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Moderate deviation principles for the current and the tagged particle in the WASEP

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Pith's one-line read WASEP current and tagged particle obey path-level moderate deviation principles with explicit rates.

desk verdict A useful extension of the authors' SSEP MDP results to WASEP with a cleaner contraction-based proof, but the exponential tightness lemma has a repairable gap that must be fixed before the sample-path result is fully established. read the letter →

arxiv 2506.03924 v1 pith:DSE4JXFJ submitted 2025-06-04 math.PR

Xiaofeng Xue , Linjie Zhao This is my paper
classification math.PR MSC 60K3560F10
keywords moderatedeviationstaggedparticlecurrentweaklyasymmetricsimpleexclusionprocesssamplepathdensityfluctuationfieldcontractionprinciplefractionalBrownianmotion
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

This paper proves sample-path moderate deviation principles for two central observables of the one-dimensional weakly asymmetric simple exclusion process (WASEP): the particle current across a bond and the position of a tagged particle, when the system starts from a stationary product measure. For deviation sizes $a_n$ lying between $\sqrt{n\log n}$ and $n$, the probability that the rescaled process follows a path $h$ decays as $\exp(-(a_n^2/n)J_\beta(h))$, with explicit rate functions $J_\beta$ for the current and $\rho^2J_\beta$ for the tagged particle. The result covers the regimes $\beta<1$, $\rho\neq 1/2$ and $\beta>1$, where the rate function is the quadratic energy of a Brownian motion or of a fractional Brownian motion with Hurst parameter $1/4$. The authors also aim to simplify the method: they replace a Fourier-based variational calculation used in their earlier symmetric-process papers by a contraction-principle argument.

What carries the argument

The load-bearing object is the covariance function $a(t,s)$ of the limiting Gaussian fluctuations of the current, given in (2.1): for $\beta<1$ it is the covariance of Brownian motion, for $\beta>1$ of fractional Brownian motion with Hurst parameter $1/4$, and for $\beta=1$ it has a variational expression. The sample-path MDP is built from two lemmas: finite-dimensional MDPs with rate $\frac{1}{2} r^T A^{-1}r$, where $A=(a(t_i,t_j))$, and exponential tightness of the rescaled current process; the general sample-path MDP theorem cited as [4, Theorem 4.28] then lifts the finite-dimensional principle to a principle on path space with the corresponding Gaussian rate function. The proof simplifies earlier work by noting that the rate function $Q^\beta$ of the density fluctuation field and the rate function of its Gaussian limit coincide, so contraction from $Q^\beta$ yields the covariance matrix directly.

What would settle it

Compute, for the WASEP with $\beta<1$ and $\rho\neq1/2$, the finite-dimensional moderate-deviation rate for the pair $(\bar J^n_{-1,0}(t_1)/a_n, \bar J^n_{-1,0}(t_2)/a_n)$ by direct expansion of the moment generating function at scale $a_n^2/n$; if it differs from $\frac12 r^T A^{-1}r$ with $A_{ij}=a(t_i,t_j)$ from (2.1), then Lemma 4.2 is false. Alternatively, simulate the process on a large ring and compare the empirical log-probability of a path deviation with the claimed rate; a mismatch at any time would falsify Theorem 2.4.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.4: under the stationary measures $\nu_\rho$ (for the current) and $\nu_\rho^*$ (for the tagged particle), the rescaled processes $\{\bar J^n_{-1,0}(t)/a_n\}$ and $\{\bar X^n(t)/a_n\}$ satisfy moderate deviation principles on the Skorokhod space $D([0,T],\mathbb{R})$ with decay rate $a_n^2/n$. For $\beta<1$ and $\rho\neq 1/2$ the rate function for the current is $J_\beta(h) = \|\dot h\|^2_{L^2}/(2\chi(\rho)\alpha|1-2\rho|)$; for $\beta>1$ it is $\sqrt{\pi}/(2\sqrt{2}\chi(\rho))\,\|\dot h\|^2_{L^2}$, expressed through the kernel $K(t,s)$ of fractional Brownian motion with Hurst parameter $1/4$. The tagged-particle rate function is $\rho^2 J_\beta$. The proof obtains the finite-dimensional rate function by a contraction principle from the moderate deviations of the density fluctuation field, solving the variational problem with the covariance of the limiting Gaussian process rather than by Fourier calculation.

