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Integrability and exact large deviations of the weakly-asymmetric exclusion process

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Pith's one-line read Exact large-deviation statistics are derived for the current and tracer in the weakly asymmetric exclusion process, interpolating between SSEP and weak-noise KPZ and revealing the integrability of the underlying macroscopic fluctuation…

desk verdict Exact WASEP large deviations with a clean integrability structure, but the central rate function rests on an unproven self-averaging step that a referee should pin down. read the letter →

arxiv 2505.12034 v1 pith:HR4DJKJT submitted 2025-05-17 cond-mat.stat-mech cond-mat.dis-nnmath-phmath.MPmath.PRnlin.SI

Alexandre Krajenbrink , Pierre Le Doussal This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nnmath-phmath.MPmath.PRnlin.SI
keywords weaklyasymmetricexclusionprocessmacroscopicfluctuationtheorylargedeviationsintegratedcurrenttracerparticleLaxpairLandau-LifshitzmodelKardar-Parisi-Zhangcrossover
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 establishes that the weakly asymmetric exclusion process (WASEP), the driven one-dimensional lattice gas with hopping asymmetry scaled as $1/\sqrt{T}$, has exact large-deviation statistics rather than only perturbative cumulants. The core object is a rate function $\Psi(u)$ written as a dilogarithm contour integral over the initial densities; from it, by Legendre-type transforms, the paper obtains the rate functions and all cumulants of the time-integrated current and of a tracer's position. These formulas interpolate between two known worlds: the symmetric exclusion process when the driving $\nu\to0$ and the short-time weak-noise Kardar-Parisi-Zhang equation when $\nu\to\infty$, including the crossover of the cubic tail to the $5/2$ and $3/2$ tail exponents. The paper also proves that the macroscopic fluctuation theory of the WASEP is classically integrable by exhibiting an explicit Lax pair through a mapping to the anisotropic Landau-Lifshitz spin chain. If correct, this turns the WASEP into a reference model where full current statistics are available at any driving strength.

What carries the argument

The load-bearing machinery is the first-cumulant reduction of the ASEP Fredholm determinant (SA.1): in the WASEP limit $\eta=2\nu\varepsilon\to0$, the log-determinant is replaced by the trace of the kernel (SE.9), whose saddle-point evaluation produces the dilogarithm integral (5). Around this sits a calculus that converts the unusual generating function (4) into ordinary cumulants: the auxiliary variable $\omega$ with weight $F(\omega)$ is eliminated by a saddle point, and the Legendre system (10)-(12) turns $\Psi(u)$ into $\Phi(J)$ and $\phi(P)$. On the integrability side, the key object is the change of variables (32) taking the MFT fields $(q,p)$ to $(Z,R)$; the matrix (35) realises these fields as a stereographic parametrisation of a complex anisotropic Landau-Lifshitz spin, and the zero-curvature pair (36)-(37) is the Lax representation whose compatibility reproduces the MFT equations.

What would settle it

Compute the next-order correction in $\eta=2\nu\varepsilon$ to the log-determinant in Eq. (SE.9): if corrections appear at order $\eta^0$ or $\eta^1$ rather than vanishing, the saddle-point integral (SE.21) is incomplete. Alternatively, simulate the ASEP with rates $1\pm\nu\varepsilon$ up to time $T/\varepsilon^2$, measure the generating function (4) for several small $\varepsilon$, and check whether $\varepsilon\log(\cdot)$ converges to $-\Psi(u)$; a non-vanishing slope in $\varepsilon$ at fixed $u$ would falsify the claimed rate function.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the generating function (4), built from the exact ASEP Fredholm determinant, collapses in the WASEP scaling to the simple integral $\Psi(u)$ in Eq. (5), and that the parametric system (10)-(12) then yields the complete large-deviation functions $\Phi(J)$ and $\phi(P)$ for the integrated current, from which the tracer position statistics follow by setting $J=0$. The same formula is shown to carry known results: the $\nu\to0$ limit reproduces the SSEP formulas of [44,56] and the $\nu\to\infty$ limit reproduces the weak-noise KPZ rate function for Brownian initial data [50], with an explicit crossover tail function between the $J^3$ and $J^{5/2}$ tails and a $3/2$ tail of the upper current distribution. In addition, the MFT saddle-point equations (31) are shown to be integrable: the change of variables (32) maps them to a complex extension of the anisotropic Landau-Lifshitz model, with Lax pair (36)-(37), so the authors claim integrability for all MFTs with quadratic mobility and their duals.

