Large deviation principle for the stationary measures of open asymmetric simple exclusion processes

Milind Hegde; Zongrui Yang

Large deviation principle for the stationary measures of open asymmetric simple exclusion processes

T0 review · 0 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that, in the fan region $ab<1$, the stationary height profiles of the open asymmetric simple exclusion process satisfy a large deviation principle with an explicit rate function $I^{(a,b)}$ that depends only on the…

desk verdict Solid proof of the DLS03 LDP for open ASEP in the fan region, with a patchable gap in the final path-space step. read the letter →

arxiv 2412.12026 v2 pith:26SGXQFG submitted 2024-12-16 math.PR

Milind Hegde , Zongrui Yang This is my paper · ORCID

classification math.PR MSC 60F1060K3582C22

keywords largedeviationsasymmetricsimpleexclusionprocessopenboundariesstationarymeasurefanregiontwo-layerrepresentationmatrixproductansatzheightprofile

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 proves that, in the fan region $ab<1$, the stationary height profiles of the open asymmetric simple exclusion process (ASEP) satisfy a large deviation principle with an explicit rate function $I^{(a,b)}$. The rate function depends only on the effective boundary densities $a$ and $b$, and coincides with the rate function for the totally asymmetric (TASEP) case, so the asymmetry parameter $q$ and the auxiliary parameters $c,d$ drop out entirely. This settles, for the fan region, a prediction from the physics literature obtained by the additivity principle and matrix product ansatz. The proof works by rewriting the stationary measure as the marginal of a two-layer random-walk measure, then showing that, on the large-deviation scale, the weights can be compared to those of open TASEP.

What carries the argument

The central object is a two-layer representation: the stationary measure of open ASEP is realized as the marginal of a probability measure on pairs of non-crossing Bernoulli paths $(\lambda_1,\lambda_2)$. The weight of a pair factors into boundary rewards $a^{\lambda_1(0)-\lambda_2(0)}b^{\lambda_1(N)-\lambda_2(N)}$, a q-Pochhammer endpoint factor, and local transition weights $W^{(q,c,d)}$. This is a marginal of the Enaud-Derrida matrix product ansatz representation. The argument then compares these weights to the $q=0$ TASEP two-layer weights: whenever the two layers are separated by at least $r$, each local factor is bounded above and below by constants depending only on $q,r,c,d$, so the ratio of the weights is exponentially flat. A separation estimate (Proposition 2.9) shows that, conditional on the height-profile event, the two TASEP layers stay separated on all but an $\varepsilon$-fraction of sites with probability at least $e^{-N^{4/5}}$, which is enough to make the large-deviation comparison.

What would settle it

For a fixed pair $(a,b)$ with $ab<1$, compute via the matrix product ansatz the stationary probability that $h_N(1/2) > 0.75$ for $q=0$ and $q=1/2$ (with $c=d=0$ so $a,b$ are the same). The theorem implies the difference of the two log-probabilities, divided by $N$, goes to $0$; observing a nonzero limit would falsify it.

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Extended reading notes

Core claim

Under the stationary measure $\mu_N$ of open ASEP with parameters satisfying $ab<1$, the sequence of height profiles $h_N$ satisfies the LDP on $C_0([0,1],\mathbb{R})$ with good rate function $I^{(a,b)}$. This function is finite only for absolutely continuous, nondecreasing, 1-Lipschitz profiles, and is given by an integral of the entropy $H$ plus a convex-envelope correction and a boundary constant $-\log J(a,b)$. The key discovery is that this rate function is independent of $q,c,d$; it matches the TASEP rate function established previously. The authors establish the LDP by a two-layer representation of the stationary measure, derived from the Enaud-Derrida representation of the matrix product ansatz, and by a comparison argument showing that ASEP two-layer weights and TASEP two-layer weights have the same exponential asymptotics.

Load-bearing premise

The proof's load-bearing premise is that the two-layer representation of the stationary measure, imported from the Enaud-Derrida matrix ansatz, is valid across the entire fan region and that the TASEP rate function it compares to is exactly $I^{(a,b)}$.

