On the quenched functional CLT in 2d random sceneries, examples

Guy Cohen; Jean-Pierre Conze

REVIEW 4 major objections 5 minor 20 references

On the quenched functional CLT in 2d random sceneries, examples

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In two dimensions, sums of a random field along a random walk satisfy a quenched functional central limit theorem under only a finite second moment.

desk verdict Solid continuation of the authors' own program: removes the log-moment condition in the quenched 2d RWRS FCLT and adds two genuinely new FCLT models, with the main caveat that everything inherits the cited self-intersection LLN (12) and the Lorentz part is only a sketch. read the letter →

arxiv 1908.03777 v1 pith:TUT6LJWR submitted 2019-08-10 math.PR

Guy Cohen , Jean-Pierre Conze This is my paper · ORCID

classification math.PR MSC 60F0528D0522D4060G5047B1537A2537A30

keywords quenchedfunctionalCLTrandomwalkinscenery2Dself-intersectionsofatoralautomorphismsS-unitequationsassociatedvariablesLorentzprocess

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 proves quenched functional central limit theorems for sums of a stationary random field sampled along a two-dimensional random walk: for almost every realization of the walk, the partial-sum process normalized by $\sqrt{n\log n}$ converges weakly to a Wiener process. In the iid case, the only moment assumption on the field is $E(X_0^2)=1$, which removes the logarithmic moment condition required by an earlier quenched result [17]. The same conclusion is extended to moving averages of iid fields and to fields generated by a totally ergodic $\mathbb{Z}^2$-action of commuting toral automorphisms, and a quenched version for a planar Lorentz process in random scenery is sketched. The result matters because the $\sqrt{n\log n}$ normalization and the explicit variance $(\pi\sqrt{\det\Sigma})^{-1}$ quantify how two-dimensional recurrence, with the walk repeatedly revisiting sites, dominates the fluctuations.

What carries the argument

The central object is the self-intersection count of the walk, $V(\omega,I,J,p)=\#\{(u,v)\in I\times J : Z_u-Z_v=p\}$, especially $V_n(\omega)=V(\omega,[0,n),[0,n),0)$. Its almost-sure law of large numbers, $V(\omega,[1,n],p)/(C_0 n\log n)\to1$ for each $p\in\mathbb{Z}^2$ with $C_0=(\pi\sqrt{\det\Sigma})^{-1}$ in the strongly aperiodic case, makes the summation weights $\delta_0$-regular and fixes the normalization and asymptotic variance. Around this sit two auxiliary mechanisms: asymptotic orthogonality of the cross terms between separated time intervals, which yields the finite-dimensional convergence with the correct Wiener covariance, and maximal inequalities — for associated random variables in the iid and moving-average cases, and a fourth-moment maximal inequality for partial sums in the algebraic case. In the algebraic case the fourth-moment bound is controlled by counting solutions of the S-unit equation $\alpha^{\ell_1}-\alpha^{\ell_2}+\alpha^{\ell_3}=1$, whose solution set is finite by a known result on S-unit equations; that finiteness is what gives tightness.

What would settle it

One concrete check is to simulate a strongly aperiodic centered two-dimensional walk with finite variance and test whether $V(\omega,[1,n],p)/(\pi\sqrt{\det\Sigma}\,n\log n)$ converges almost surely to 1 for $p=0$ and for $p\neq0$; a single $p$ where the ratio does not converge would break the variance normalization on which Theorems 2.2 and 6.3 rest. Alternatively, an iid scenery with $E(X_0^2)=1$ but with $E|X_0|^2(\log^+|X_0|)^\chi$ infinite whose normalized sums fail to converge weakly would show the logarithmic moment condition is more than technical.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 2.2: if $(X_\ell)$ is a centered iid random field on $\mathbb{Z}^2$ with $E(X_0^2)=1$, then for almost every walk path $\omega$ the process $t\mapsto S_{\lfloor nt\rfloor}^{\omega,X}(x)/\sqrt{n\log n}$ satisfies a functional CLT with asymptotic variance $\sigma^2=(\pi\sqrt{\det\Sigma})^{-1}$. This is a quenched statement: the randomness of the scenery is integrated out with respect to the measure $\mu$ on the field, while the walk path is held fixed. It sharpens the earlier quenched FCLT of [17] by dropping its logarithmic moment assumption. Theorem 6.3 extends the same conclusion to random fields generated by a totally ergodic $\mathbb{Z}^2$-action of commuting automorphisms of a torus, for observables with absolutely convergent Fourier series and non-zero asymptotic variance. A quenched FCLT for the Lorentz process in random scenery is also indicated, using the law of large numbers for self-intersections of the billiard map.