Load-bearing premise

The whole result rests on replacing the log of the ASEP Fredholm determinant by the trace of its kernel in the limit $\varepsilon\to0$; if that self-averaging step has non-vanishing subleading corrections, the central rate function $\Psi(u)$ is not exact.

Editorial extensions

If this is right

  • Every cumulant of the integrated current and of the tracer can be written in closed form; the coefficients $\Psi_n(X,T)$ obey the order-$n$ heat equation (6), so the formulas propagate from $T=0$ data.
  • The $\nu\to0$ limit reproduces the SSEP current and tracer CGFs, including formulas at arbitrary observation point $X$, and the $\nu\to\infty$ limit reproduces the weak-noise KPZ rate function, so the WASEP is a single model connecting both universality classes.
  • The right tail of the current remains cubic for fixed driving, and as the driving grows the tail crosses over, through the explicit function (28)-(29), to the KPZ $5/2$ tail; the upper tail shows a $3/2$ wall solution.
  • Integrability extends to the WASIP, weakly asymmetric KMP, and their duals, giving exact current and tracer statistics for all these models at once.
  • The explicit Lax pair opens the way to inverse-scattering solutions of the WASEP MFT, which would cover more general initial conditions and observables than the two-sided Bernoulli case solved here.

Reading between the lines

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

  • If the first-cumulant self-averaging is exact, the same trace-only reduction should apply to other $q$-deformed Fredholm determinants in the same scaling, giving analogous rate functions for $q$-TASEP-like models.
  • The explicit crossover tail is a quantitative experimental prediction for driven single-file systems: the rescaled current PDF should collapse onto the universal function $\Phi_+$ with no intermediate exponent.
  • The soliton-branch continuation for $J<J_c$ suggests a dynamical phase transition in the WASEP current statistics at large $\nu^2T$; locating the non-analyticity of $\Phi(J)$ would test whether the integrable structure, not a saddle-point artefact, controls the transition.
  • Because the Lax equation is gauge-equivalent to standard integrable nonlinear Schrödinger-type flows, spectral methods may yield Riemann-Hilbert or Fredholm representations of the rate function and connect these MFT large deviations to random-matrix edge statistics.
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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

4 major / 4 minor

Summary. The manuscript claims to derive exact large-deviation quantities for the weakly asymmetric exclusion process (WASEP) for a two-sided Bernoulli initial condition: a rate function Ψ(u) for the integrated current, the associated cumulants, the tracer-position statistics, and the SSEP/KPZ crossovers. The derivation starts from an ASEP Fredholm determinant formula of Aggarwal, converts it to a WASEP generating function in the limit ε→0, and then extracts cumulants and tails via Legendre-type parametric relations. In a second part, the paper transforms the macroscopic fluctuation theory (MFT) equations into a complex anisotropic Landau-Lifshitz spin system and exhibits an explicit Lax pair, thereby claiming classical integrability of the WASEP MFT and of related quadratic-mobility MFTs. The main quantitative results are benchmarked against independent perturbative and known limiting results. The central formula for Ψ(u) depends, however, on a first-cumulant approximation of a Fredholm determinant in the supplement, for which no error estimate is provided.

