Asymmetric Simple Exclusion Process with Open Boundaries and Quadratic Harnesses

Bryc, Włodzimierz; Wesołowski, Jacek

doi:10.1007/s10955-017-1747-5

Cited by 18 publications

(27 citation statements)

References 42 publications

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“…The open KPZ stationary measures that we construct in this work will be characterized via a form of duality with another stochastic process which we call the continuous dual Hahn process (denoted below by T s ). This is a special limit of the Askey-Wilson processes constructed by Bryc and Wesołowski [BW17]; see Section 7 for more on this. The continuous dual Hahn process depends on two parameters u, v ∈ R which are assumed throughout to satisfy the relation u + v > 0.…”

Section: The Open Asepmentioning

confidence: 99%

“…This expression has a very algebraic flavor and for asymptotic problems it is natural to look for more analytic formulas. This was accomplished by Bryc and Wesołowski [BW17] who rephrased the matrix product ansatz in terms of the Askey-Wilson processes. We record their main result as Proposition 2.2 in Section 2.3.2 (see Section 7 for definitions).…”

Section: The Open Asepmentioning

confidence: 99%

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Stationary measure for the open KPZ equation

Corwin¹,

Knizel²

2021

Preprint

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We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0, 1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When u + v ≥ 0 we uniquely characterize the constructed stationary measures through their multipoint Laplace transform which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process. Contents 1. Introduction 1 2. Discussion and key ideas for Theorem 1.3 7 3. Open ASEP 13 4. Weak asymmetry limit to the open KPZ equation 18 5. Attractive coupling of open ASEPs with different boundary parameters 22 6. Proof of Theorem 1.3 (1)-(4) 27 7. Askey-Wilson process and the open ASEP invariant measure 27 8. Asymptotics of the ASEP generating function and proof of Theorem 1.3 (5) 39 9. Proof of Proposition 8.1 41 10. Asymptotics of q-Pochhammer symbols 64 References 82

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Section: The Open Asepmentioning

confidence: 99%

Section: The Open Asepmentioning

confidence: 99%

Stationary measure for the open KPZ equation

Corwin¹,

Knizel²

2021

Preprint

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“…A similar duality holds for Brownian excursions [39] and can be obtained as a limit of ( 22) as L → +∞, as we shall see in the sequel. Note also that an analog of ( 22) has been established for ASEP [18], and it would be interesting to study the connection to discrete variants of LQM [40]. It would be also very interesting to know if such dualities extend to other solvable models in the KPZ class.…”

Section: Model -The Kpz Equation For the Height Field H(x T) Readsmentioning

confidence: 96%

“…Similarly a direct calculation of P L (Y ) is possible for u + v = −n, using (18). To this aim let us define a Brownian bridge B(x) such that B(L) = Y , in terms of the standard Brownian bridge B(x), as…”

Section: Pdf Of the Total Height Differencementioning

confidence: 99%

“…This Laplace transform formula relates the stationary measure to an auxiliary stochastic process called continuous dual Hahn process. This construction corresponds to the KPZ equation limit of formulas with a similar structure obtained in [18] for the stationary measure of ASEP. However, obtaining a characterization of the process allowing to study properties of the stationary measure remains a challenge.…”

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confidence: 92%

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Steady state of the KPZ equation on an interval and Liouville quantum mechanics

Barraquand,

Doussal

2021

Preprint

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We obtain a simple formula for the stationary measure of the height field evolving according to the Kardar-Parisi-Zhang equation on the interval [0, L] with general Neumann type boundary conditions and any interval size. This is achieved using the recent results of Corwin and Knizel (arXiv:2103.12253) together with Liouville quantum mechanics. Our formula allows to easily determine the stationary measure in various limits: KPZ fixed point on an interval, half-line KPZ equation, KPZ fixed point on a half-line, as well as the Edwards-Wilkinson equation on an interval.

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Stationary measure for the open KPZ equation

Corwin,

Knizel

2023

Comm Pure Appl Math

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We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When , we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic process that we call the continuous dual Hahn process. Our work relies on asymptotic analysis of Bryc and Wesołowski's Askey–Wilson process formulas for the open ASEP stationary measure (which in turn arise from Uchiyama, Sasamoto and Wadati's Askey‐Wilson Jacobi matrix representation of Derrida et al.'s matrix product ansatz) in conjunction with Corwin and Shen's proof that open ASEP converges to open KPZ under weakly asymmetric scaling.

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Asymmetric Simple Exclusion Process with Open Boundaries and Quadratic Harnesses

Cited by 18 publications

References 42 publications

Stationary measure for the open KPZ equation

Stationary measure for the open KPZ equation

Steady state of the KPZ equation on an interval and Liouville quantum mechanics

Stationary measure for the open KPZ equation

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