Load-bearing premise

The paper assumes, without reproving it, the almost-sure law of large numbers for self-intersection counts, $V(\omega,[1,n],p)/(C_0 n\log n)\to1$ for every $p\in\mathbb{Z}^2$, quoted from the authors' earlier work; if this ratio failed to converge for some $p$, the $\sqrt{n\log n}$ normalization would not give a non-degenerate Wiener limit.

Editorial extensions

If this is right

For iid sceneries in two dimensions, only the second moment of the field is needed for the quenched FCLT; the normalization is $\sqrt{n\log n}$ and the limiting variance is $(\pi\sqrt{\det\Sigma})^{-1}$.
For almost every walk path, the normalized partial-sum process converges to the same Wiener process, so the limit law is genuinely quenched rather than merely annealed.
Moving averages of an iid field with absolutely summable coefficients and non-zero coefficient sum satisfy the same quenched FCLT, with variance $|\sum_q a_q|^2(\pi\sqrt{\det\Sigma})^{-1}$.
Random fields of the form $f\circ A^\ell$ under a totally ergodic $\mathbb{Z}^2$-action of commuting toral automorphisms satisfy the quenched FCLT for any observable in $AC_0$ with non-zero asymptotic variance.
When the random walk is replaced by a planar Lorentz process in a random scenery, the same argument gives a quenched FCLT once the self-intersection LLN for the billiard map holds, as sketched in the paper.

Reading between the lines

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

The proof scheme suggests that any recurrent two-dimensional walk whose local times satisfy the per-displacement self-intersection LLN and asymptotic orthogonality will inherit the quenched FCLT; a concrete testable case is a centered random walk in a two-dimensional random environment once the needed local-time LLN is established.
In the algebraic setting, the absolutely-convergent-Fourier-series assumption is used for approximation by trigonometric polynomials and for spectral-density control; a natural extension to probe is whether indicator functions or other $L^2$ observables with continuous spectral density still satisfy the quenched FCLT.
For dimensions $d>2$ the walk is transient and self-intersections are an order of magnitude smaller, so the same mechanism should give a quenched CLT with $\sqrt{n}$ normalization; the paper only notes that higher dimensions are easier, leaving this as an implicit next case.
The Lorentz-process example transfers the argument from walk self-intersections to billiard self-intersections, so the same route may cover other recurrent dynamical trajectories for which a self-intersection LLN is known.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real news is Theorem 2.2: for iid 2d random sceneries, quenched FCLT under just E(X0^2)<∞, with the explicit variance (π√det Σ)^{-1}. That improves Guillotin-Plantard–Poisat–Soares, which needed a logarithmic moment, and the proof via Newman–Wright association plus truncation is clean. The moving-average FCLT (Theorem 4.1) and the toral-automorphism FCLT (Theorem 6.3) are also new functional versions of earlier CLTs; the cumulant machinery is used honestly, and the Móricz-based tightness criterion in Section 5 is a useful standalone tool. The variance asymptotics in Section 1 — especially the asymptotic orthogonality of disjoint increments — is careful and appears correct.