Significance. If the main formula (5) is valid, this is a significant advance: exact current and tracer large deviations for the WASEP across the full SSEP-to-KPZ crossover would be a natural benchmark for driven diffusive systems and macroscopic fluctuation theory. The paper ships a large amount of explicit, checkable material, including closed expressions for cumulants, parametric tail formulae, and a concrete Lax pair in Eqs. (36)-(37) whose zero-curvature condition is stated to be verified by direct computation. The consistency checks are strong and appropriately credited: cumulants match perturbative results up to n=3 (n=4 for ν=0), the SSEP limit matches [44,56], the weak-noise KPZ limit matches [50] up to fourth order, and the tracer variance matches [43, Eq. (S135)]. The value of the contribution is therefore high, but the central rate function rests on an unproven self-averaging step, so the paper is currently conditional rather than established.

major comments (4)
  1. [Supplement E, Eq. (SE.9)] The replacement Det(I+K_u) ≃ exp(-Tr(ς A_2 A_3 A_1)) is the load-bearing step for the main formula (5). The exact object is an expectation over a determinantal point process of exp(-Σ_ℓ ς(a_ℓ)), and the discarded terms are the second and higher cumulants of this linear statistic. Since η=2νε is the only small parameter and the retained trace is O(1/η), a fluctuation contribution of order O(1/η)—not merely O(1)—would change Ψ(u) at leading order. The text asserts self-averaging but provides no estimate of Var(Σ ς(a_ℓ)), no rigidity statement for the point process, and no bound on the neglected cluster terms. The strong low-order and limit checks do not control the full function Ψ(u). This should be fixed by an explicit error estimate, or by a proof of the self-averaging in the relevant η→0 limit, before the word 'exact' is attached to (5).
  2. [Supplement E.2, Eqs. (SE.11)-(SE.21)] The passage from the exact trace formula to the saddle-point integral (SE.21) needs justification of two points: the integration by parts in r states 'there is no boundary term', and the subsequent saddle-point evaluation assumes a unique relevant saddle with contour deformation between v and ζ and vanishing boundary contributions at infinity. The contour C0 is described as containing −1 as an asymptotic point, so boundary contributions are not obviously negligible in this oscillatory regime. A same-order boundary term would alter the O(1/η) result (SE.21) and hence the rate function. The authors should provide the missing estimates or state precisely which contour deformations are used.
  3. [Supplement L, Eqs. (SL.47), (SL.49), (SL.55)] The general stationary cumulant formula in Appendix B is presented as an iterative arbitrary-order result, but its derivation in Appendix L relies on the conjecture (SL.47) and on the assumption (SL.49) that the jump at k=0 is exactly one-half of the jump at infinity 'for any q'. These are verified only for q=1,2 and tested in the KPZ limit. Since the claim 'arbitrary order' depends on these unproved identities, the paper should either prove them or explicitly label the general-order formula as conjectural with the stated support.
  4. [Main text, Tails paragraph; Supplement S.4-S.5, T.3] The paper claims the full large-deviation statistics including J<J_c, but the solitonic continuation is derived under a single-valued-branch assumption (SS.13), and the text states that the multi-branch phase-transition case is 'analyzed elsewhere [85]'. Also, the matching of Eq. (30) with the ASEP upper-tail result of [86] is presented as a demonstrated coincidence, while the supporting branch selection for the step initial condition is only sketched. These are limitation statements already contained in the manuscript; they should be elevated: either the missing cases are treated here, or the scope of the 'full statistics' claim should be narrowed accordingly.
minor comments (4)
  1. [Eqs. (10)-(12) and Supplement H] The variables z and J are used both for random variables and for their saddle-point values; footnote 74 acknowledges this, but the notation is still hard to follow in (10)-(12). A dedicated symbol for saddle-point values would improve readability.
  2. [Appendix B vs. Supplement L] The main text says the stationary formulae 'allow iterative calculation of the cumulants to an arbitrary order', while the supplement labels the underlying identity as conjectural. The main text should carry the same caveat.
  3. [Fig. SS.2] Panel (c) of Fig. SS.2 is printed in a dot-matrix style that makes the tri-valued branch structure nearly unreadable; a vector plot with labelled branches would be much clearer.
  4. [Eq. (4)] The symbol ∼ is used for the large-deviation equivalence, but the left-hand side is an integral over ω of an expectation value; it would help to state explicitly which variable(s) are integrated out in the rate function and that the ω-integral is evaluated by saddle point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central derivation starts from an independent ASEP Fredholm determinant and is benchmarked against external SSEP and KPZ results.

