{"id":"dbfd76b9-c804-494d-97a5-c7f24d9951be","arxiv_id":"2505.01095","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Moderate deviation principles for the equilibrium fluctuation fields of the one-dimensional facilitated exclusion process are established, with quadratic rate functions in the symmetric case and a purely initial-condition rate in the asymmetric case.","lead":"A mathematical proof that density fluctuations of a constrained particle system, the facilitated exclusion process, satisfy large-deviation-type principles at intermediate scales. The result gives exact rate functions for both symmetric and asymmetric dynamics, using a new technical tool based on logarithmic Sobolev inequalities.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Log-Sobolev inequality for FEP with boundary conditions is cited to an in-preparation reference and is load-bearing; if its constant is not O(ℓ^2), the super-exponential BG principle and the MDP upper bound fail.","rationale":"The paper's central claim (Theorem 2.1) would require a working super-exponential Boltzmann-Gibbs principle (Prop. 3.1), and the proof of that principle hinges on a logarithmic Sobolev inequality for the FEP on finite intervals with boundary conditions. The reader's weakest-assumption analysis correctly identifies this inequality as the critical unresolved input: it is cited to an in-preparation manuscript, and no proof or detailed reference is given. I checked the algebra around (3.18)–(3.22): the LSI constant enters as Cℓ^3D(f)/γ in (3.22), and the subsequent optimization γ = C a_N M ℓ^3/N^3 is chosen precisely to cancel the -N^2D(f) term in (3.18). Any departure from an O(ℓ^2) constant degrades the final bound and can violate (3.10). The asymmetric section is explicitly an outline and inherits the same LSI input plus the extra condition a_N ≫ sqrt N (log N)^2. No internal contradiction was found in the rest of the argument; the rates and rate functions appear consistent, and the zero dynamical rate in the asymmetric case is plausible from the size of the second-order term. Thus the appropriate verdict is CONDITIONAL, unchanged from the reader.","tokens_in":26657,"tokens_out":25836,"duration_ms":251273,"concrete_test":"Obtain or independently re-derive the proof of the logarithmic Sobolev inequality used in §3.3.1 (the text says it follows from the FEP-to-SSEP mapping and cites [10]) and verify that the constant is O(ℓ^2) uniformly in x, ℓ, k, a, b with a,b∈{0,1} and k/(2ℓ+1)∈(ρ-δ,ρ+δ). In particular, check the boundary terms: if fixing η_{x-ℓ-1}=a or η_{x+ℓ+1}=b degrades the constant to O(ℓ^3), then the choice γ = C a_N M ℓ^3/N^3 no longer cancels the Dirichlet term and the proof collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.1 (the super-exponential Boltzmann-Gibbs principle) is the key step for the MDP upper bound. Its proof in §3.3.1 invokes the logarithmic Sobolev inequality for the FEP on intervals with boundary conditions: H(f|π^{a,b}_{x,ℓ,k}) ≤ Cℓ^2 D^{a,b}_{x,ℓ,k}(√f;π^{a,b}_{x,ℓ,k}), attributed to the in-preparation reference [10]. This inequality is used through (3.19)–(3.22): the entropy term Cℓ^3D(f;πρ)/γ in (3.22) must be cancelled by the -N^2D term in (3.18), and the remaining terms must vanish under the choice ℓ=N a_N^{-3/4} and (3.10). If the LSI constant is worse than O(ℓ^2) — e.g. depends on the boundary values a,b or is O(ℓ^{2+ε}) — the cancellation is imperfect and the error bound (3.10) cannot be achieved, so (3.9) fails and the upper bound in Theorem 2.1 is not proven. Since [10] is not publicly available and no proof is sketched, the central estimate is currently unverifiable. The asymmetric case (§5) depends on the same type of estimate (Proposition 5.1) and is only outlined; it is therefore also conditional on the same unresolved input.