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arxiv: 2604.10509 · v1 · submitted 2026-04-12 · 🧮 math.PR

Nonequilibrium fluctuations and moderate deviations for the occupation time of the SSEP with Glauber dynamics

Linjie Zhao This is my paper

Pith reviewed 2026-05-10 16:19 UTC · model grok-4.3

classification 🧮 math.PR
keywords symmetric simple exclusion processGlauber dynamicsoccupation timecentral limit theoremmoderate deviationsnonequilibrium initial measureinteracting particle systemsfluctuation field
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The pith

Starting from a nonequilibrium measure, the occupation time of the symmetric simple exclusion process with Glauber dynamics satisfies a central limit theorem in two dimensions and a sample path moderate deviation principle in one dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves fluctuation and moderate deviation results for the occupation time in the symmetric simple exclusion process with Glauber dynamics. It shows these hold when the process begins from a nonequilibrium initial measure rather than an equilibrium one. The central limit theorem applies in dimension two and the moderate deviation principle in dimension one. The arguments combine martingale techniques with relative entropy bounds for the fluctuations and use a logarithmic Sobolev inequality to connect the occupation time to the density fluctuation field. A reader would care because such controls quantify how local particle interactions produce observable statistics even far from equilibrium.

Core claim

When the symmetric simple exclusion process with Glauber dynamics starts from a nonequilibrium measure, the occupation time satisfies a central limit theorem in dimension two and a sample path moderate deviation principle in dimension one. The fluctuations are established by the martingale method together with the sharp relative entropy method. The moderate deviations are obtained by relating the occupation time to the density fluctuation field through the logarithmic Sobolev inequality for the Glauber dynamics.

What carries the argument

The occupation time, controlled by martingales and sharp relative entropy estimates that link it to the density fluctuation field via the logarithmic Sobolev inequality of the Glauber dynamics.

If this is right

  • In two dimensions the occupation time converges in distribution to a centered Gaussian random variable after diffusive scaling.
  • In one dimension the rescaled occupation time process satisfies a large-deviation principle with speed between the central limit and full large-deviation regimes.
  • The same entropy and Sobolev controls that close the estimates remain valid under the chosen nonequilibrium initial measures.
  • The occupation time can be approximated by the integrated density fluctuation field at the level of moderate deviations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combination of martingale and entropy methods may apply to occupation times of other conservative particle systems with Glauber-type flips.
  • The results suggest that moderate-deviation controls on additive functionals can be obtained in one dimension whenever a logarithmic Sobolev inequality is available for the underlying dynamics.
  • Numerical checks of the predicted variance in small periodic domains could provide an independent test of the central limit theorem in two dimensions.

Load-bearing premise

The sharp relative entropy method and the logarithmic Sobolev inequality for Glauber dynamics extend directly to the nonequilibrium initial measure without additional restrictions.

What would settle it

A simulation in two dimensions that produces occupation time distributions whose scaled variance fails to match the Gaussian limit predicted by the martingale and entropy estimates would falsify the central limit theorem.

read the original abstract

We study the symmetric simple exclusion process with Glauber dynamics. When the process starts from a nonequilibrium measure, we prove central limit theorems for the occupation time in dimension two, and sample path moderate deviation principles in dimension one. For the fluctuations, we use the martingale method and the sharp relative entropy method from [Jara and Menezes, arXiv:1810.09526]. For the moderate deviations, the main idea is to relate the occupation time to the density fluctuation field by using the logarithmic Sobolev inequality from the Glauber dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves central limit theorems for the occupation time of the symmetric simple exclusion process with Glauber dynamics, starting from nonequilibrium initial measures, in dimension two. It also establishes sample-path moderate deviation principles in dimension one. The CLTs are obtained via the martingale method combined with sharp relative entropy estimates from Jara and Menezes (arXiv:1810.09526); the MDPs are derived by relating the occupation time to the density fluctuation field through the logarithmic Sobolev inequality for Glauber dynamics.

Significance. If the derivations hold, the results extend equilibrium fluctuation theory for interacting particle systems to nonequilibrium settings, addressing a technically demanding regime. The dimension-specific statements (CLT in d=2, sample-path MDP in d=1) and the explicit invocation of established tools (martingales, relative entropy, LSI) constitute a solid incremental contribution to the literature on hydrodynamic fluctuations and moderate deviations. The absence of new parameter fitting or ad-hoc constructions strengthens the rigor.

minor comments (3)
  1. [Abstract] The abstract and introduction should explicitly state the precise form of the nonequilibrium initial measure (e.g., whether it is a product Bernoulli measure with a slowly varying density profile) to clarify the hypotheses under which the entropy and LSI estimates are applied.
  2. [References] In the bibliography, provide the full published reference details for Jara and Menezes (arXiv:1810.09526) once available, and ensure all cited works on Glauber dynamics LSI are listed consistently.
  3. [Introduction] Notation for the occupation time functional and the density fluctuation field should be introduced with a short display equation early in the introduction to improve readability for readers unfamiliar with the specific scaling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external cited methods

full rationale

The paper derives its CLTs (d=2) and sample-path MDPs (d=1) for occupation times under nonequilibrium initial measures by applying the martingale method plus sharp relative entropy estimates from the external Jara-Menezes reference and the standard Glauber LSI. These tools are invoked as pre-established results in the literature; the paper does not redefine them, fit parameters to its own outputs, or rest its central claims on self-citations. No step reduces by construction to an input or prior result by the same author, and the extension to nonequilibrium measures is presented as a direct application within the regime where the cited bounds hold. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard domain assumptions for particle systems and techniques from prior literature; no free parameters or new entities introduced.

axioms (2)
  • domain assumption The symmetric simple exclusion process with Glauber dynamics satisfies the logarithmic Sobolev inequality used to relate occupation time to the density fluctuation field.
    Invoked for the moderate deviation principle in dimension one.
  • domain assumption The sharp relative entropy method from Jara and Menezes applies to the nonequilibrium initial measure for the central limit theorem.
    Cited directly for the fluctuation results in dimension two.

pith-pipeline@v0.9.0 · 5379 in / 1229 out tokens · 67383 ms · 2026-05-10T16:19:23.157712+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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