Stationary directed polymers and energy solutions of the Burgers equation

Jara, Milton; Flores, Gregorio R. Moreno

doi:10.1016/j.spa.2020.04.012

Cited by 12 publications

(10 citation statements)

References 44 publications

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“…Lemma 3.1 is a consequence of an expansion of W j − η j (see (7.4)), which can be deduced by a similar way as for the O'Connell-Yor polymer model given in [16]. The proof of Lemma 3.1 is postponed to Section 7.…”

Section: Proof Outlinementioning

confidence: 80%

“…Other important class from which KPZ equation is derived is directed polymers, which is introduced in [13] and mathematically analyzed in [14] for the first time. As to the stationary case, recently [16] derived the stochastic Burgers equation from free-energy fluctuation of the stationary O'Connell-Yor model ( [20]). On the other hand, a some relation between the O'Connell-Yor polymer and an interacting particle system is pointed out: the q-deformation of totally asymmetric simple exclusion process (q-TASEP, in short) with parameter q ∈ (0, 1) is introduced in [2] and moreover it is proved that the q-TASEP in some sense converges to the O'Connell-Yor polymer as q → 1.…”

Section: Introductionmentioning

confidence: 99%

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Equilibrium fluctuations for totally asymmetric interacting particle systems

Hayashi¹

2022

Preprint

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We study equilibrium fluctuations for a class of totally asymmetric zero-range type interacting particle systems. As a main result, we show that density fluctuation of our process converges to the stationary energy solution of the stochastic Burgers equation. As a special case, microscopic system we consider here is related to q-totally asymmetric simple exclusion processes (q-TASEPs) and our scaling limit corresponds to letting the quantum parameter q to be one.

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Section: Proof Outlinementioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Equilibrium fluctuations for totally asymmetric interacting particle systems

Hayashi¹

2022

Preprint

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“…The first result of this type was on the Cole-Hopf level [1]. The result in [22] shows convergence of the discrete gradients of H n to the energy solution of the stochastic Burgers equation i.e. the space-derivative of the KPZ equation.…”

Section: The Intermediate Disorder Regimementioning

confidence: 98%

“…Let H n = log Z st n . From [22], we know that H n ⇒ H as n → ∞ in the locally uniform topology. This is known as intermediate disorder regime.…”

Section: The Intermediate Disorder Regimementioning

confidence: 99%

Aging for the stationary Kardar-Parisi-Zhang equation and related models

Deuschel¹,

Flores²,

Orenshtein³

2020

Preprint

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We study aging properties for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution of the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. The key ingredient to our proofs is a covariance-to-variance reduction which is a characteristic of space-time stationarity and allows to deduce the asymptotic behavior of the correlations of two points by the one of the variance at one point. We formulate several open problems.

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“…Since its introduction in [17], the semi-discrete polymer has been the subject of much investigation, revealing a rich algebraic structure far beyond the invariant measure statement contained in Proposition 1. See for example [3,4,8,9,10,11,13,14,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

Central moments of the free energy of the O'Connell-Yor polymer

Noack,

Sosoe

2020

Preprint

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Seppäläinen and Valkó showed in [19] that for a suitable choice of parameters, the variance growth of the free energy of the stationary O'Connell-Yor polymer is governed by the exponent 2/3, characteristic of models in the KPZ universality class.We develop exact formulas based on Gaussian integration by parts to relate the cumulants of the free energy, log Z θ n,t , to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, s0. We then use these formulas to obtain estimates for the k-th central moment of log Z θ n,t as well as the k-th annealed moment of s0 for k > 2, with nearly optimal exponents (1/3)k + ǫ and (2/3)k + ǫ, respectively.

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Stationary directed polymers and energy solutions of the Burgers equation

Cited by 12 publications

References 44 publications

Equilibrium fluctuations for totally asymmetric interacting particle systems

Equilibrium fluctuations for totally asymmetric interacting particle systems

Aging for the stationary Kardar-Parisi-Zhang equation and related models

Central moments of the free energy of the O'Connell-Yor polymer

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