{"id":"ed8b22a7-741c-4e82-8715-7f0fc512e26a","arxiv_id":"1908.03777","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Quenched FCLTs in 2D random sceneries are proved under only a second moment for iid fields, and for moving averages and toral automorphism fields.","lead":"This paper proves quenched functional central limit theorems for sums of random fields along a 2D random walk, covering iid random sceneries, moving averages, and random fields generated by commuting torus automorphisms. The result improves the known moment condition for iid sceneries from a logarithmic moment to a plain second moment, and extends quenched CLTs to algebraic random fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (12) is the load-bearing input: if the cited a.s. LLN for shifted self-intersection counts fails, the sqrt(n log n) normalization and the Wiener limit in Theorems 2.2 and 6.3 lose their deterministic variance.","rationale":"The reader's weakest assumption identifies the correct load-bearing point. Theorem 2.2's claimed a.s. deterministic variance and the independence of increments in the limiting Wiener process both reduce to equation (12), which is imported from [4] rather than proved here. The other ingredients -- Lindeberg for iid scenery given the local-time bounds, the block selection in Lemma 1.5, the Newman-Wright maximal inequality, and the cumulant-vanishing argument for algebraic actions -- are standard and internally consistent. No fabricated entities, fitted parameters, or obvious internal contradictions were found. The Lorentz-process section is only sketched, but it is secondary to the strongest claim. Since the central argument is sound conditional on the cited self-intersection LLN, the conditional verdict should stand unchanged.","tokens_in":78,"tokens_out":30506,"duration_ms":467928,"concrete_test":"Independently re-derive equation (12) from the local-time strong law for a general aperiodic centered 2D random walk, verifying that the convergence is almost sure for every fixed p in Z^2, with the same constant C0, and on a full-measure set that can be intersected with the sets used in Lemma 1.5. If the re-derivation requires additional hypotheses on the walk or only gives annealed convergence, then the hypotheses of Theorems 2.2 and 6.3 are insufficient as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central results rest on the a.s. convergence V(omega,[1,n],p)/(C0 n log n) -> 1 for each p in Z^2, stated as equation (12) and cited from the authors' prior work [4, Theorem 3.13]. This is not a minor technicality: it supplies the asymptotic variance C0 = (pi sqrt(det Sigma))^{-1}, the a.s. variance normalization (15), and the asymptotic orthogonality of disjoint increments (22) used for the finite-dimensional convergence. If (12) held only in an annealed sense, or only for p=0, then for typical omega the conditional variance would fluctuate and the increments of Y_n would not be asymptotically independent, so the quenched FCLT with a deterministic sigma^2 would fail. The rest of the proof -- Lindeberg with weights w_n(omega,ell), the Newman-Wright maximal inequality, and the truncation argument for tightness -- is coherent conditional on (12). The paper explicitly does not reprove (12), so the central claim is no stronger than the cited input. This is a dependence on a premise, not an internal contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25151,"tokens_out":22374,"duration_ms":241123,"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":[{"comment":"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.","section":"Section 1.2, Eq. (12)"},{"comment":"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.","section":"Theorem 2.2, Section 2.1"},{"comment":"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":"Theorem 4.1, Section 4, item 2) Tightness"},{"comment":"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.","section":"Section 2.2, Lorentz process model"}],"minor_comments":[{"comment":"The phrase 'satisfies a FCLT holds' is ungrammatical; 'holds' should be deleted.","section":"Theorem 6.3 statement"},{"comment":"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":"Introduction"},{"comment":"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.","section":"Section 1.2, footnote 6"},{"comment":"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":"Lemma 1.2"},{"comment":"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.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is readable and the main line of argument appears plausible. The main reason for major revision is that the proofs of several load-bearing claims, especially the moving-average tightness and the status of the self-intersection LLN (12), are either sketched or imported. These issues are fixable within the paper's scope, and I do not see a fatal flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nCitation 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).