Load-bearing premise

The argument depends on showing that the current's random path cannot leave large windows too often, at a cost that grows like the moderate-deviation rate; this exponential-tightness step is only sketched and requires the deviation size $a_n$ to be much larger than $\sqrt{n\log n}$, so if that proof fails the theorem loses its path-level conclusion.

Editorial extensions

If this is right

  • The moderate-deviation speed is $a_n^2/n$ for both the current and the tagged particle, so a deviation of size $a_n$ costs $\exp(-(a_n^2/n)J_\beta(h))$ on path level.
  • For $\beta<1$ and $\rho\neq1/2$, deviations are priced by the $L^2$ energy of the path derivative, reflecting Brownian current fluctuations with diffusivity $\chi(\rho)\alpha|1-2\rho|$; constant paths have zero cost.
  • For $\beta>1$, the rate function is that of fractional Brownian motion with Hurst parameter $1/4$, so moderate deviations inherit the covariance structure of the $\sqrt{n}$-scale fluctuations.
  • The tagged-particle rate function is $\rho^2$ times the current rate, so a tagged particle's path deviations are exponentially cheaper by that density-dependent factor.
  • The proof route through contraction from density-fluctuation MDPs can also simplify the occupation-time MDP proof for the exclusion process, as the authors note.

Reading between the lines

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

  • If the exponential-tightness estimate can be extended to the full regime $\sqrt{n}\ll a_n\ll n$, the same proof should yield Theorem 2.4 without the $\sqrt{n\log n}$ restriction; the blocking step is the coupling estimate in the proof of Lemma 4.1.
  • The covariance-driven variational solution suggests that the $\beta=1$ rate function, which the paper leaves as an infimum over time points, should equal the quadratic form of the Gaussian process with covariance (2.1).
  • The relation $X_\beta=\rho^2 J_\beta$ likely holds more generally for exclusion-type systems in equilibrium, because the tagged particle's displacement is the current minus a superexponentially small local-density term.
  • The observation that the fluctuation-field rate function coincides with that of its Gaussian limit could be tested on other interacting particle systems with a linear limiting SPDE, where Fourier evaluation is heavy.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the weakly asymmetric simple exclusion process (WASEP) on Z with generator n^γ(L_s + α n^{-β} L_a), starting from a stationary measure ν_ρ (or ν*_ρ for the tagged particle). It proves sample-path moderate deviation principles (MDPs) for the total current across the bond (-1,0) and for the tagged particle position, with speed a_n^2/n and explicit rate functions: for β<1 the rate is the L²-norm of the derivative (Brownian case), for β>1 it is the corresponding fractional Brownian (H=1/4) rate, and for β=1 it is given by a variational formula. The proof strategy is to relate current and tagged particle to the density fluctuation field, use a prior MDP for the field (Proposition 3.4), and then solve the resulting variational problem through a comparison with the Gaussian covariance of stationary fluctuations, simplifying earlier Fourier-based arguments for the SSEP.

Significance. If correct, this is a valuable contribution: it extends sample-path MDP results from the symmetric exclusion process to the weakly asymmetric case, covers three scaling regimes, and gives closed-form rate functions in the non-critical cases β≠1. The contraction-principle approach is elegant and simplifies earlier proofs. However, the paper is not self-contained: Proposition 3.4 is quoted without proof, Lemma 4.1 (exponential tightness) is only sketched and contains a concrete error in Eq. (4.7), and Lemma 4.4 is proved only for β=1 while the main theorem covers β<1 and β>1. The technical restriction a_n ≫ √(n log n) is also stronger than the natural MDP scale. These issues affect the proof of the central claim but appear fixable.