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives moderate deviation principles (MDPs) for the rescaled density fluctuation fields of the one-dimensional facilitated exclusion process (FEP) started from its stationary measure, in both the symmetric and asymmetric cases, with speed a_N^2/N where sqrt(N) << a_N << N. In the symmetric case the rate function is Qsym = Qini + Qdyn, with Qdyn a quadratic functional of the path; in the asymmetric case the rate function is Qasym = Qini + Qasym_dyn, and Qasym_dyn is proved to be zero whenever finite. The proof follows the standard exponential-martingale/entropy strategy: an upper bound via exponential martingales and a new super-exponential Boltzmann-Gibbs principle, and a lower bound via a hydrodynamic limit for a tilted dynamics. The main technical input is a logarithmic Sobolev inequality for the FEP on finite intervals with boundary conditions, cited to an in-preparation work, together with equivalence-of-ensembles results from a prior paper by the author.","tokens_in":26986,"tokens_out":11838,"duration_ms":118976,"significance":"If the logarithmic Sobolev inequality input is valid, the paper makes a solid contribution: it extends moderate deviation theory beyond product invariant measures, provides explicit parameter-free rate functions, and introduces an LSI-based method for the Boltzmann-Gibbs principle that may apply to other conservative lattice gases. The symmetric case is treated in considerable detail, the initial-condition MDP (Lemma 3.2) is proved in full, and the estimates are quantified. However, the central LSI estimate is not independently verifiable, and the asymmetric case is only outlined; these issues currently make the main theorem conditional on unpublished material.","major_comments":[{"comment":"The logarithmic Sobolev inequality H(f|π^{a,b}_{x,ℓ,k}) ≤ Cℓ² D^{a,b}_{x,ℓ,k}(√f; π^{a,b}_{x,ℓ,k}) is cited to the in-preparation reference [10]. This inequality is load-bearing: the entropy term Cℓ³D(f;πρ)/γ in (3.22) is cancelled against the −N²D term in (3.18) by the choice γ = C a_N M ℓ³/N³, and the cancellation requires the LSI constant to be O(ℓ²) uniformly in the boundary values a,b and the density parameter k. The manuscript gives no proof or sketch of this inequality, and [10] is not publicly available. Since this step is essential for the super-exponential Boltzmann-Gibbs principle (Proposition 3.1) and hence for the MDP upper bound, the central claim of the paper is not verifiable as written.","section":"§3.3.1, LSI display after Eq. (3.22)"},{"comment":"The asymmetric case is only outlined. The proof of Proposition 5.1 is stated to be 'similar to Proposition 3.1', but the different time scaling (N instead of N²) changes the Feynman-Kac term and the final error terms; the displayed final bound (ℓ²/(N a_N) + N(log ℓ)²/(a_N ℓ) + ℓ²/N + N³/(a_N³ ℓ)e^{−Cℓ} + N/(a_N ℓ)) is given without derivation, and the choice ℓ = ε√N is not accompanied by the detailed verification that all five terms vanish under the assumption a_N ≫ √N (log N)². Moreover, the hydrodynamic limit for the tilted dynamics in Proposition 5.2 is stated without proof. Since Theorem 2.1 explicitly covers the asymmetric case, this half of the main result is incomplete.","section":"§5, Proposition 5.1 and the bound after it"},{"comment":"The bound E[e^{αX}] ≤ Cα² for X = (2ℓ′+1)^{−1/2} ∑_{|y|≤ℓ′} (τ_y g − E[τ_y g]) under the conditioned measure π^{a,b}_{ℓ,k} is essential for the γ²/ℓ term in (3.22), but it is only justified heuristically and cited to [29, Eq. (5.21)]. The cited result concerns lattice gases with mixing conditions; the verification that it applies to the FEP conditioned measures, with the stated uniformity in a,b,k, is not supplied. This estimate is needed for the error bound (3.10) to hold, so this is another unverified load-bearing input.","section":"§3.3.1, the Gaussian/exponential moment estimate after Eq. (3.23)"}],"minor_comments":[{"comment":"There are several typographical errors and ambiguous formulas: 'which will go to infinity at last' in §2.1, 'boundary v alues' in §3.3.1, and the exponential-martingale factor in (3.2) is hard to parse; the factor appears to be N/a_N² rather than N a_N², and missing parentheses in the definition of πN_{ρ,φ} in Section 4 make the