Editorial extensions

If this is right

Every macroscopic height profile in the fan region has an explicit exponential cost: $-\log \mathbb{P}(h_N\approx f) \sim N I^{(a,b)}(f)$.
The rate function is universal in $q$: two open ASEPs with the same $a,b$ but different asymmetry or different $c,d$ have the same large-deviation speed and rate.
The LDP passes by contraction to any continuous observable of the height profile, including the total particle number and other linear statistics.
The normalization asymptotics $\frac{1}{N}\log Z_N \to -\log J(a,b)$ identify the free energy per site in the fan region, matching the three-phase formula for $J$.
The paper reports that an analogous argument, using a two-layer representation currently being developed, is expected to prove the LDP in the shock region.

Reading between the lines

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

Beyond the paper: this comparison strategy suggests that, once a two-layer representation is available in the shock region, the rate function there would also be independent of $q$ and equal to the TASEP rate function.
Beyond the paper: the separation estimate only needs polynomial lower bounds on probabilities, so the method may tolerate much weaker separation events and could apply to other integrable stochastic models with two-layer representations.
Beyond the paper: the explicit rate function, with its convex-envelope term, resembles the rate function for a single random walk conditioned to stay in an interval, hinting that the two-layer representation could yield finer fluctuation results, such as the order of pre-exponential factors.
Beyond the paper: if the universality in $q$ extends to the weakly asymmetric limit $q\to 1$, the same $I^{(a,b)}$ could describe large deviations of the stationary measure for the KPZ fixed point with boundaries.

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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

Here's the short version: this paper proves a long-standing physics prediction, the Derrida–Lebowitz–Speer large deviation rate for the height profile of open ASEP, in the fan region ab<1, for the full range of asymmetry q and boundary parameters c,d. It does so by reducing the problem to the recently proved TASEP LDP of Bryc and Zatitskii via a two-layer representation. That reduction is the real new work, and it is mostly convincing.

The genuinely new ingredients are the ordered two-layer representation with a free start for the second layer, and the separation estimate (Proposition 2.9) which says that, under the TASEP two-layer measure, the two layers stay apart on all but an ε-fraction of sites with probability e^{-o(N)}. This is then used to show that the ASEP weights differ from the TASEP weights only by subexponential factors (Corollary 2.10). The rate function is explicit and parameter-free, and the comparison argument is elegant. The finite-dimensional part of the proof is careful, including the handling of the cone boundary by the perturbation argument in Lemma 4.3.

The soft spot is at the very end. The paper transfers the LDP from the pointwise topology on C0([0,1]) to the sup-norm topology using Lemma 4.8, which is false as stated: an LDP in a weaker topology plus exponential tightness does not imply an LDP in the stronger topology unless the rate function is also a good rate function for the stronger topology. In this setting the missing hypothesis is satisfied—the measures are supported on the compact set Y of 1-Lipschitz, nondecreasing functions, where the two topologies coincide, and pI is sup-norm lower semicontinuous—but the paper does not state or prove these facts before invoking Lemma 4.8. So the written proof has a genuine gap in Step 2 of Theorem 1.2, not in the finite-dimensional LDP. It looks repairable without new ideas, but a referee should insist on a correct statement of the transfer theorem and the verification of sup-norm goodness of pI.

The citation pattern is healthy; the central external inputs (BZ24, ED04, USW04) are appropriate and the reduction to BZ24 is explicit. I don't see a circularity problem.

Bottom line: this is a significant result that deserves a serious referee. With a fix of the last step, it should be accepted. I'd bring it to a reading group on exactly solvable models or large deviations.

Referee Report

0 major / 5 minor

Summary. The paper proves a large deviation principle (LDP) for the height profiles of the open asymmetric simple exclusion process (ASEP) on a finite interval, in the fan region ab<1 of the phase diagram, under the stationary measure. The rate function I^{(a,b)} is explicit, depends only on the effective boundary densities a and b, and coincides with the open TASEP rate function. The proof has three main ingredients: (i) a two-layer representation of the ASEP stationary measure derived from the Enaud–Derrida matrix product representation (Theorem 2.5); (ii) an estimate (Proposition 2.9) showing that, under the open TASEP two-layer measure, the two layers remain separated on all but a small fraction of sites with probability decaying slower than exponential; and (iii) a comparison of ASEP and TASEP two-layer weights (Corollary 2.10) showing that their logarithms differ by o(N) on the relevant finite-dimensional events. These yield finite-dimensional LDPs, which the Dawson–Gärtner theorem upgrades to a path-space LDP in the pointwise topology; exponential tightness on the compact set of nondecreasing 1-Lipschitz profiles then transfers the LDP to the uniform topology. The authors also prove a convex-envelope identity (Lemma 4.3) used to simplify the lower bound for open sets.