major comments (3)
  1. [§4.2, Eq. (4.7)] The superexponential estimate for F_l = G̃_l − G_l is false as stated. By (3.5), the field µ^n_t is uncentered, so E⟨µ^n_t, F_l⟩ = (ρ/a_n) Σ_x F_l(x/n) ≈ ρ (n/a_n) ∫ F_l. The conditions on G̃_l — L²-close to G_l and supported in [−2l,2l] — do not imply ∫F_l = 0. Since a_n ≪ n, this expectation diverges for any fixed l with ∫F_l ≠ 0, so the probability in (4.7) actually tends to 1 and the superexponential bound cannot hold. The proof must impose the centering condition ∫G̃_l = ∫G_l (so ∫F_l = 0) and then use a Bernstein-type bound; without that, Lemma 4.1 is not proved.
  2. [§4.2, paragraph after Eq. (4.7)] The Poisson variable ξ bounding particle displacement is taken with parameter αT n^{-1}. For β>1, γ=2, so the symmetric jump rate is n² and the total jump count of a particle in time T n^{-3} has mean T n^{-1} (plus a lower-order asymmetric correction), not αT n^{-1}; when α<1 the stochastic domination with parameter αT n^{-1} fails. The proof should use a bound of the form C T n^{-1} with C ≥ 1 independent of α. This is easily corrected but indicates that the sketched argument is not self-contained.
  3. [§4.3.2, Lemma 4.4] Lemma 4.4 is stated for any β≥0, but the proof explicitly treats only the case β=1 and asserts that the remaining cases follow from the same analysis. Since Theorem 2.4 includes β<1 and β>1, the lower bound for the finite-dimensional MDP is not actually proved in the regimes covered by the main theorem. The authors should either provide the analogues of Lemma 4.5 and the approximation argument for β<1 and β>1, or give a detailed reduction showing how those cases differ (e.g., the degenerate transport semigroup for β<1 and the Brownian kernel for β>1).
minor comments (5)
  1. [§3, Proposition 3.4] Proposition 3.4 is stated without proof and with reference to [27]. Since it is a key input to Lemmas 4.1 and 4.2, the authors should state precisely which theorem in [27] is being invoked and confirm that the scaling and the definition of Q^β match the present setting.
  2. [§4.3.2, proof of Lemma 4.4] After Eq. (4.16), the convergence of r_ε to r is only asserted, not demonstrated. This convergence is essential for the limiting lower bound, and the details should be provided for all β regimes, not just for β=1.
  3. [§2, Remark 2.6] The technical condition a_n ≫ √(n log n) is a genuine restriction relative to the natural MDP scale √n ≪ a_n ≪ n. Since this condition is assumed in the main theorem, it should be stated prominently in the introduction or abstract rather than only in a remark.
  4. [§4.4, Eqs. (4.17)–(4.19)] The relation between current and tagged particle is stated for x>0 and with separate sign cases; a short derivation of these identities and a comment on how the sign of the current is handled would improve readability.
  5. [Throughout] There are several typographical issues, including 'MODERA TE DEVIATION PRINCIPLES' in the title, 'the we get' in Section 2, and inconsistent use of ℓ and l in Section 4.2; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rate functions are computed from the limiting Gaussian covariance via contraction, and the cited prior works are independent ingredients rather than restatements of Theorem 2.4.

full rationale

The derivation chain for Theorem 2.4 goes through Lemmas 4.1 and 4.2. Lemma 4.2 is proved by contracting the MDP for the rescaled density field (Proposition 3.4, delegated to the independent prior work [27]) and the Gaussian large deviation principle for the limiting field (Lemma 3.3, from the standard stationary-fluctuation result Proposition 3.1). The variational infimum is then evaluated directly: Lemma 4.3 identifies the contraction as the Gaussian quadratic form with covariance Sigma, (4.8) identifies the limit of Sigma as the covariance a(t,s) from Proposition 2.1, and Lemma 4.4 gives the matching lower bound for the same quadratic form. None of these steps assumes the conclusion of Theorem 2.4: the rate function J_beta is not used as an input, and the empirical-measure MDP is a different statement from the current/tagged-particle MDP. The heavy reliance on the authors' earlier papers [25,26,27] is self-citation, but those works are prior, independently stated results used as lemmas and proof templates, not as disguised versions of the target theorem. The proof has acknowledged gaps (the exponential-tightness argument in Lemma 4.1 is only sketched and Remark 2.6 records the extra condition a_n >> sqrt(n log n)), but these are completeness or correctness concerns, not circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the empirical-measure MDP (Proposition 3.4, from the authors' preprint [27]) and on Gaussian fluctuation results whose proofs are omitted. No free parameters are fitted; all constants come from the model. The main ad hoc element is the technical scaling condition on a_n.