full rationale

The main result (5) is obtained by taking the WASEP limit of an independent ASEP Fredholm determinant theorem (Aggarwal, ref. [99]) and then evaluating the determinant through an explicitly written first-cumulant trace approximation in the supplement. No parameter is fitted to the target observable, and no equation is re-inserted as its own input: the q-Pochhammer asymptotics and saddle-point integration are displayed, and the resulting rate function is used to derive the cumulants and large deviation functions through the displayed Legendre-type relations (10)-(12). The limits to the SSEP and to the weak-noise KPZ regime are checked against independent results of Derrida-Gerschenfeld, Mallick-Moriya-Sasamoto, and earlier short-time KPZ work. Self-citations appear mainly for the first-cumulant method and for supplemental computational details, but these are techniques or in-paper derivations rather than a target result imported as evidence; the approximation itself is stated openly and is not a hidden reduction. The Lax pair (36)-(37) is verified by explicit zero-curvature calculation and inherits integrability from the anisotropic Landau-Lifshitz model, an external integrable system. The uncontrolled first-cumulant self-averaging step is a genuine mathematical-rigor concern and could affect correctness, but it is not a circularity: the derivation does not define its output as its input or rename a fitted quantity as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No data fitting is performed: nu, rho_1, rho_2, X, T and u are model or observational parameters, and constants such as alpha and omega_u are fixed functions of them. The paper relies on known ASEP formulas, standard hydrodynamic scaling, q-Pochhammer asymptotics, a heuristic first-cumulant step, and the classical integrability of the Landau-Lifshitz chain. The invented entities are auxiliary mathematical constructs with no independent physical evidence.

assumptions (6)
  • domain assumption Aggarwal's ASEP Fredholm determinant (Theorem 4.8 in [99]) characterizes the distribution of the discrete current with two-sided Bernoulli initial condition.
    External exact result used as the starting point of the WASEP limit (Appendix A, Eq. SA.1).
  • domain assumption The WASEP hydrodynamic scaling limit with R=1+epsilon nu, L=1-epsilon nu, t=T/epsilon^2, x=epsilon m converges to fluctuating hydrodynamics (1) with D0=1 and sigma(rho)=2 rho(1-rho).
    Standard result from [3,15,18,36,66-68]; the paper builds on it without reproving it.
  • standard math The q-Pochhammer asymptotics (SF.3): log(x;q)_infty tends to -Li2(x)/(1-q) + (1/2) log(1-x) as q tends to 1.
    Used in (SA.6)-(SA.8) to evaluate the auxiliary measure and g(v) in the WASEP limit.
  • ad hoc to paper The first-cumulant approximation (self-averaging of the linear statistics) (SE.9): Det(I+Ku) is asymptotic to exp(-Tr(ςA)) in the limit eta=2 nu epsilon to 0, with no subleading corrections.
    Load-bearing asymptotic step; no error bound is given, and the rate function Psi(u) in (5)/(SA.9) rests on it.
  • ad hoc to paper The saddle-point evaluation of the resulting two-dimensional integral (SE.15)-(SE.21) has a unique relevant saddle with the stated contour, and boundary terms at infinity vanish.
    Standard in the first-cumulant method but not fully justified here; the contour conditions are asserted in Appendix E.
  • standard math The anisotropic Landau-Lifshitz model is integrable, with the Lax pair from [92] adjusted by the magnetic-field term -nu^2 sigma_3/2 for the complex extension (SN.35)-(SN.47).
    The zero-curvature condition is checked by explicit algebra, relying on the classical integrability established in [87,88,92].
invented entities (2)
  • Auxiliary random variable omega in [1,infty), with PDF proportional to exp(-F(omega)/(2 nu epsilon))
    purpose: Encodes the q-deformed geometric distribution of the ASEP in the WASEP limit and defines the generalized generating function (4).
    A mathematical auxiliary variable, not a physical entity; it has no handle outside the paper except through the derived rate functions.
  • Complex spin S with S^2=Id and stereographic components (S_z,S_+,S_-) built from R and Z
    purpose: Maps the MFT equations (33) to the anisotropic Landau-Lifshitz spin chain, exposing the Lax pair.
    A representation change, not a new physical field; integrability of the target model is already known.