The soft spots are real but not disabling. First, everything rests on the a.s. LLN for shifted self-intersection counts, equation (12), cited from the authors' own [4]. That is load-bearing: it supplies the deterministic variance and the orthogonality used for fidi convergence. I do not regard this as circular — [4] is published and independent — but a referee should verify that (12) is stated there exactly as used, including the uniformity over p∈Z^2. If there is any gap in [4], it would propagate directly into Theorems 2.2, 4.1, and 6.3. Second, the Lorentz-process section is explicitly a sketch; it is not a proved theorem. That is fine as an advertisement, but it should be labeled as open or as a sketch in the abstract/final version, not listed as a proved result. Third, the fidi convergence for the iid case is deferred to Bolthausen's argument rather than written out; a few more details would make the paper more self-contained, though the missing part is standard. The toral section also has some minor notational roughness (e.g., the garbled line in the proof of Theorem 6.3), but nothing that blocks understanding.

Citation pattern: the reliance on [4] and [5] is appropriate — these are the authors' own published tools, and the new contribution is the functional form and the wider class of fields. No fitted parameters, no invented objects; the mathematics is standard and technically sound conditional on (12).

Who is this for? People working on random walks in random sceneries, quenched limit theorems, and Z^d-actions on tori. It deserves a serious referee; I would send it out, with instructions to check the statement of (12) in [4] and to ask the authors to either prove the Lorentz version or clearly mark it as a sketch. Recommend minor-to-moderate revision rather than rejection.

Referee Report

4 major / 5 minor

Summary. The paper proves quenched functional central limit theorems for sums of a stationary random field along a two-dimensional random walk. The main results are: (i) a quenched FCLT for iid random sceneries under only a finite second moment, improving a result of Guillotin-Plantard et al. that required an extra logarithmic moment; (ii) a quenched FCLT for moving averages of iid variables; and (iii) a quenched FCLT for random fields generated by a totally ergodic Z^2-action by commuting toral automorphisms with observables in AC0. The proofs use an almost-sure law of large numbers for self-intersections of the random walk to identify the variance normalization, the Newman--Wright maximal inequality for associated variables to obtain tightness in the independent and moving-average cases, and cumulant estimates together with S-unit bounds and a Móricz maximal inequality for the algebraic case. A brief sketch is given for a Lorentz-process version.

Significance. If the results are correct, the paper makes a genuine improvement by removing the logarithmic moment condition in the iid random-scenery FCLT and by extending quenched FCLTs to non-iid fields arising from toral automorphisms. The proof strategy is modular and the variance constants are explicit. The reliance on the S-unit theorem to control fourth moments in the algebraic setting is elegant. The main caveats are that the central results inherit a deep cited input, the a.s. self-intersection LLN (12), and that some load-bearing parts of the proofs are presented as sketches or references rather than complete arguments.

major comments (4)

[Section 1.2, Eq. (12)] Equation (12) is load-bearing: it supplies the a.s. normalization V(omega,[1,n],p)/(C0 n log n) -> 1 that determines the deterministic variance in Theorems 2.2, 4.1, and 6.3 and the asymptotic orthogonality in Lemma 1.5. The paper states that (12) is a theorem from the authors' prior work [4, Theorem 3.13] and does not reprove it. This is not in itself an error, but the Introduction's claim that the independent-case proof is 'short and self-contained' is misleading. The manuscript should state prominently that Theorems 2.2 and 6.3 are conditional on (12), or include a proof or precise reference for it.
[Theorem 2.2, Section 2.1] The finite-dimensional convergence for the iid random scenery is not actually proved in this section. The proof says it follows 'as in Bolthausen' or, alternatively, 'based on truncation and cumulants, is like the more general case of moving averages in Section 4'. Since the improvement over [17] is precisely the removal of the logarithmic moment condition, the truncation argument for the fidi part should be written out rather than left as an analogy, especially because the convergence must be quenched, not annealed.
[Theorem 4.1, Section 4, item 2) Tightness] The tightness proof for moving averages is reduced to a single sentence: 'The proof is like the proof of tightness in Theorem 2.2'. For a moving-average field the variables are associated but not independent, and the variance of a block sum S_J^Xi is not simply V(omega,J); a fourth-moment bound analogous to (32) must be derived using the coefficients a_q and the self-intersection bounds. This is a load-bearing step for Theorem 4.1 and should be presented in detail.
[Section 2.2, Lorentz process model] The Lorentz-process part contains no formal theorem statement and no proof; it is a short sketch that cites [25, Proposition 7] and [26, Corollary 4]. If this is intended to be one of the paper's advertised examples, the abstract and introduction overstate what is proved. The section should either give a precise statement and a complete argument, or be explicitly labeled as a heuristic announcement.