assumptions (4)
  • domain assumption MDP for the rescaled density field μ_t^n (Proposition 3.4) holds with rate function Q_β.
    The paper's proof of the current/tagged particle MDP rests on this result via the contraction principle. The proof is omitted and attributed to [27], an unpublished preprint by the second author.
  • ad hoc to paper Technical scaling condition a_n ≫ √(n log n) in addition to the natural MDP scale √n ≪ a_n ≪ n.
    Required for exponential tightness of the current; the paper states it does not know how to remove it (Remark 2.6).
  • domain assumption The limiting Gaussian field Y_t in Proposition 3.1 satisfies the listed SPDEs and has increment covariance a(t,s).
    Used to identify the matrix A in the finite-dimensional rate functions and hence the sample path rate functions. Proposition 3.1 is stated without proof as 'standard'.
  • standard math Standard LDP contraction principle and the Feng-Kurtz sample path LDP criterion [4, Theorem 4.28].
    Used to lift finite-dimensional MDPs and exponential tightness to sample path MDPs in the proof of Theorem 2.4.

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Cite this review

Pith. "Pith review of Moderate deviation principles for the current and the tagged particle in the WASEP." pith.science (2026). https://pith.science/paper/DSE4JXFJ

@misc{pith2026250603924,
  author       = {Pith},
  title        = {Pith review of: Moderate deviation principles for the current and the tagged particle in the WASEP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DSE4JXFJ}},
  note         = {Machine review of arXiv:2506.03924}
}
read the original abstract

We study the weakly asymmetric simple exclusion process in one dimension. We prove sample path moderate deviation principles for the current and the tagged particle when the process starts from one of its stationary measures. We simplify the proof in our previous works [Xue and Zhao, Electronic Journal of Probability, 2024] and [Xue and Zhao, Stochastic Processes and their Applications, 2023], where the same problem was investigated in the symmetric simple exclusion process.

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Works this paper leans on

28 extracted references · 24 canonical work pages

  1. [27]

    L. J. Zhao. Moderate deviation principles for the W ASEP.arXiv preprint arXiv:2405.16151, 2024

    work page Pith review arXiv 2024

  2. [1]

    Bal´ azs and T

    M. Bal´ azs and T. Seppalainen. Order of current variance and diffusivity in the asymmetric simple exclusion process.Ann. of Math., 2:1237–1265, 2010

    work page 2010

  3. [2]

    Decreusefond and A.S

    M. Decreusefond and A.S. ¨Ust¨ unel. Stochastic analysis of the fractional Brownian motion.Potential Anal- ysis, 10:177–214, 1999

    work page 1999

  4. [3]

    J. D. Deuschel and D. W. Stroock.Large Deviations. Academic Press, 1989

    work page 1989

  5. [4]

    Feng and T

    J. Feng and T. G. Kurtz.Large deviations for stochastic processes. American Mathematical Soc., 2006

    work page 2006

  6. [5]

    F. Q. Gao and J Quastel. Moderate deviations from the hydrodynamic limit of the symmetric exclusion process.Science in China Series A: Mathematics, 46(5):577–592, 2003. 18 XIAOFENG XUE AND LINJIE ZHAO

    work page 2003

  7. [6]

    Gon¸ calves

    P. Gon¸ calves. Central limit theorem for a tagged particle in asymmetric simple exclusion.Stochastic Processes and their Applications, 118(3):474–502, 2008

    work page 2008

  8. [7]

    M. Jara, C. Landim, and S. Sethuraman. Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensional zero-range processes.Probability theory and related fields, 145(3):565–590, 2009

    work page 2009

Show all 28 references
  1. [8]

