Moderate deviations for the facilitated exclusion process in equilibrium

Linjie Zhao

REVIEW 3 major objections 5 minor 31 references

Moderate deviations for the facilitated exclusion process in equilibrium

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.01095 v1 pith:HSZPNVMQ submitted 2025-05-02 math.PR

Linjie Zhao This is my paper · ORCID

classification math.PR MSC 60K3560F1082C22

keywords facilitatedexclusionprocessmoderatedeviationsfluctuationfieldsBoltzmann-GibbsprinciplelogarithmicSobolevinequalitynon-productinvariantmeasuresequivalenceofensembleshydrodynamiclimit

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $Q_{\mathrm{ini}}$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.

The 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.

This 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.

My 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§3.3.1, LSI display after Eq. (3.22)] 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.
[§5, Proposition 5.1 and the bound after it] 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.
[§3.3.1, the Gaussian/exponential moment estimate after Eq. (3.23)] 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.

minor comments (5)

[§2.1 and §3] 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.
[§3.3, opening paragraph] 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.
[Equations (3.11)–(3.12)] 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).
[§3.2.2] 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.
[General] 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].

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the MDP rate functions are explicit density-dependent expressions, and the cited LSI and equivalence-of-ensembles results are external inputs rather than restatements of the target theorem.

full rationale

The paper's derivation chain is not circular. The moderate deviation rate functions are written explicitly as Qsym(μ)=Qini(μ)+Qdyn(μ) and Qasym(μ)=Qini(μ)+Qasym_dyn(μ), with constants A(ρ), B(ρ), h̄′(ρ) computed directly from the invariant measure πρ. There is no fitted parameter that is later relabeled as a prediction. The main technical step, the super-exponential Boltzmann–Gibbs principle (Proposition 3.1), is proved via the entropy inequality, the logarithmic Sobolev inequality for the FEP, and the equivalence-of-ensembles estimate (3.23). The logarithmic Sobolev inequality is introduced as an external fact: 'The logarithm Sobolev inequality states that there exists some constant C independent of x, ℓ, k, a, b such that H(f|π^{a,b}_{x,ℓ,k}) ≤ Cℓ²D^{a,b}_{x,ℓ,k}(√f;π^{a,b}_{x,ℓ,k}). ... see [10].' The equivalence of ensembles is cited to [15, Proposition 5.6]. Both are inputs with stated assumptions — densities in (1/2,1), boundary conditions, and the FEP on intervals — and neither of those assumptions includes the target moderate deviation result. No equation in the paper is shown to be identical to another by construction. The proof is not fully self-contained: [10] is listed as 'in preparation' and the ℓ² LSI constant is load-bearing for the cancellation in (3.18)–(3.22), while the asymmetric case is only outlined. That is a completeness and verifiability concern, not circularity. The same-author citation [15] is used only for the previously established equivalence-of-ensembles bound, not to define the MDP rate function or to derive the MDP itself, so it does not make the central claim circular.

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

No parameters are fitted to data; all constants A(rho), B(rho), d(rho) are explicit functions of the density rho. The central claim rests on the logarithmic Sobolev inequality (unpublished citation) and the equivalence of ensembles from prior work.

assumptions (5)

standard math Feynman-Kac formula and exponential martingale identities hold for the FEP
Used in Section 3 to represent the process via mean-one exponential martingales.
domain assumption Logarithmic Sobolev inequality for the FEP restricted to intervals with boundary conditions: H(f|pi) <= C ell^2 D(sqrt(f); pi)
Load-bearing for Proposition 3.1, cited to [10], an in-preparation manuscript; no proof given in this paper.
domain assumption Equivalence of ensembles for the FEP canonical invariant measures with error C(log ell)^2/ell (Proposition 5.6 of [15])
Used in (3.23) to bound fluctuations of block averages; proven in a previous paper by the author and a coauthor.
domain assumption Exponential decay of correlations and Markov property of the grand-canonical measure pi_rho
Used throughout Sections 3.3 and 3.4, e.g., Eq. (3.25) and the induction arguments.
standard math Sub-Gaussian tail bound for block averages (from [21])
Used in (3.16) to bound density deviation probabilities.

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Cite this review

Pith. "Pith review of Moderate deviations for the facilitated exclusion process in equilibrium." pith.science (2026). https://pith.science/paper/HSZPNVMQ

@misc{pith2026250501095,
  author       = {Pith},
  title        = {Pith review of: Moderate deviations for the facilitated exclusion process in equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSZPNVMQ}},
  note         = {Machine review of arXiv:2505.01095}
}

read the original abstract

We derive the moderate deviation principles for the fluctuation fields of the facilitated exclusion process (FEP) in one dimension when the process starts from its stationary measure, both in the symmetric and asymmetric cases. The main step is to prove a super-exponential version of the Boltzmann-Gibbs principle, which relies on the logarithmic Sobolev inequality for the FEP.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [10]

Da Cunha and L

H. Da Cunha and L. Xu. The asymmetric facilitated exclus ion process with open boundaries. In preparation. 26 LINJIE ZHAO

work page

[2] [15]

Erignoux and L

C. Erignoux and L. Zhao. Stationary ﬂuctuations for the facilitated exclusion process. Electronic Journal of Prob- ability, 29:1–41, 2024

work page 2024

[3] [31]

L. Zhao. Moderate deviation principles for the W ASEP. arXiv preprint arXiv:2405.16151 , 2024. School of Mathematics and Statistics & Hubei Key Laboratory o f Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wu han 430074, China. Email address : linjie zhao@hust.edu.cn

arXiv 2024

[4] [1]

Ayre and P

J. Ayre and P. Chleboun. Mixing times for the facilitated exclusion process. arXiv preprint arXiv:2402.18999 , 2024

arXiv 2024

[5] [2]

Stationa ry states of the one-dimensional facilitated asymmetric exclusion process

A Ayyer, S Goldstein, JL Lebowitz, and ER Speer. Stationa ry states of the one-dimensional facilitated asymmetric exclusion process. volume 59, pages 726–742. Institut Henr i Poincar´ e, 2023

work page 2023

[6] [3]

J. Baik, G. Barraquand, I. Corwin, and T. Suidan. Facilit ated exclusion process. In The Abel Symposium , pages 1–35. Springer, 2016

work page 2016

[7] [4]

Weakly asymmetric facilitated exclusion process

G. Barraquand, O. Blondel, and M. Simon. W eakly asymmetr ic facilitated exclusion process. arXiv preprint arXiv:2301.04689, 2023

work page Pith review arXiv 2023

[8] [5]

Blondel, C

O. Blondel, C. Erignoux, M. Sasada, and M. Simon. Hydrody namic limit for a facilitated exclusion process. Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 56(1):667–714, 2020

work page 2020