formula difficult to read.","section":"§2.1 and §3"},{"comment":"The reduction removing the supremum over time in Proposition 3.1 is delegated to [30, Proof of Lemma 3.1]; a brief explanation of the Garsia-Rodemich-Rumsey step would improve readability.","section":"§3.3, opening paragraph"},{"comment":"The notation ~g(ηℓ_x(s)) is not defined explicitly; it should be stated that ~g is evaluated at the local average density ηℓ_x(s).","section":"Equations (3.11)–(3.12)"},{"comment":"The display using the Garsia-Rodemich-Rumsey inequality contains an unclear expression 'Cδ^{1/3 − 1/6 B^{1/12}}'; this should read C δ^{1/3} B^{1/12} (or similar), and the constants should be specified.","section":"§3.2.2"},{"comment":"The dependence on the in-preparation reference [10] should be flagged clearly in the introduction, and the authors should either include a proof of the LSI in an appendix or state that Theorem 2.1 is conditional on [10].","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the unpublished logarithmic Sobolev inequality. If the author can supply a proof or a public preprint of [10], the symmetric case is likely acceptable. The asymmetric case needs to be expanded from an outline to a verifiable proof. I recommend major revision rather than rejection because the approach is credible and the symmetric case is written in detail."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Linjie Zhao proves the first moderate deviation principles for the facilitated exclusion process, in equilibrium, in both the symmetric and asymmetric regimes, with explicit rate functions and no fitted constants. The genuinely new step is a super-exponential Boltzmann-Gibbs principle proved via a logarithmic Sobolev inequality for the non-product invariant measure; this is a real methodological contribution, since earlier MDPs for WASEP and reaction-diffusion relied on product measures. The LSI route should transfer to other non-product systems like zero-range processes.\n\nThe symmetric case is treated in detail. The rate function splits into an initial part and a dynamical part, the latter quadratic with diffusion coefficient tied to the derivative of the current. Lemma 3.2, the MDP for the stationary measure, is self-contained and uses martingale differences plus concentration arguments. The lower bound follows the perturbed-hydrodynamics template and is competently handled. The asymmetric outline is brief but honest, and the conclusion that the finite dynamical rate vanishes is a nice observation.\n\nThe soft spot is exactly the one the stress-test identifies, and I confirm it after reading the proof. Proposition 3.1 is load-bearing for the upper bound. The proof in §3.3.1 invokes the logarithmic Sobolev inequality on intervals with boundary conditions, H(f|π) ≤ Cℓ²D(√f;π), with C independent of boundary data. That inequality is attributed to the in-preparation reference [10] and no proof is sketched. If the constant is worse than O(ℓ²), the entropy term Cℓ³D in (3.22) is not cancelled by the N²D term in (3.18), the error bound (3.10) fails, and the moderate deviation upper bound falls. The asymmetric Proposition 5.1 uses the same input. So the central estimate is currently unverifiable by a referee.\n\nThis is addressable, not a fundamental error. The citations to [15] and [31] are to public sources; the only missing input is [10]. The author openly acknowledges the debt to Da Cunha's notes, so the gap is not concealed.