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Pith. "Pith review of Integrability and exact large deviations of the weakly-asymmetric exclusion process." pith.science (2026). https://pith.science/paper/HR4DJKJT

@misc{pith2026250512034,
  author       = {Pith},
  title        = {Pith review of: Integrability and exact large deviations of the weakly-asymmetric exclusion process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HR4DJKJT}},
  note         = {Machine review of arXiv:2505.12034}
}
abstract

The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities $\rho_1,\rho_2$ on either side of the origin, so that for $\rho_1=\rho_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulae for the cumulant generating functions, and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process (SSEP) and the weak noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the $5/2$ and $3/2$ KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions.

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  1. Full stochastic dynamics of a tracer in a dense single-file system

    cond-mat.stat-mech 2025-05 accept novelty 8.0 of 10

    In the dense limit of the symmetric exclusion process, all n-time tracer cumulants reduce to integrals over a Brownian walker that stays positive.

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Works this paper leans on

31 extracted references · 31 canonical work pages · cited by 1 Pith paper

  1. [1]

    KPZ limit of the large deviation rate function In the KPZ limit, we rescale the generalized Laplace parameteru and the partition functionz as u = 4˜ueν2T−νX ν2 , z=Ze−ν2T+νX (SP.3) and then take the limitν→∞ . We proceed to a change of variable in the integrand of the rate function (5) y = 1 2 + ˜y 2ν, ˜y = ˜δ + ik (SP.4) and one finds that at leading ord...

    work page

  2. [2]

    We consider the height fieldH =−2νJ and use the parameterization (SP.1)

    KPZ limit of the cumulants of the current We now check the KPZ limit on the cumulants of the current. We consider the height fieldH =−2νJ and use the parameterization (SP.1). The average ofH reads, from (SK.12) ⟨H⟩ =νX−ν2T + log   e˜ϱ1(˜ϱ1T+X)Erfc ( 2˜ϱ1T+X 2 √ T ) +e˜ϱ2(˜ϱ2T+X)Erfc ( − 2˜ϱ2T+X 2 √ T ) 2   (SP.15) which is exact and does not involve...

    work page

  3. [3]

    Let us define the matrix spinS S = 1 1 +RZ (1−RZ 2R 2Z −(1−RZ) ) := (Sz S+ S− −Sz ) (SN.33) 43 The direct properties of this representation areS2 =Id, DetS =−1 and S−S+ +S2 z = 1

    Mapping the WASEP to the anisotropic Landau-Lifshitz model The last change of variable we now introduce allows us to represent (SN.20) as a complex extension of the dynamics of the classical anisotropic Landau-Lifshitz model in its stereographic frame [87, 88]. Let us define the matrix spinS S = 1 1 +RZ (1−RZ 2R 2Z −(1−RZ) ) := (Sz S+ S− −Sz ) (SN.33) 43 ...

    work page

  4. [4]

    Lax pair representation of the MFT The Lax pair for the anisotropic Landau-Lifshitz magnet has been known for a few decades, see e.g., Ref. [92]. To obtain a Lax pair for the MFT of the WASEP we use the conventions of this reference, with minor changes: we setλ =−k/2 and since the time there, which we denote byτ, is τ = it (see above), we setM =−iM1− ν2 2...