minor comments (5)

[Theorem 6.3 statement] The phrase 'satisfies a FCLT holds' is ungrammatical; 'holds' should be deleted.
[Introduction] The phrase 'Our proof is short and self-contained' should be qualified because equation (12) is imported from [4, Theorem 3.13] and is not proved in this paper.
[Section 1.2, footnote 6] The formula sigma^2 = (pi sqrt(det Sigma))^{-1} in Theorems 2.2 and 4.1 is stated without qualification, while footnote 6 says this constant holds for strongly aperiodic walks and refers to [4] for the general aperiodic case. The theorem statements should either include the strong aperiodicity assumption or state the condition under which the displayed constant is valid.
[Lemma 1.2] In the proof of Lemma 1.2, the local limit theorem is applied to bound sums over i0,i1,i2,i3 by (i1 i2 i3)^{-1}; the contribution of small index differences is not discussed and a sentence clarifying the treatment of that range would improve readability.
[Section 5] The notation for the block size is inconsistent: the text uses both Delta_n = n^{1/2}(log n)^{-2} and n^{1/2}(ln n)^{-2}. Please standardize the logarithm notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cited self-intersection LLN is a genuine input, not an equivalent reformulation of the quenched FCLT.

full rationale

The paper's central claims are not equivalent to their inputs by construction. Equation (12), the a.s. law of large numbers for shifted self-intersection counts V(omega,[1,n],p)/(C0 n log n) -> 1, is load-bearing: it fixes the normalization sqrt(n log n), the variance constant C0, and the asymptotic orthogonality used for finite-dimensional convergence, and it is cited from the authors' prior paper [4, Theorem 3.13] rather than reproved. This is not circular, however. The cited theorem is a published, parameter-free statement about self-intersections of a random walk whose assumptions do not include the quenched FCLT, and the FCLT conclusion is strictly stronger than that LLN. The paper supplies additional and independent work: Cramer-Wold and Lindeberg-type arguments for fidi convergence, Newman-Wright maximal inequality and truncation for tightness in Theorem 2.2, cumulant estimates and approximation arguments in Sections 4 and 6, and Moricz's moment inequality in Proposition 5.2. The cumulant criterion of [4, Theorem 6.2] similarly is a prior mathematical result, not a restatement of the target theorem. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors to force a choice, and no known result is merely renamed. The self-citations are relied upon, but they are genuine external support under ordinary mathematical practice, so no circular step can be exhibited. The proof is not literally self-contained, but dependence on a legitimate prior theorem is not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs rest on standard probabilistic and ergodic-theoretic inputs: the LLN for self-intersections of a 2D recurrent random walk, the local limit theorem, the Newman-Wright and Móricz maximal inequalities, the cumulant method, and the Evertse-Schlickewei-Schmidt theorem on S-unit equations. The authors prove the necessary moment bounds (Lemma 1.2) and variance asymptotics (Lemma 1.5) in the paper, but the LLN for self-intersections (12) is cited from prior work, including the authors' own [4]. No free parameters are fitted and no new entities are postulated.