    M. D. Jara and C. Landim. Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. InAnnales de l’IHP Probabilit´ es et statistiques, volume 42, pages 567–577, 2006

    2006

  2. [9]

    M. D. Jara and C. Landim. Quenched non-equilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder. InAnnales de l’IHP Probabilit´ es et statistiques, volume 44, pages 341–361, 2008

    2008

  3. [10]

    Johansson

    K. Johansson. Shape fluctuations and random matrices.Communications in mathematical physics, 209(2):437–476, 2000

    2000

  4. [11]

    C. Kipnis. Central limit theorems for infinite series of queues and applications to simple exclusion.The Annals of Probability, 14(2):397–408, 1986

    1986

  5. [12]

    Kipnis and C

    C. Kipnis and C. Landim.Scaling limits of interacting particle systems, volume 320. Springer Science & Business Media, 2013

    2013

  6. [13]

    Kipnis, S

    C. Kipnis, S. Olla, and S. R. S. Varadhan. Hydrodynamics and large deviation for simple exclusion processes.Communications on Pure and Applied Mathematics, 42(2):115–137, 1989

    1989

  7. [14]

    Kipnis and S

    C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions.Communications in Mathematical Physics, 104(1):1–19, 1986

    1986

  8. [15]

    Komorowski, C

    T. Komorowski, C. Landim, and S. Olla.Fluctuations in Markov processes: time symmetry and martingale approximation, volume 345. Springer Science & Business Media, 2012

    2012

  9. [16]

    T. M. Liggett.Interacting particle systems, volume 276. Springer Science & Business Media, 2012

    2012

  10. [17]

    T. M. Liggett.Stochastic interacting systems: contact, voter and exclusion processes, volume 324. springer science & Business Media, 2013

    2013

  11. [18]

    Rezakhanlou

    F. Rezakhanlou. Evolution of tagged particles in non-reversible particle systems.Communications in Mathematical Physics, 165(1):1–32, 1994

    1994

  12. [19]

    E. Said. A limit theorem for the position of a tagged particle in a simple exclusion process. Technical report, 1987

    1987

  13. [20]

    Sethuraman

    S. Sethuraman. Diffusive variance for a tagged particle ind≤2 asymmetric simple exclusion.ALEA Lat. Am. J. Probab. Math. Stat., 1:305–332, 2006

    2006

  14. [21]

    Sethuraman and S

    S. Sethuraman and S. R. S. Varadhan. Large deviations for the current and tagged particle in 1d nearest- neighbor symmetric simple exclusion.The Annals of Probability, 41(3A):1461–1512, 2013

    2013

  15. [22]

    Sethuraman and S.R.S

    S. Sethuraman and S.R.S. Varadhan. Atypical behaviors of a tagged particle in asymmetric simple exclu- sion.arXiv preprint arXiv:2311.07800, 2023

    2023 arXiv

  16. [23]

    Sethuraman, SRS Varadhan, and H

    S. Sethuraman, SRS Varadhan, and H. T. Yau. Diffusive limit of a tagged particle in asymmetric simple exclusion processes.Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 53(8):972–1006, 2000

    2000

  17. [24]

    S. R. S. Varadhan. Self diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. InAnnales de l’IHP Probabilit´ es et statistiques, volume 31, pages 273–285, 1995

    1995

  18. [25]

    X. F. Xue and L. J. Zhao. Sample path MDP for the current and the tagged particle in the SSEP. Electronic Journal of Probability, 29:1–39, 2024

    2024

  19. [26]

    Xue and L.J

    X.F. Xue and L.J. Zhao. Moderate deviations for the current and tagged particle in symmetric simple exclusion processes.Stochastic Processes and their Applications, (104218), 2023

    2023

  20. [28]

    L. J. Zhao. The motion of the tagged particle in the asymmetric exclusion process with long jumps. Bernoulli, 30(3):2399–2422, 2024. MODERATE DEVIATION PRINCIPLES 19 School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China. Email address:xfxue@b...

    2024

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