    work page

  5. [5]

    Comment on gauge equivalences There is no uniqueness of the Lax pair to represent (SN.20) and thus we cannot exclude that other gauge equivalent representations of this pair can be convenient to solve a scattering problem. A gauge 45 transformation on the Lax matrices is defined as the map(L,M )↦→(˜L, ˜M) involving an invertible gaugeG so that ˜L =G−1LG−G...

    work page

  6. [6]

    We recall that the general MFT depends on two functionsD(q) and σ(q)

    Extension to the MFT of asymmetric models with quadratic mobility We now extend the integrability argument we have derived to the MFT of asymmetric models with quadratic mobility. We recall that the general MFT depends on two functionsD(q) and σ(q). With the choiceσ(q) =σA,B(q) := 2Aq(B−q), D(q) = 1, the MFT reads ∂tq =∂x [ ∂xq− 2Aq(B−q)(∂xp +ν) ] , (SN.4...

    work page

  7. [7]

    height field

    Duality in the MFT There is a general duality property of the MFT models, see [43, 63] for more details. Consider a MFT model with density fieldϱ(x,t ) and parametersD(ϱ),σ (ϱ),ν. The dual MFT model describes a density field 47 ˜ϱ(x,t ) and parameters ˜D(ϱ), ˜σ(ϱ), ˜ν, with the following involutive relations ϱ(x,t ) = 1 ˜ϱ(k(x,t ),t ) , ∂ xk(x,t ) =ϱ(x,t ...

    work page

  8. [8]

    free energy density

    Optimal density at initial timeT = 0 At timeT = 0 the density fieldϱ(x) of the WASEP is the coarse-grained version of the density field of the ASEP with a double sided i.i.d. Bernoulli initial condition, i.e., for eachx one can writeϱ(x) =∑1/ε i=1ni whereni = 0, 1 with probabilities1− ¯ϱ(x) and ¯ϱ(x), respectively. Its probability distribution thus decoup...

    work page

Show all 31 references
  1. [10]

    KPZ limit of the MFT equations Finally, we obtain the KPZ limit of the MFT equations by two methods: first directly on the stochastic version of the MFT equation (as in the text) and then on the nonlinear dynamical saddle point equations

  2. [11]

    The regime εKPZ≪ 1 corresponds to the weak noise theory of KPZ, which is also its short-time regime

    The stochastic MFT equation of the WASEP reads ∂tϱ =∂x(∂xϱ−νσ(ϱ) + √ εσ(ϱ)η) (SP.20) We first rescale the density around1/2, i.e., also the maximum ofσ(ϱ), as ϱ = 1 2 + ˜ϱ 2ν , σ (ϱ) = 2ϱ(1−ϱ) = 1 2 ( 1− ˜ϱ2 ν2 ) (SP.21) At largeν, the dynamics of˜ϱ is governed by the Burgers ...

  3. [12]

    We now show that they simply converge to the weak noise equations of KPZ

    The MFT equations directly describe the weak noise regime of the stochastic hydrodynamic equation of the WASEP. We now show that they simply converge to the weak noise equations of KPZ. One starts from the pair ∂tq =∂x [ ∂xq− 2q(1−q)(∂xp +ν) ] , −∂tp =∂2 xp + (1− 2q)∂xp(∂xp + ...

  4. [13]

    Derivatives of Ψ(u) Starting from the expression of the derivatives ofΨ(u) (SJ.10), we obtain the lowest two orders asν→ 0. Ψ(n)(0) =− ϱn 1 (1−ϱ2)n 4νn [dn−1 dyn−1  (y(1−y))n−1e2νny(2νT (y−1)+X) (y−ϱ1)n Erfc ( − √n 4T (2νT (2y− 1) +X) ) |y=ϱ2 + (−1)n−1 dn−1 dyn−1  (y(1−y...