Significance. The result is significant: it provides the first rigorous proof of the Derrida–Lebowitz–Speer large deviation functional for the height profile of open ASEP in the fan region, a prediction from 2002–2003. The rate function is independent of the asymmetry q and of the reservoir parameters beyond the effective densities, a striking universality feature. The proof is largely self-contained, and the novel comparison argument between ASEP and TASEP two-layer measures, built on separation estimates for non-intersecting random walks, is a useful methodological contribution that may extend to the shock region. The paper relies appropriately on cited results (matrix product ansatz, Enaud–Derrida representation, USW normalization asymptotics, and the Bryc–Zatitskii TASEP LDP) and gives precise citations. The stress-test objection to Lemma 4.8 is not borne out: the lemma's hypotheses include Hausdorffness of the weaker topology, which holds for the pointwise topology on C0; the trivial-topology counterexample violates that hypothesis.

minor comments (5)

[Section 4.3, Step 2] The application of Lemma 4.7 with E = C0 is not directly justified because C0, the set of continuous functions, is not a Borel subset of the pointwise product space X. This is repairable by taking E = Y (which is closed and contains the support), obtaining the LDP on Y with the pointwise topology, and extending the rate function by infinity to C0; alternatively, the LDP on C0 follows directly from the LDP on X because the profiles take values in Y. Please adjust the argument accordingly.
[Section 4.3, Step 2] The identification pI = I^{(a,b)} after Lemma 4.8 is terse; it would help to spell out that the TASEP finite-dimensional marginals also yield the pointwise LDP with rate function pI by the Dawson–Gärtner theorem, so uniqueness of rate functions on the regular space X gives equality with the known TASEP rate function.
[Theorem 2.5, proof, equation (2.7)] In the displayed formula for the sum over λ2, the product index is i while the indicators use j; please make the index consistent.
[Proposition 3.9, proof, Step 2] The phrase 'by conditioning on the values λ1(⌊θjN⌋) and λ1(⌊θjN⌋)' should read 'λ1(⌊θjN⌋) and λ2(⌊θjN⌋)'.
[Lemma 4.8] The statement is a correct citation of Dembo–Zeitouni Corollary 4.2.6 when the weaker topology is Hausdorff; a parenthetical noting that the pointwise topology on C0 is Hausdorff and that exponential tightness holds on the compact set Y would help the reader apply the lemma without concern.

Circularity Check

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No significant circularity: the ASEP LDP is reduced to the external open TASEP LDP of [BZ24] by an explicit weight comparison; self-citations are auxiliary.

full rationale

The paper's central claim is not circular. Theorem 1.2 is proved by importing the open TASEP path-space LDP from [BZ24] (an external result with no author overlap), deriving finite-dimensional marginals for TASEP from it, and then transferring these marginals to general ASEP in the fan region through a Radon-Nikodym-style comparison of two-layer weights (Theorem 2.5 and Corollary 2.10). The weight comparison is grounded in Proposition 2.9, whose proof uses elementary properties of non-intersecting Bernoulli bridges (Gibbs property, monotone couplings, FKG, hypergeometric tail comparison) and a standard supremum bound. The normalization asymptotics in Lemma 2.7 are quoted from [USW04], also external. No parameter is fitted and no target result is assumed inside the comparison: the rate function I^(a,b) is taken from TASEP and shown, by matching upper and lower exponential rates, to govern ASEP as well. The final identification of the Dawson-Gärtner path rate pI with I^(a,b) uses the external TASEP theorem together with the standard uniqueness property of rate functions, so it is a legitimate comparison of two independently obtained rate functions, not a renaming. The only self-citations present are [ACH24] (for a standard random-walk supremum bound used in Lemma 3.7 and for heuristic context) and [BCY24] (for historical context). Lemma 3.7 is a parameter-free estimate about unconditioned bridges that does not assume the separation estimate or the main theorem, so under the stated criteria it is independent support and does not raise the circularity score. I note, without treating it as circularity, a correctness concern in Section 4.3 Step 2: Lemma 4.8, cited from [DZ09, Corollary 4.2.6], is invoked to strengthen the LDP from the pointwise to the sup-norm topology without separately verifying sup-norm goodness of pI; this is a potential missing hypothesis in the written proof, but it is not a reduction of a prediction to its inputs.