\n\nMy recommendation: send to peer review, with the condition that the LSI proof be included or a public preprint of [10] be supplied. Once that is available, the paper is solid. Until then, readers should treat Theorem 2.1 as conditional. Bring it to a reading group after the LSI preprint appears; not much point discussing the key step from hearsay.","headline":"First moderate deviations for the facilitated exclusion process, with a genuinely new LSI-based Boltzmann-Gibbs principle, but the key log-Sobolev inequality is cited to an in-preparation paper and must be supplied before the result is fully verified.","tokens_in":27424,"tokens_out":4031,"would_cite":false,"duration_ms":39267,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35","60F10","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Moderate deviations of the facilitated exclusion process are governed by a quadratic rate in the symmetric case and by the initial measure alone in the asymmetric case.","keywords":["facilitated exclusion process","moderate deviations","fluctuation fields","Boltzmann-Gibbs principle","logarithmic Sobolev inequality","non-product invariant measures","equivalence of ensembles","hydrodynamic limit"],"falsifier":"Compute the optimal logarithmic Sobolev constant of the FEP on an interval of length $\\ell$ at a fixed density $\\rho\\in(1/2,1)$. If it grows faster than $C\\ell^2$, the length scale used in Section 3.3 cannot make the error in (3.22) vanish, so Proposition 3.1 and Theorem 2.1 would fail; if it grows as $\\ell^2$ uniformly in the boundary conditions, the paper's main hypothesis is verified.","tokens_in":26492,"feed_emoji":"📉","tokens_out":12256,"duration_ms":114512,"temperature":0.7,"pith_summary":"The paper derives moderate deviation principles for the density fluctuation field of the one-dimensional facilitated exclusion process (FEP), a lattice gas in which a particle may jump only when a neighboring particle is present, so that equilibrium states are not product measures. Starting from the stationary measure of density $\\rho \\in (1/2,1)$, fluctuations of amplitude $a_N$ with $\\sqrt{N} \\ll a_N \\ll N$ satisfy a large-deviation estimate with speed $a_N^2/N$, and the paper identifies the rate function exactly. In the symmetric case the rate is $Q_{\\mathrm{ini}}+Q_{\\mathrm{dyn}}$, with $Q_{\\mathrm{dyn}}$ quadratic in the gradient of the test profile and coefficients computed from the model; in the asymmetric case any finite dynamical cost is zero, so deviations come only from the initial measure. The main step is a super-exponential Boltzmann-Gibbs principle proved through a logarithmic Sobolev inequality for the FEP, replacing the polynomial arguments used for the simple exclusion process.","feed_headline":"Facilitated exclusion process: deviations are quadratic or zero","feed_subtitle":"In equilibrium, symmetric fluctuations pay a quadratic cost; asymmetric ones only through the initial measure.","key_machinery":"The central object is the exponential martingale $M_t^N(H)$ associated with the fluctuation field; the FEP's gradient structure rewrites its exponent as $\\ell_T(\\mu,H)-A(\\rho)\\int_0^T\\|\\nabla H_t\\|^2dt$ plus a remainder that is super-exponentially negligible, converting the upper bound into a variational problem. The load-bearing mechanism is the super-exponential Boltzmann-Gibbs principle (Proposition 3.1): for any local function $g$, the time-integrated field of $\\sum_x \\tau_x(g(\\eta)-\\bar g(\\rho)-\\bar g'(\\rho)(\\eta_0-\\rho))H(x/N)$, rescaled by $a_N$, is super-exponentially small. Since the invariant measures are not product, the proof cannot exploit polynomial structure; it uses a logarithmic Sobolev inequality on finite intervals with boundary conditions, $H(f|\\pi^{a,b}_{x,\\ell,k})\\le C\\ell^2 D_{a,b}^{x,\\ell,k}(\\sqrt{f};\\pi^{a,b}_{x,\\ell,k})$, together with an equivalence-of-ensembles estimate of order $(\\log \\ell)^2/\\ell$, and a length scale $\\ell$ chosen so that $N/a_N\\ll \\ell/(\\log \\ell)^2$ and $\\ell\\ll\\sqrt{N a_N}$. These bounds make the entropy and exponential-moment remainders vanish at speed $a_N^2/N$. The same replacement principle, applied under a perturbed dynamics, identifies the heat or transport equation that drives the lower bound.","core_discovery":"The central claim is Theorem 2.1. For densities $1/2<\\rho<1$, take the FEP with generator accelerated by $N^2$ in the symmetric case or by $N$ in the asymmetric case, started from its stationary measure, and rescale the density field as $\\mu_t^N(H)=a_N^{-1}\\sum_x(\\eta_x(t)-\\rho)H(x/N)$ with $\\sqrt{N}\\ll a_N\\ll N$. The theorem states that $(\\mu_t^N)_{0\\le t\\le T}$ satisfies a moderate deviation principle with speed $a_N^2/N$ and rate function $Q^{\\mathrm{sym}}$ in the symmetric case, and $Q^{\\mathrm{asym}}$ in the asymmetric case under the additional hypothesis $a_N\\gg\\sqrt{N}(\\log N)^2$. The rate functions split as $Q^{\\mathrm{sym}}=Q_{\\mathrm{ini}}+Q_{\\mathrm{dyn}}^{\\mathrm{sym}}$ and $Q^{\\mathrm{asym}}=Q_{\\mathrm{ini}}+Q_{\\mathrm{dyn}}^{\\mathrm{asym}}$, where $Q_{\\mathrm{ini}}$ is the Gaussian cost of the initial fluctuation with variance $B(\\rho)=(2\\rho-1)\\rho(1-\\rho)$, and $Q_{\\mathrm{dyn}}^{\\mathrm{sym}}$ is the quadratic cost $A(\\rho)\\int_0^T\\|\\nabla H_t\\|^2_{L^2(\\mathbb{R})}dt$ with $A(\\rho)=(1-\\rho)(2\\rho-1)/\\rho$, expressed through the diffusion coefficient $\\bar h'(\\rho)$ appearing in the linear functional $\\ell_T$. In the asymmetric case the paper proves that if $Q_{\\mathrm{dyn}}^{\\mathrm{asym}}$ is finite, it is zero: on the hyperbolic time scale the dynamics contributes no bulk deviation, and the whole cost is $Q_{\\mathrm{ini}}$.","pith_inferences":["The zero dynamical rate in the asymmetric case suggests that accelerating by $N^{3/2}$ should produce a nontrivial rate function built from the same transport term; testing whether the exponential martingale remains controlled at that scale would be a direct next step.","The LSI-based proof is a template for other non-product equilibrium systems: for the zero-range process or kinetically constrained models, the only new input needed is the uniform $\\ell^2$ log-Sobolev constant, and the structure of the proof would carry over with the model-specific constants.","The paper's density cut-off away from $1/2$ and $1$ suggests the boundary densities are the singular cases; one could test numerically whether moderate deviations persist at $\\rho=1/2$, where the facilitation constraint can freeze particles and the equivalence-of-ensembles estimates change."],"forward_implications":["For any intermediate scale $a_N$, the probability that the symmetric FEP fluctuation field visits a closed set $C$ decays as $\\exp\\{-(a_N^2/N)\\inf_C Q^{\\mathrm{sym}}\\}$, so the moderate scale interpolates between the $\\sqrt{N}$ central-limit scale and the hydrodynamic scale.","In the asymmetric case, the hyperbolic time scale produces no dynamical deviations at the moderate level: the only cost is the initial randomness, so any nontrivial dynamical large deviations must be sought at the longer $N^{3/2}$ time scale, which the paper explicitly leaves open.","The proof's replacement principle works without product invariant measures, so the logarithmic Sobolev route applies to any conservative lattice gas with uniform $\\ell^2$ log-Sobolev estimates, such as the zero-range process.","The explicit constants $A(\\rho)$ and $\\bar h'(\\rho)$ in the symmetric rate tie the moderate deviations to the FEP's transport coefficients, giving an exponential-asymptotic signature of the diffusion constant."],"supporting_citations":[{"why":"Defines the facilitated exclusion process and its grand-canonical equilibrium states, and supplies the exponential decay of correlations used throughout the estimates.","marker":"[5]"},{"why":"In-preparation work cited for the logarithmic Sobolev inequality on finite intervals with boundary conditions, the entropy control behind the Boltzmann-Gibbs principle.","marker":"[10]"},{"why":"Supplies the equivalence-of-ensembles bound for canonical FEP states and the stationary-fluctuation setup that this paper builds on.","marker":"[15]"},{"why":"Supplies the moderate-deviation estimate for martingales and mixing sequences used to prove the initial-measure MDP (Lemma 3.2).","marker":"[16]"},{"why":"Gives the standard moderate-deviation framework for exclusion processes, including exponential tightness criteria, the minimax argument, and rate-function properties.","marker":"[17]"},{"why":"Provides the general toolbox—exponential martingales, Feynman-Kac, entropy inequality, Dirichlet forms, and the Boltzmann-Gibbs principle—that structures