  5. [14]

    Determination of the leading order ofΨ(u) As we now show, to obtain the leading order of theν→ 0 limit of Ψ(u), it is sufficient to setν = 0 inside the integrand in (5). To see this, first note that from (SH.4) we have the following relation at the saddle point Ψ′(u) = 1 2ν lo...

  6. [15]

    We will show a perfect matching with [56, Eq

    Determination of the subleading order ofΨ(u) To obtain the expression ofΨ0(u), we proceed to a resummation of its derivatives obtained in (SQ.16). We will show a perfect matching with [56, Eq. (6.35)] for anyX. We recall that we use the notations 57 Ω = uϱ1(1−ϱ2), ξ = √ X/(4T ...

  7. [16]

    J→ +∞ tail for the WASEP For generalν >0, let us examine the equations (SI.1) asu→ +∞. In that limit one hasuΨ′(u)→ +∞ logarithmically inu and ωu = 1 + α u +O(1/u2), thus the various expansions read ζ(u) =e2νuΨ′(u)− (1 +α) +αe−2νuΨ′(u) (SR.1) J =−uΨ′(u) + 1 2ν logu + 1 +α 2ν e...

  8. [17]

    First, by parity we restrict the integral to [0,∞[ and we split the domain of integration on[0,k 0] and [k0,∞[

    We now study the limitν fixed and u→ +∞, which implies thatk0→ +∞ since one has k2 0 = log(u) 4ν2T + 1 4 (log(4) ν2T − 1 ) + 1 4−w2 log(u) +O ( 1 log(u) )2 (SR.5) In that limit it is convenient to split the integral in (SR.3) in two parts. First, by parity we restrict the inte...

  9. [18]

    We show in (SQ.2) that the rate functionΨ(u) admits the following expansion uΨ′(u) = log(1 +uωu) 2ν +uΨ′ 0(u) +O(ν) (SR.17) where Ψ0(u) is given explicitly in (SQ.8)

    J→ +∞ tail in the SSEP limit To obtain the tail of the current distribution in the SSEP limit, we first proceed to the limitν→ 0 and then takeu→ +∞. We show in (SQ.2) that the rate functionΨ(u) admits the following expansion uΨ′(u) = log(1 +uωu) 2ν +uΨ′ 0(u) +O(ν) (SR.17) wher...

  10. [19]

    We use the results and notations of Section P1

    J→ +∞ tail in the KPZ limit To obtain the tail of the current distribution in the KPZ limit, we first proceed to the limitν→∞ and only later take the rescaled Laplace parameter˜u→ +∞. We use the results and notations of Section P1. In the limitν→∞ at fixed ˜u the large deviati...

  11. [20]

    Crossover between the J 3 tail of the WASEP and theJ 5/2 tail of KPZ For a largeν, it seems natural to expect a crossover between the exponents3 and 5/2 in the right tail of the current distribution. The crossover between the two tails occurs whenJ−νT/ 2∼ νT and we thus 62 def...

  12. [21]

    with k0 fixed, i.e., with logu ν2T = 1 + 4k2 0 =O(1) (SR.36) A crossover will occur ask0 increases fromk0→ 0 (KPZ) andk0→ +∞ (WASEP). Neglecting the pre- exponential factors in the argument of the logarithm in (SR.35), it is easy to see that the leading estimate at largeν is u...

  13. [22]

    2 π arctan(2k0) (SR.41) which is a strictly increasing function ofk0 with ˜J∼k2 0 for k0≪ 1 and ˜J∼ 2k0/π for k0≫ 1. The second equation in (SR.38) then shows that in the crossover regionΦ(J) will take the scaling form Φ(J)≃T 2(2ν)3Φ+( ˜J) (SR.42) and one has Φ′ +( ˜J) = uΨ′(u...

  14. [23]

    2 π arctan(2k0) (SR.45) From this representation it is easy to obtain the expansion ofΦ+( ˜J) for small ˜J (corresponding to the KPZ limit, and smallk0) and for large˜J (corresponding to the WASSEP limit, and largek0). One find • For ˜J≪ 1, we have Φ+( ˜J) = 16 ˜J5/2 15π + 32 ...