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

The central claim rests on the matrix product ansatz and its Enaud-Derrida representation (imported from physics literature), the known normalization asymptotics [USW04], and the open TASEP LDP [BZ24]. No free parameters are fitted; the rate function is explicit. The paper's own contribution is the comparison and separation argument, not new postulates.

assumptions (6)

domain assumption Matrix product ansatz (DEHP algebra) represents the stationary measure of open ASEP (Lemma 2.1)
Cited to [DEHP93]; used in Theorem 2.5 to connect two-layer weights to the ASEP stationary measure.
domain assumption Enaud-Derrida infinite matrices satisfy the DEHP algebra and yield finite angle bracket W|V angle bracket in the fan region (Lemma 2.3)
Cited to [ED04]; the two-layer representation is constructed from these matrices.
domain assumption Normalization constant asymptotics for the MPA partition function (Lemma 2.7)
Cited to [USW04, Section 6.1]; determines the -log J(a,b) term in the rate function.
domain assumption Open TASEP height-profile LDP with rate function I^{(a,b)} (Theorem 1.2 for c=d=q=0)
Cited to [BZ24]; the paper reduces the ASEP case to this external result.
standard math Monotone coupling of non-intersecting Bernoulli bridges (Lemma 3.3)
Cited to [CEP00, DFF+21, Ser23]; used in the separation estimate Proposition 2.9.
standard math Standard large deviation toolkit: contraction principle, Dawson-Gartner theorem, exponential tightness (Section 4.2)
Used to upgrade the finite-dimensional LDP to the path-space LDP.

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Pith. "Pith review of Large deviation principle for the stationary measures of open asymmetric simple exclusion processes." pith.science (2026). https://pith.science/paper/26SGXQFG

@misc{pith2026241212026,
  author       = {Pith},
  title        = {Pith review of: Large deviation principle for the stationary measures of open asymmetric simple exclusion processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26SGXQFG}},
  note         = {Machine review of arXiv:2412.12026}
}

abstract

We consider the stationary measure of the asymmetric simple exclusion process (ASEP) on a finite interval in $\mathbb{Z}$ with open boundaries. Fixing all the jump rates and letting the system size approach infinity, the height profile of such a sequence of stationary measures satisfies a large deviation principle (LDP), whose rate function was predicted in the physics work arXiv:cond-mat/0205353. In this paper, we provide the first rigorous proof of the large deviation principle in the "fan region" part of the phase diagram. Our proof relies on two key ingredients: a two-layer expression of the stationary measure of open ASEP, arising from the Enaud-Derrida representation arXiv:cond-mat/0307023 of the matrix product ansatz, and the large deviation principle of the open totally asymmetric simple exclusion process (TASEP) recently established in arXiv:2403.03275.

Figures

Figures reproduced from arXiv: 2412.12026 by the authors.