\n\nWho 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.","headline":"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.","tokens_in":25706,"tokens_out":1519,"would_cite":true,"duration_ms":17954,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","28D05","22D40","60G50","47B15","37A25","37A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["quenched functional CLT","random walk in random scenery","2D random walk","self-intersections of a random walk","toral automorphisms","S-unit equations","associated random variables","Lorentz process"],"falsifier":"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.","tokens_in":24697,"feed_emoji":"🎲","tokens_out":9432,"duration_ms":89385,"temperature":0.7,"pith_summary":"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.","feed_headline":"2D random-walk sums in random scenery converge to a Wiener process","feed_subtitle":"For almost every walk path, the √(n log n)-normalized process converges; no extra log-moment condition.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the annealed-CLT framework and the variance estimates $E(V_n)\\sim C_0 n\\log n$, plus the local-time bound $\\sup_\\ell w_n(\\omega,\\ell)=o(n^\\varepsilon)$ used throughout.","marker":"[2]"},{"why":"Provides the almost-sure law of large numbers for self-intersection counts $V(\\omega,[1,n],p)/(C_0 n\\log n)\\to1$, which fixes the normalization and asymptotic variance.","marker":"[4]"},{"why":"The earlier quenched FCLT whose logarithmic moment condition Theorem 2.2 removes; it serves as the baseline the paper improves on.","marker":"[17]"},{"why":"Gives the maximal inequality for sums of associated random variables that yields tightness in the iid and moving-average cases.","marker":"[24]"},{"why":"Provides the moment inequality for maxima of partial sums used to prove tightness for the algebraic examples.","marker":"[22]"},{"why":"The finiteness theorem for solutions of S-unit equations that bounds the number of quadruples in the fourth-moment estimate for characters.","marker":"[15]"},{"why":"Supplies the law of large numbers for self-intersections of the Lorentz billiard map, the key step in the quenched Lorentz version.","marker":"[26]"},{"why":"Supplies the annealed FCLT and normalization for the Lorentz process in random scenery on which the quenched version is modeled.","marker":"[25]"}],"fun_headline_variants":["Quenched FCLT for 2D random walks: no log moment required","Dropping log moments: quenched FCLT for 2D random scenery","2D random sceneries: quenched FCLT without log-moment assumption","Quenched FCLT for random scenery sums: torus and Lorentz cases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Quenched FCLT for 2D random walks: no log moment required","Dropping log moments: quenched FCLT for 2D random scenery","2D random sceneries: quenched FCLT without log-moment assumption","Quenched FCLT for random scenery sums: torus and Lorentz cases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001554,"raw_usage":{"total_tokens":6173,"prompt_tokens":867,"completion_tokens":5306,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":5218}},"tokens_in":483,"tokens_out":5306,"duration_ms":37778,"temperature":1.0,"reasoning_tokens":5218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:03:29.852542+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the annealed-CLT framework and the variance estimates $E(V_n)\\sim C_0 n\\log n$, plus the local-time bound $\\sup_\\ell w_n(\\omega,\\ell)=o(n^\\varepsilon)$ used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the almost-sure law of large numbers for self-intersection counts $V(\\omega,[1,n],p)/(C_0 n\\log n)\\to1$, which fixes the normalization and asymptotic variance."},{"cited_title":", A quenched functional central limit theorem for planar random walks in random sceneries","cited_arxiv_id":null,"evidence_quote":"The earlier quenched FCLT whose logarithmic moment condition Theorem 2.2 removes; it serves as the baseline the paper improves on."},{"cited_title":"M., Wright, A","cited_arxiv_id":null,"evidence_quote":"Gives the maximal inequality for sums of associated random variables that yields tightness in the iid and moving-average cases."},{"cited_title":"Wahrscheinlichkeitstheorie und Verw","cited_arxiv_id":null,"evidence_quote":"Provides the moment inequality for maxima of partial sums used to prove tightness for the algebraic examples."},{"cited_title":"P., Schmidt, W","cited_arxiv_id":null,"evidence_quote":"The finiteness theorem for solutions of S-unit equations that bounds the number of quadruples in the fourth-moment estimate for characters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the law of large numbers for self-intersections of the Lorentz billiard map, the key step in the quenched Lorentz version."},{"cited_title":"IHP, Probab","cited_arxiv_id":null,"evidence_quote":"Supplies the annealed FCLT and normalization for the Lorentz process in random scenery on which the quenched version is modeled."}],"review_version":1}