a random union process is a finitary factor of an IID process, proving both directions and showing the remaining gap is real.

desk verdict Strong, mostly coherent answer to two open questions; one indexing typo in the central construction needs fixing, but the paper is worth a serious referee. read the letter →

arxiv 2506.04797 v1 pith:UMJ2ZJXL submitted 2025-06-05 math.PR

Yinon Spinka This is my paper
classification math.PR MSC 60G1060G5560K3537A35
keywords PoissonrepresentableprocessfinitaryfactorofIIDstochasticdominationBernoullipercolationexponentialmomentcodingradiuscountable-stateMarkovchain
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

The paper studies random subsets of $\mathbb{Z}^d$ built by activating each finite set $A$ with a translation-invariant probability $p_A$ and taking the union of the activated sets. It establishes that whether the resulting union process can be generated from an IID process by a finitary code is governed by the exponential moments of the size of the activated sets: finite values for every $\lambda>0$ are sufficient, and finiteness for at least one $\lambda>0$ is necessary. In the case where only pairs are activated, this resolves an open question about Poisson representable processes. A companion result shows the same exponential-moment condition is exactly what is needed for the union process to be stochastically dominated by a non-trivial Bernoulli percolation on $\mathbb{Z}^d$.

What carries the argument

The machinery has several load-bearing pieces. A Poisson representable process is $\bar X$ built from independent Bernoulli variables indexed by finite subsets, and the finitary coding problem is whether $\bar X$ can be computed by an equivariant map of IID randomness whose output at a site depends on a finite random coding radius. The main construction decomposes the family of allowed sets into blocks of bounded diameter, represents each block process as a block factor whose coding radius is zero with high probability, and then uses Borel–Cantelli to make all radii finite simultaneously. This domination is supplied by a sequential stochastic-domination lemma that bounds the conditional probability at a site by $\varepsilon$ plus a moment-sum factor, allowing a per-site sprinkling. A random partition lemma produces finite partition classes with a uniform bound on the number of classes meeting any ball, which keeps the construction equivariant and finitary. For the one-dimensional theorem, the process $\check X$ recording sets crossing a site is shown to be a hidden Markov chain with exponential return times, and known results on such chains provide finitary coding and finitary isomorphism.

What would settle it

Check whether any translation-invariant family $(p_A)$ with $\sum_{A\ni 0} p_A e^{\lambda|A|}=\infty$ for all $\lambda>0$ can be finitarily coded from IID; Theorem 1.2 says none can, so a single such family that is finitarily codeable would settle the central claim false.

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Extended reading notes

Core claim

Let $\bar X$ be the union of the activated sets. Theorem 1.1 states that if $\sum_{A\ni 0} p_A e^{\lambda|A|}<\infty$ for every $\lambda>0$, then $\bar X$ is a finitary factor of an IID process; Theorem 1.2 states that if $\bar X$ is a finitary factor of an IID process, then such a sum is finite for some $\lambda>0$. These two statements frame the central claim that the exponential moment of the activated-set size is the codeability threshold. The paper further shows the threshold cannot be collapsed to a single condition: Example 1.6 gives a process satisfying one exponential moment that is not finitarily codeable, while Theorem 1.3 shows that when singletons have positive probability, the moment needed is only at $\lambda=-\log p_{\{0\}}$. On the domination side, Theorem 1.5 characterizes, on $\mathbb{Z}^d$, domination of $\bar X$ by a non-trivial IID Bernoulli process by the existence of some finite exponential moment. In one dimension, under an exponential moment on the diameter, the richer process recording the activated sets crossing a site is a countable-state mixing Markov chain with exponential return times, hence finitarily isomorphic to an IID process.

Load-bearing premise

The load-bearing premise is the no-infinite-descent claim behind the greedy random partition lemma: almost surely there is no infinite chain of sites in which the assigned priorities strictly decrease at every step, so every site's cell is decided after finitely many steps.