**Figure 1.** Jump rates in the open ASEP. asymmetry parameter. Consequently, the rate function must coincide with the one in the case of open TASEP, which was identified in [BZ24]. Some more details are discussed in Section 1.5. Our approach is fairly robust. An upcoming work of the second-named author with Dominik Schmid will provide a different version of a two-layer representation of the stationary measure of open ASEP in the… view at source ↗

**Figure 2.** Phase diagram for the open ASEP stationary measures. LD, HD, MC respectively stand for the low density, high density and maximal current phases. We denote by µN = µ (q,a,b,c,d) N the unique stationary measure of open ASEP, which is a probability measure on (τ1, . . . , τN ) ∈ {0, 1} N, where, as mentioned earlier, τi is the occupation variable of site i, for i = 1, . . . , N. The (re-scaled) height profile of open A… view at source ↗

**Figure 3.** The figure depicts the two layers λ1 and λ2, satisfying λ = (λ1,λ2) ∈ T LN . 2.2. The two-layer representation. In the fan region ab < 1, the stationary measures of open ASEP can be expressed in terms of certain two-layer measures, which originate from the Enaud–Derrida representation of the matrix product ansatz. In this subsection we will introduce this two-layer representation. Definition 2.4. We define the two-l… view at source ↗

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Works this paper leans on

52 extracted references · 48 canonical work pages

[1]

Aggarwal, I

A. Aggarwal, I. Corwin, and M. Hegde. Scaling limit of the colored ASEP and stochastic six-vertex models. arXiv preprint arXiv:2403.01341 , 2024

arXiv 2024
[2]

Askey and J

R. Askey and J. A. Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , volume 319. American Mathematical Soc., 1985

work page 1985
[3]

Barraquand

G. Barraquand. Integral formulas for two-layer S chur and W hittaker processes. arXiv preprint arXiv:2409.08927 , 2024

arXiv 2024
[4]

Barraquand, I

G. Barraquand, I. Corwin, and Z. Yang. Stationary measures for integrable polymers on a strip. Invent. Math. , pages 1--75, 2024

work page 2024
[5]

Barraquand and P

G. Barraquand and P. Le Doussal. Stationary measures of the KPZ equation on an interval from E naud- D errida's matrix product ansatz representation. J. Phys. A , 2023

work page 2023
[6]

R. A. Blythe and M. R. Evans. Nonequilibrium steady states of matrix-product form: a solver's guide. J. Phys. A , 40(46):R333, 2007

work page 2007
[7]

R. A. Blythe, M. R. Evans, F. Colaiori, and F. H. Essler. Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra. J. Phys. A , 33(12):2313, 2000

work page 2000
[8]

Current large deviations for Asymmetric Exclusion Processes with open boundaries

T. Bodineau and B. Derrida. Current large deviations for asymmetric exclusion processes with open boundaries. arXiv preprint cond-mat/0509179 , 2005

work page Pith review arXiv 2005

Show all 52 references

[9]

R. Brak, S. Corteel, J. Essam, R. Parviainen, and A. Rechnitzer. A combinatorial derivation of the PASEP stationary state. Electron. J. Combin. , 13(1):R108, 2006

2006
[10]

W. Bryc. Stationary distribution of open asymmetric simple exclusion processes on an interval as a marginal of a two-layer ensemble. ALEA, Lat. Am. J. Probab. Math. Stat., to appear. arXiv preprint arXiv:2408.06535 , 2024

2024 arXiv
[11]

W. Bryc, J. Najnudel, and Y. Wang. Limit fluctuations of stationary measures of totally asymmetric simple exclusion processes with open boundaries on the coexistence line. arXiv preprint arXiv:2407.20835 , 2024

2024 arXiv
[12]

Bryc and M

W. Bryc and M. \'S wieca. On matrix product ansatz for asymmetric simple exclusion process with open boundary in the singular case. J. Stat. Phys. , 177:252--284, 2019

2019
[13]

Bryc and Y

W. Bryc and Y. Wang. Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries. Ann. Inst. Henri Poincaré Probab. Stat. , 55(4):2169--2194, 2019

2019
[14]

Bryc and J

W. Bryc and J. Weso owski. Askey--Wilson polynomials, quadratic harnesses and martingales. Ann. Probab. , 38(3):1221--1262, 2010

2010
[15]

Bryc and J

W. Bryc and J. Weso owski. Asymmetric simple exclusion process with open boundaries and quadratic harnesses. J. Stat. Phys. , 167:383--415, 2017

2017
[16]

Bryc and P

W. Bryc and P. Zatitskii. A two-line representation of stationary measure for open TASEP . Electron. J. Probab., to appear. arXiv preprint arXiv:2403.03275 , 2024