the proof.","marker":"[22]"},{"why":"Provides the normal-approximation exponential estimate used to bound moments of block averages inside the Boltzmann-Gibbs proof.","marker":"[29]"},{"why":"Shows how to remove the supremum over time in the super-exponential Boltzmann-Gibbs principle via the Garsia-Rodemich-Rumsey inequality.","marker":"[30]"},{"why":"Gives the previous super-exponential Boltzmann-Gibbs derivation for the weakly asymmetric exclusion process, whose structure is adapted here to non-product invariant measures.","marker":"[31]"}],"fun_headline_variants":["Symmetric FEP pays quadratic cost, asymmetric pays none","Facilitated exclusion: asymmetric deviations only from initial state","MDP for FEP: quadratic dynamic cost in symmetric, zero in asymmetric","In equilibrium, FEP deviations are quadratic or just Gaussian initial","Asymmetric FEP: no extra deviation cost beyond initial Gaussian"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the logarithmic Sobolev inequality for the FEP on finite intervals, which says relative entropy is controlled by a constant times $\\ell^2$ times the Dirichlet form; if that constant grows faster than $\\ell^2$, the error estimate (3.10) fails and the moderate-deviation upper bound collapses.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric FEP pays quadratic cost, asymmetric pays none","Facilitated exclusion: asymmetric deviations only from initial state","MDP for FEP: quadratic dynamic cost in symmetric, zero in asymmetric","In equilibrium, FEP deviations are quadratic or just Gaussian initial","Asymmetric FEP: no extra deviation cost beyond initial Gaussian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000663,"raw_usage":{"total_tokens":3045,"prompt_tokens":976,"completion_tokens":2069,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":592,"completion_tokens_details":{"reasoning_tokens":1982}},"tokens_in":592,"tokens_out":2069,"duration_ms":13444,"temperature":1.0,"reasoning_tokens":1982,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:25:53.448055+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the optimal logarithmic Sobolev constant of the FEP on an interval of length $\\ell$ at a fixed density $\\rho\\in(1/2,1)$. If it grows faster than $C\\ell^2$, the length scale used in Section 3.3 cannot make the error in (3.22) vanish, so Proposition 3.1 and Theorem 2.1 would fail; if it grows as $\\ell^2$ uniformly in the boundary conditions, the paper's main hypothesis is verified.","supporting_citations":[{"cited_title":"Blondel, C","cited_arxiv_id":null,"evidence_quote":"Defines the facilitated exclusion process and its grand-canonical equilibrium states, and supplies the exponential decay of correlations used throughout the estimates."},{"cited_title":"Da Cunha and L","cited_arxiv_id":null,"evidence_quote":"In-preparation work cited for the logarithmic Sobolev inequality on finite intervals with boundary conditions, the entropy control behind the Boltzmann-Gibbs principle."},{"cited_title":"Erignoux and L","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence-of-ensembles bound for canonical FEP states and the stationary-fluctuation setup that this paper builds on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the moderate-deviation estimate for martingales and mixing sequences used to prove the initial-measure MDP (Lemma 3.2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the standard moderate-deviation framework for exclusion processes, including exponential tightness criteria, the minimax argument, and rate-function properties."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the normal-approximation exponential estimate used to bound moments of block averages inside the Boltzmann-Gibbs proof."},{"cited_title":"Moderate deviation principles for a reaction diffusion model in non-equilibrium","cited_arxiv_id":"2408.11633","evidence_quote":"Shows how to remove the supremum over time in the super-exponential Boltzmann-Gibbs principle via the Garsia-Rodemich-Rumsey inequality."}],"review_version":1}