Editorial extensions

If this is right

  • In the pair-only case, every translation-invariant such process is a finitary factor of an IID process, settling the open question for that class.
  • A union process whose activated-set sizes have no finite exponential moment cannot be finitarily coded from IID randomness, no matter how the coding is designed.
  • On $\mathbb{Z}^d$, stochastic domination of $\bar X$ by a non-trivial Bernoulli percolation holds if and only if some exponential moment of the activated-set size is finite.
  • Varying the singleton intensity while fixing all other probabilities produces a phase transition for the finitary-factor property, with a critical value $p_c$ that is always strictly below one when some exponential moment exists, and can be positive or zero.
  • In one dimension, an exponential moment on the diameter gives more than a code: the crossing process is finitarily isomorphic to an IID process.

Reading between the lines

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

  • A natural test bed for closing the gap is the one-parameter family $p_{\{0,n\}}\sim e^{-n}$; the paper's theorems locate the finitary-factor boundary somewhere inside the class of processes with exactly one exponential moment, and a decisive computation there would map the true threshold.
  • The sprinkled-coupling domination lemma is stated for finite-set intensities, but the same conditional-probability bound should localize to intensities with infinite sets, where the exponential moment would be replaced by a tail condition on the local mass.
  • The one-dimensional isomorphism statement uses the hidden Markov structure of the crossing process; in higher dimensions no such structure is available, so the corresponding question is whether exponential moments on connected size imply finitary isomorphism rather than merely finitary coding.
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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

6 major / 5 minor

Summary. The paper studies translation-invariant Poisson representable processes on Z^d constructed by taking a union of independently selected finite sets with translation-invariant probabilities p_A. The main results are: (1) Theorem 1.1, a finitary factor of IID (FFIID) construction under the condition that all exponential moments of |A| are finite; (2) Theorem 1.2, a converse showing that some exponential moment is necessary for being FFIID; (3) Theorem 1.3, an improved sufficient condition when the singleton probability is positive; (4) Theorem 1.4, an FFIID construction with exponential coding tail under a connected-size exponential moment condition; (5) Theorem 1.5, a stochastic-domination characterization by nontrivial IID processes in terms of existence of an exponential moment. The paper also proves, in one dimension, that a related hidden Markov chain is finitarily isomorphic to an IID process. The overall strategy is to decompose the intensity into pieces, use monotone couplings and random partitions to obtain finitary codings, and use stochastic-domination bounds to control coding radii.

Significance. If the proofs are completed as indicated, the paper answers Question 7 of Forsström-Gantert-Steif for pair processes and gives a clean exponential-moment divide for finitary codeability and for stochastic domination by IID processes. The techniques, including the sprinkling-based stochastic domination lemmas, the random-partition construction, and the use of hyperfinite exhaustions, are innovative and likely useful beyond this paper. The manuscript is generally well organized, and the main derivations for Theorems 1.1, 1.2, 1.3 and 1.5 are detailed and coherent. However, several load-bearing points in the proofs are currently incomplete or incorrect as written: an indexing error in the central construction of Theorem 1.1, an invalid inequality in Example 1.6, an omitted proof of a key fact in Lemma 3.2, a sketched final step in Theorem 1.4, a non-equivariant tie-breaking in the non-free case of Theorem 1.4, and a summability gap in Lemma 3.3. These issues preclude acceptance in the present form.