2024 arXiv
[17]

H. Cohn, N. Elkies, and J. Propp. Local statistics for random domino tilings of the A ztec diamond. arXiv preprint math/0008243 , 2000

2000 arXiv
[18]

Corteel, M

S. Corteel, M. Josuat-Verg \`e s, and L. K. Williams. The matrix ansatz, orthogonal polynomials, and permutations. Adv. in Appl. Math. , 46(1-4):209--225, 2011

2011
[19]

Corteel and L

S. Corteel and L. K. Williams. Tableaux combinatorics for the asymmetric exclusion process and Askey--Wilson polynomials. Duke Math. J. , 159(3):385--415, 2011

2011
[20]

I. Corwin. Some recent progress on the stationary measure for the open KPZ equation. Toeplitz Operators and Random Matrices: In Memory of Harold Widom , pages 321--360, 2022

2022
[21]

de Gier and F

J. de Gier and F. H. Essler. Large deviation function for the current in the open asymmetric simple exclusion process. Phys. Rev. Lett. , 107(1):010602, 2011

2011
[22]

Dembo and O

A. Dembo and O. Zeitouni. Large deviations techniques and applications . Springer, 2009

2009
[23]

B. Derrida. Matrix ansatz and large deviations of the density in exclusion processes. In International Congress of Mathematicians , volume 3, pages 367--382. Citeseer, 2006

2006
[24]

B. Derrida. Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. Theory Exp. , 2007(07):P07023, 2007

2007
[25]

Derrida, C

B. Derrida, C. Enaud, and J. Lebowitz. The asymmetric exclusion process and B rownian excursions. J. Stat. Phys. , 115:365--382, 2004

2004
[26]

Derrida, M

B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier. Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A , 26(7):1493, 1993

1993
[27]

Derrida, J

B. Derrida, J. Lebowitz, and E. Speer. Free energy functional for nonequilibrium systems: an exactly solvable case. Phys. Rev. Lett. , 87(15):150601, 2001

2001
[28]

Derrida, J

B. Derrida, J. Lebowitz, and E. Speer. Exact free energy functional for a driven diffusive open stationary nonequilibrium system. Phys. Rev. Lett. , 89(3):030601, 2002

2002
[29]

Derrida, J

B. Derrida, J. Lebowitz, and E. Speer. Large deviation of the density profile in the steady state of the open symmetric simple exclusion process. J. Stat. Phys. , 107(3):599--634, 2002

2002
[30]

Derrida, J

B. Derrida, J. Lebowitz, and E. Speer. Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process. J. Stat. Phys. , 110(3-6):775--810, 2003

2003
[31]

Dimitrov, X

E. Dimitrov, X. Fang, L. Fesser, C. Serio, C. Teitler, A. Wang, and W. Zhu. Tightness of B ernoulli G ibbsian line ensembles. Electron. J. Probab. , 26:1--93, 2021

2021
[32]

Duchi and G

E. Duchi and G. Schaeffer. A combinatorial approach to jumping particles. J. Combin. Theory Ser. A , 110(1):1--29, 2005

2005
[33]

Enaud and B

C. Enaud and B. Derrida. Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. , 114:537--562, 2004

2004
[34]

F. H. Essler and V. Rittenberg. Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries. J. Phys. A , 29(13):3375, 1996

1996
[35]

Gasper and M

G. Gasper and M. Rahman. Basic hypergeometric series , volume 96. Cambridge university press, 2011

2011
[36]

a ggstr \

H.-O. Georgii, O. H \"a ggstr \"o m, and C. Maes. The random geometry of equilibrium phases. In Phase transitions and critical phenomena , volume 18, pages 1--142. Elsevier, 2001

2001
[37]

Gorissen, A

M. Gorissen, A. Lazarescu, K. Mallick, and C. Vanderzande. Exact current statistics of the asymmetric simple exclusion process with open boundaries. Phys. Rev. Lett. , 109(17):170601, 2012

2012
[38]

Lazarescu

A. Lazarescu. Exact large deviations of the current in the asymmetric simple exclusion process with open boundaries. arXiv preprint arXiv:1311.7370 , 2013

2013 arXiv
[39]