  15. [24]

    We show in this Section that this range can be extended tou∈ [uc,∞[, withuc < 0, and additionally thatΨ(u) possesses several branches

    Extension of the domain of definition ofΨ(u) The range of definition ofΨ(u) covers naturallyu∈ [0,∞[. We show in this Section that this range can be extended tou∈ [uc,∞[, withuc < 0, and additionally thatΨ(u) possesses several branches. The main branch is given by (5). The sec...

  16. [25]

    Its Jacobian reads du u = g′(κ) g(κ) dκ = (−8Tν 2κ + κ ( 2− 8w2) (4κ2− 1) (w2−κ2))dκ (SS.7) Let us study the structure of the solutions, see Fig

    Location of solitons To study potential solitons, we need to solve fork∈ iR the equation 1 +uh(k) = 0, equivalent to u uc = ( w2−κ2) e−4κ2ν2T (1− 4κ2)w2 :=g(κ) (SS.6) Note that since we restrict here toX = 0, we only need to considerk∈ iR, forX̸= 0, the solitons might have a m...

  17. [26]

    [50] (see also [53] for subsequent work)

    Analytical continuation of uΨ′(u) Knowing the solitonic structure, we can now obtain the analytical continuation ofuΨ′(u) using the same derivation as in Ref. [50] (see also [53] for subsequent work). We start by proceeding to an integration by 66 part on the integral definiti...

  18. [27]

    Continuation of the parametric representation ofΦ(J) for J <Jc Let us recall that the main branch (SI.1) corresponds tou∈ [uc, +∞[ and J > Jc. Let us now give our result for the parametric representation ofΦ(J) for J <Jc, where againu∈ [uc, +∞[ Φ′(J) = 2νu(Ψ′(u) + ∆′(u)) J =− ...

  19. [28]

    We report the plots in Fig

    Plots of Φ(J) and Φ′(J) We plot in this Section the functionsΦ(J) and Φ′(J) using the parametric representations in the main branch (SI.1) and in the second branch (SS.16). We report the plots in Fig. SS.3. Appendix T: Tail forJ→−∞ (or J→ 0 for step initial condition) This tai...

  20. [29]

    We will use the results from Section R1 for the asymptotics ofuΨ′(u), which we will complement with the one ofu∆′(u) obtained above

    Tail for J→−∞ for the WASEP (double-sided Bernoulli initial condition) The upper tail of the WASEP forα> 0 (which excludes the step initial condition) is obtained by taking u→ +∞ adding the contribution of the jump to the large deviation function. We will use the results from ...

  21. [30]

    One can ask about the asymptotic behavior near the wall

    Tail near the wallJ→ 0 for the WASEP (step initial condition) In the case of the step initial condition there is a "wall" atJ = 0, since Φ(J) is defined only forJ > 0. One can ask about the asymptotic behavior near the wall. It corresponds tou→− 1− and κ(u)→ 1/2−. Setting u =−...

  22. [31]

    In that regime we will find that Jc≃⟨J⟩≃ νT 2

    Large ν limit and crossover from the wall to the typical value (step initial condition) Here we study the step initial condition in the regime whereν→ +∞. In that regime we will find that Jc≃⟨J⟩≃ νT 2 . We study here the regionJ <Jc. Interestingly, it describes a crossover bet...

  23. [87]

    The mappings between symmetric exclusion processes (including the SSEP) and isotropic spin chains was considered in [89], and [44, 45]

    for the analysis of the Landau-Lifshitz model in the stereographic frame. The mappings between symmetric exclusion processes (including the SSEP) and isotropic spin chains was considered in [89], and [44, 45]. For a recent appearance of the Landau-Lifshitz model in the context...

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Reviewed August 15, 2026 · model on record in the stance chip above.