major comments (6)
  1. [Section 3.3] The definition 'For A∈A_k, define X_A := X^{min A}_A' is not well-posed: immediately before, X^u was defined only for u∈M, the set of lexicographic minima of the partition classes, and for a generic finite set A, min A is not the minimum of its own partition class. Consequently X^{min A} is not among the objects constructed. The law argument that follows is only valid if each A is assigned to the unique u∈M such that min A∈L_u, namely u=M_{min A}. The construction should be re-indexed accordingly; as printed, the central step of the proof of Theorem 1.1 is undefined for most A.
  2. [Example 1.6] The inequality e^{-λk} = p_{A_k} ≤ P(\bar X_{A_k}≡1) ≤ (P(\bar X_0=1)+P(R≥k/2))^k is not justified and is in general false for a finitary factor with pointwise coding radius R. A correct bound using disjointness of the balls of radius k/2 around the k points of A_k is P(\bar X_{A_k}≡1) ≤ P(\bar X_0=1)^k + k P(R≥k/2). This only implies P(R≥k/2) ≥ e^{-λk}/k, which is compatible with R being finite almost surely. Therefore the example does not establish that condition (1.2) is not sufficient for being FFIID, and the claimed gap between Theorems 1.1 and 1.2, as well as the later phase-transition discussion, loses its supporting counterexample as written.
  3. [Lemma 3.2] The proof of Lemma 3.2 relies on the assertion that 'there are no infinite r-paths with descending W-values almost surely' to guarantee that the greedy construction terminates locally, and hence that the coding radius is finite almost surely. This fact is load-bearing for the finitariness of the random partition used in Theorem 1.1, but it is only asserted. The fact is true and can be proved by a standard first-moment argument over self-avoiding paths, but the proof should be included in the paper.
  4. [Section 3.5, Theorem 1.4] The final step of the proof of Theorem 1.4, passing from the finitary factor Y with exponential coding tail to the process \bar X^1, is presented only as a sketch. The claims that (\tilde X, Y) is decoupled by zeros of Y and that \tilde X can be obtained as a finitary factor of Y plus an independent IID process are asserted without proof. The exponential-tail assertion for the composed coding radius also requires quantitative control on the cluster-size distribution of Y and on the composition of coding radii. Since Theorem 1.4 is one of the main theorems, this sketch must be replaced by a complete proof.
  5. [Section 3.5, non-free case] In the paragraph removing the free-action assumption, the proposed equivariant definition of c(A) via minimizing ∑_{v∈A'} W_v is not Γ-equivariant: applying a group element γ permutes the elements of A', and for a fixed realization W the sum ∑_{v∈γ(A')} W_v is not equal to ∑_{v∈A'} W_v, so γ(c(A)) need not equal c(γ(A)). An equivariant choice can be obtained by minimizing the sorted tuple of W-values (the order statistics), which is invariant under permutations of A', so the construction is repairable, but as written the non-free case is not established.
  6. [Lemma 3.3] The definition of ε_{v,L} contains the factor (1-p_A)^{-1}. Property 3 of the lemma, namely that P(Z^L_0=*) = ε_{0,L} tends to zero as L↑Z^d, does not follow from assumption (1.3) with this factor, because summability of ∑ p_A p_0^{-|A|} does not imply summability of ∑ p_A(1-p_A)^{-1} p_0^{-|A|} when p_A can approach 1. The proof is repairable: in the displayed conditional-probability estimate, the identity P(X_A=1|...)=p_A/(p_A+(1-p_A)p_0^N) actually implies the sharper bound p_A p_0^{-N}, so ε_{v,L} can be defined without the factor (1-p_A)^{-1}. The authors should make this change.
minor comments (5)
  1. [Example 1.6] The definition of A_k appears to contain a typo: '{0, k, k, . . . ,(k−1)k}' should presumably be '{0, k, 2k, . . . ,(k−1)k}'.
  2. [Sections 2 and 3] Cross-references to lemmas are inconsistent: Lemma 2.1 is called Theorem 2.1, Lemma 2.2 is called Theorem 2.2, and Lemma 3.1 is called Theorem 3.1. Please standardize the numbering references.
  3. [Section 3.4] In the proof of Lemma 3.3, the sentence 'which in turn equals the sum of p_A over all A⋐M having v∈A and A̸⊂L' should read 'is at least' rather than 'equals', because of the factor (1-p_A)^{-1} in ε_{v,L}; the subsequent union bound works with the at-least statement.
  4. [Section 3.3] The notation M_v is used both for the minimum of the partition class containing v and for the set M_v := {u∈M : v∈L_u^*}; these should be denoted by distinct symbols to avoid confusion.
  5. [Theorem 1.5] In the first part of the proof of Theorem 1.5, the decomposition into A and A' for general Γ is described via an enumeration of orbits, but the proof should explicitly state that the tail of the enumeration is chosen so that ∑_{A∈A', v∈A} p_A e^{λ|A|} is at most 1 for a fixed v; the current wording leaves this implicit.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the main theorems are proved from the model definition and standard external results; the few self-citations are supporting and not load-bearing.