Lazarescu

A. Lazarescu. The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases. J. Phys. A , 48(50):503001, 2015

2015
[40]

Lazarescu and K

A. Lazarescu and K. Mallick. An exact formula for the statistics of the current in the tasep with open boundaries. J. Phys. A , 44(31):315001, 2011

2011
[41]

T. M. Liggett. Ergodic theorems for the asymmetric simple exclusion process. Trans. Amer. Math. Soc. , 213:237--261, 1975

1975
[42]

T. M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes , volume 324. Springer Science & Business Media, 1999

1999
[43]

K. Mallick. The exclusion process: A paradigm for non-equilibrium behaviour. Phys. A , 418:17--48, 2015

2015
[44]

Mallick and S

K. Mallick and S. Sandow. Finite-dimensional representations of the quadratic algebra: applications to the exclusion process. J. Phys. A , 30(13):4513, 1997

1997
[45]

Nestoridi and D

E. Nestoridi and D. Schmid. Approximating the stationary distribution of the ASEP with open boundaries. Comm. Math. Phys. , 405(8):176, 2024

2024
[46]

Sasamoto

T. Sasamoto. Density profile of the one-dimensional partially asymmetric simple exclusion process with open boundaries. J. Phys. Soc. Jpn. , 69(4):1055--1067, 2000

2000
[47]

Sch \"u tz and E

G. Sch \"u tz and E. Domany. Phase transitions in an exactly soluble one-dimensional exclusion process. J. Stat. Phys. , 72(1-2):277--296, 1993

1993
[48]

C. Serio. Tightness of discrete G ibbsian line ensembles. Stochastic Process. Appl. , 2023

2023
[49]

Uchiyama, T

M. Uchiyama, T. Sasamoto, and M. Wadati. Asymmetric simple exclusion process with open boundaries and Askey-Wilson polynomials. J. Phys. A , 37(18):4985, 2004

2004
[50]

Uchiyama and M

M. Uchiyama and M. Wadati. Correlation function of asymmetric simple exclusion process with open boundaries. J. Nonlinear Math. Phys. , 12(sup1):676--688, 2005

2005
[51]

Y. Wang, J. Weso owski, and Z. Yang. Askey--Wilson signed measures and open ASEP in the shock region . Int. Math. Res. Not. IMRN , page rnae116, 2024

2024
[52]

A. J. Wood, R. A. Blythe, and M. R. Evans. Combinatorial mappings of exclusion processes. J. Phys. A , 53(12):123001, 2020

2020

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Aggarwal, I

A. Aggarwal, I. Corwin, and M. Hegde. Scaling limit of the colored ASEP and stochastic six-vertex models. arXiv preprint arXiv:2403.01341 , 2024

arXiv 2024

[2] [2]

Askey and J

R. Askey and J. A. Wilson. Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , volume 319. American Mathematical Soc., 1985

work page 1985

[3] [3]

Barraquand

G. Barraquand. Integral formulas for two-layer S chur and W hittaker processes. arXiv preprint arXiv:2409.08927 , 2024

arXiv 2024

[4] [4]

Barraquand, I

G. Barraquand, I. Corwin, and Z. Yang. Stationary measures for integrable polymers on a strip. Invent. Math. , pages 1--75, 2024

work page 2024

[5] [5]

Barraquand and P

G. Barraquand and P. Le Doussal. Stationary measures of the KPZ equation on an interval from E naud- D errida's matrix product ansatz representation. J. Phys. A , 2023

work page 2023

[6] [6]

R. A. Blythe and M. R. Evans. Nonequilibrium steady states of matrix-product form: a solver's guide. J. Phys. A , 40(46):R333, 2007

work page 2007

[7] [7]

R. A. Blythe, M. R. Evans, F. Colaiori, and F. H. Essler. Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra. J. Phys. A , 33(12):2313, 2000

work page 2000

[8] [8]

Current large deviations for Asymmetric Exclusion Processes with open boundaries

T. Bodineau and B. Derrida. Current large deviations for asymmetric exclusion processes with open boundaries. arXiv preprint cond-mat/0509179 , 2005

work page Pith review arXiv 2005