assumptions (5)

domain assumption For a 2D aperiodic centered random walk with finite variance, the self-intersection counts satisfy V(ω,[1,n],p)/(C0 n log n) → 1 for P-a.e. ω and every p∈Z² (equation (12)).
Used in Lemma 1.5 and Lemma 1.6 to derive the asymptotic variance of quenched sums; cited from [4, Theorem 3.13] and [2].
domain assumption Absolute summability of the series of decorrelations ∑_{k∈Z^d}|⟨T^k f, f⟩| < ∞ (Condition (1)).
Ensures existence and continuity of the spectral density, the basis for the asymptotic variance formula (5). For iid and moving averages it follows from the iid structure; for the algebraic case with f∈AC0 it is proved in Proposition 6.1.
domain assumption For a totally ergodic Z²-action by commuting automorphisms of a torus, the action is mixing of all orders.
Used in Section 6 to conclude that cumulants of trigonometric polynomial observables vanish for sufficiently separated indices (Proposition 3.2 and Step 1a of Theorem 6.3).
standard math Evertse-Schlickewei-Schmidt theorem: for a finite-rank subgroup Γ of (K*)^r, the equation a1x1+...+arxr=1 has finitely many solutions with no vanishing proper subsum, with a bound depending only on the rank (Theorem 6.2).
Used in the tightness proof for algebraic models to bound the number of solutions of α^{ℓ1}-α^{ℓ2}+α^{ℓ3}=1, giving the H estimate in Section 6.2.
standard math Newman-Wright maximal inequality (31) and Móricz moment inequality (Theorem 5.1).
Provide the tightness criteria for the iid/moving-average cases and the algebraic case, respectively.

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Pith. "Pith review of On the quenched functional CLT in 2d random sceneries, examples." pith.science (2026). https://pith.science/paper/TUT6LJWR

@misc{pith2026190803777,
  author       = {Pith},
  title        = {Pith review of: On the quenched functional CLT in 2d random sceneries, examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUT6LJWR}},
  note         = {Machine review of arXiv:1908.03777}
}

read the original abstract

We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a 2d-random walk in different situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

Cohen, G., Conze, J.-P.: CLT for random walks of commutin g endomorphisms on compact abelian groups, J. Theoret. Probab. 30 (2017), no. 1, 143-195

work page 2017

[2] [5]

Cohen, G., Conze, J.-P.: Almost mixing of all orders and C LT for some Zd-actions on subgroups of FZd p (2016), https://arxiv.org/abs/1609.06484

work page Pith review arXiv 2016

[3] [17]

, A quenched functional central limit theorem for planar random walks in random sceneries

Guillotin-Plantard, N., Poisat, J., Renato Soares, S. , A quenched functional central limit theorem for planar random walks in random sceneries. Electron. Commun. Probab. 19 (2014), no. 3, 9 pp

work page 2014

[4] [1]

John Wiley & Sons, Inc., NY, 1999

Billingsley, P.: Convergence of probability measures, 2d edition. John Wiley & Sons, Inc., NY, 1999. 24 GUY COHEN AND JEAN-PIERRE CONZE

work page 1999

[5] [2]

Bolthausen, E.: A central limit theorem for two-dimensi onal random walks in random sceneries, Ann. Probab. 17, no. 1, 108-115 (1989)

work page 1989

[6] [3]

Cohen, G., Conze, J.-P.: Central limit theorem for commu tative semigroups of toral endomorphisms (2013), https://arxiv.org/abs/1304.4556

work page Pith review arXiv 2013

[7] [8]

Graduate Texts in Mathematics, 138

Cohen, H.: A course in computational algebraic number th eory. Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin (1993). doi: 10.1007/978-3-662- 02945-9

doi:10.1007/978-3-662- 1993

[8] [9]

London Math

Cramér, H., Wold, H.: Some Theorems on Distribution Func tions, J. London Math. Soc. 11 (1936), no. 4, 290-294

work page 1936