full rationale

The main derivation chain for Theorems 1.1 and 1.2 is self-contained. Xbar is defined from independent Bernoulli variables X_A, and the sufficient direction constructs a finitary factor using stochastic domination lemmas (Lemmas 2.1-2.2, 3.1) and a finitary random partition (Lemma 3.2); the necessary direction uses only positive association of Xbar and the exponential mean-ergodic theorem. No fitted parameter is renamed as a prediction and no displayed equation is equal to its conclusion by construction. The self-citations that occur are in supporting results only: Theorem 1.4 invokes [5] after verifying its decoupling and density hypotheses, and Corollary 3.11 imports Markov-chain coding and isomorphism theorems from [1,7,8]; these prior theorems have stated assumptions that do not include the present target results, so they do not make the argument circular. The reader-flagged technical issues (the undefined X_A := X^{min A}_A for typical A, and Lemma 3.2's asserted no-infinite-descent fact) are correctness gaps or expositional shortcuts, not reductions of outputs to inputs. The score reflects the presence of minor non-load-bearing self-citations rather than any actual circular step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data; the p_A are the input model and constants such as lambda, beta and N_k are proof parameters. The listed axioms are external theorems and standing definitions that the claims import. The auxiliary processes Xhat and Xcheck are constructions, not invented physical entities.

assumptions (7)
  • domain assumption X_A for finite A are independent Bernoulli with parameters p_A.
    This defines the Poisson representable process with intensity concentrating on finite sets. Introduced in Section 1, not proved.
  • domain assumption p_A < 1 for all A and sum over A containing v of p_A is finite for each v.
    Standing assumption in Section 1, equivalent to P(Xbar_v = 1) < 1. The main theorems quantify over this class.
  • standard math Xbar is positively associated.
    Used in Section 3.1, proof of Theorem 1.2, to multiply probabilities over disjoint sets. Cited to [4, Theorem 2.4].
  • standard math Every FFIID process satisfies the mean ergodic theorem with exponential rate on boxes.
    Used in Section 3.1 to contradict P(Xbar_[N]^d = 1) >= e^{-o(N^d)}. Cited to [3].
  • standard math A decoupled-by-zeros process on a graph of degree Delta with P(Y_v = 1 | neighbors) below 1/(3Delta-1) is FFIID with exponential coding radius.
    This is the engine of Theorem 1.4 in Section 3.5, imported from [5, Theorem 4.1]; [5] is an arXiv preprint by Ray and Spinka, not reproduced here.
  • standard math Countable-state mixing Markov chains with exponential return times are finitary factors of IID and finitarily isomorphic to IID.
    Used in Section 3.6 via [1], [7] and [8] to obtain Corollary 3.11; [8] is the author's separate preprint arXiv:2506.04069.
  • standard math There exists a finitary hyperfinite exhaustion of Z^d by finite partitions, refining to singletons.
    Used in the proof of Theorem 1.3 in Section 3.4; the paper calls the existence standard.

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Cite this review

Pith. "Pith review of Finitary codings and stochastic domination for Poisson representable processes." pith.science (2026). https://pith.science/paper/UMJ2ZJXL

@misc{pith2026250604797,
  author       = {Pith},
  title        = {Pith review of: Finitary codings and stochastic domination for Poisson representable processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UMJ2ZJXL}},
  note         = {Machine review of arXiv:2506.04797}
}
abstract

Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with positive probability are pairs, such a random set is a finitary factor of an IID process, answering a question of Forsstr\"om, Gantert and Steif. More generally, we show that this is the case whenever the distribution induced by the size of the selected sets has sufficient exponential moments, and that the existence of some exponential moment is necessary. We further show that such a random set is stochastically dominated by a non-trivial Bernoulli percolation if and only if there is a finite exponential moment, thereby partially answering another question of Forsstr\"om et al. We also give a partial answer to a third question regarding a form of phase transition. These results also hold on $\mathbb{Z}^d$ with $d \ge 2$. In the one-dimensional case, under the condition that the distribution induced by the diameter of the selected sets has an exponential moment, we further show that such a random set is finitarily isomorphic to an IID process.

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Works this paper leans on

8 extracted references · 8 canonical work pages

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