Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles

Corwin, Ivan; Dimitrov, Evgeni

doi:10.1007/s00220-018-3139-3

Cited by 38 publications

(75 citation statements)

References 88 publications

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“…We will approach the half-line ASEP through a scaling limit of another integrable model, the stochastic six-vertex model in a half-quadrant, which we believe is also interesting in its own right. On the whole line Z, this approach was recently used for studying ASEP in [AB16,Agg16,BO16,CD17]. Our half-space model is closely related to off-diagonally symmetric alternating sign matrices, whose weighted enumeration was computed in [Kup02].…”

Section: Introductionmentioning

confidence: 99%

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Barraquand¹,

Borodin²,

Corwin³

et al. 2018

Duke Math. J.

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100

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We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ ) according to the Tracy-Widom GOE distribution on the τ 1/3 scale. This is the first example of KPZ asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall-Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to ASEP) using a Yang-Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogs via a refined Littlewood identity.study the statistics of the number of particles in the system when started empty, and with boundary conditions tuned to this critical point. We prove that after a very long time τ , the random variable scales (around its law of large numbers centering) like τ 1/3 and converges in this scale weakly to the GOE Tracy-Widom distribution (Theorem A). Further, our results also shed light on the distribution of the solution to the KPZ equation with Neumann boundary condition (Theorem B), which arises as a limit of the height function of weakly asymmetric half-line ASEP around this critical point [CS16]. This is the first proof of (KPZ / random-matrix-theoretic) asymptotics in a non free-fermionic half-space model. Free-fermionic full-space systems have been well-studied via robust mathematical approaches, in particular the Schur processes [OR03]. These are determinantal systems, meaning that correlation functions are written as determinants of a large matrix. The half-space analog of such systems are Pfaffian Schur processes [Rai00, BR01a, BR05, SI04], whose correlation functions are given via Pfaffians. The full and half-space TASEP (where jumps only go in one direction) and a small handful of other models fit into the free-fermionic framework.ASEP and many other important models do not fit into the free-fermionic framework. In the last decade, starting from the work of Tracy and Widom on ASEP (on the full line) [TW09], many KPZ type limit theorems have been obtained for non-free fermionic models in a full-space. These results have helped refine and expand the notion of KPZ universality. Some attempts have been made to study similar half-space systems, but until now no method has yielded rigorous distributional asymptotics without a Pfaffian structure. Among the existing works on non freefermionic half-space systems, [OSZ14] studied the Log-Gamma directed polymer in a half-quadrant using properties of the geometric RSK algorithm on symmetric matrices, and conjectured integral formulas, but these are presently not amenable for asymptotic analysis. Using...

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Section: Introductionmentioning

confidence: 99%

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Barraquand¹,

Borodin²,

Corwin³

et al. 2018

Duke Math. J.

Self Cite

100

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“…Ensembles of non-intersecting random lines, both in the discrete and continuous setups, as well as their scaling limits as the linear size of the system grows, play a significant role in the probabilistic analysis of various problems in random matrices, interacting particle systems and effective interface models; see e.g. [10,17,22,8,2,1,18,24,7,9] and references therein.…”

mentioning

confidence: 99%

Confinement of Brownian polymers under geometric area tilts

Caputo¹,

Ioffe²,

Wachtel³

2019

Electron. J. Probab.

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“…In this section we introduce the notion of a Bernoulli line ensemble and the Schur Gibbs property. Our discussion will parallel that of [6,Section 3.1], which in turn goes back to [8,Section 2.1].…”

Section: Bernoulli Gibbsian Line Ensemblesmentioning

confidence: 79%

“…For line ensembles whose underlying path structure is Brownian it appeared in the seminal work of [7] and more recently in [4,5]. For discrete Gibbsian line ensembles (more general than the one studied in this paper) it appeared in [6] and for line ensembles related to the inverse gamma directed polymer in [32]. Theorem 1.1 indicates that in order to ensure the existence of subsequential limits for L N as in (1.2) it suffices to ensure tightness of the one-point marginals of the top curves L N 1 in a sufficiently uniform sense.…”

Section: Resultsmentioning

confidence: 97%

Tightness of Bernoulli Gibbsian line ensembles

Dimitrov

Fang

Fesser

et al. 2021

Electron. J. Probab.

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A Bernoulli Gibbsian line ensemble L = (L1, . . . , LN ) is the law of the trajectories of N −1 independent Bernoulli random walkers L1, . . . , LN−1 with possibly random initial and terminal locations that are conditioned to never cross each other or a given random up-right path LN (i.e. L1 ≥ • • • ≥ LN ). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles L N = (L N 1 , . . . , L N N ) when the number of walkers tends to infinity. We prove that if one has mild but uniform control of the one-point marginals of the lowest-indexed (or top) curves L N 1 then the sequence L N is tight in the space of line ensembles. Furthermore, we show that if the top curves L N 1 converge in the finite dimensional sense to the parabolic Airy2 process then L N converge to the parabolic Airy line ensemble.

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Transversal Fluctuations of the ASEP, Stochastic Six Vertex Model, and Hall-Littlewood Gibbsian Line Ensembles

Cited by 38 publications

References 88 publications

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Confinement of Brownian polymers under geometric area tilts

Tightness of Bernoulli Gibbsian line ensembles

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