Tracy–Widom asymptotics for $q$-TASEP

Ferrari, Patrik L.; Vető, Bálint

doi:10.1214/14-aihp614

Cited by 50 publications

(80 citation statements)

References 25 publications

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“…Proof of Theorem 5.2. The proof uses Laplace's method and follows the style of [FV13] (similar proofs can be found in [Bar15] for q-TASEP with slow particles, in [BCF14] for the semidiscrete directed polymer, and in [Vet15] for the q-Hahn TASEP). Fix q ∈ (0, 1), ν = q, R > L 0 with R + L = 1 and θ > 0 such that κ(θ) 0.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…. in the definition of the contour D, as in [FV13]. The rest of the asymptotic analysis would remain almost unchanged provided one is able to prove that for any W ∈ C α and k 1 such that Proof.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…This is stated as Theorem 5.2 and Theorem 5.4 and proves Theorem 1.1 and Theorem 1.3 (see Section 5). The asymptotic analysis here is an instance of the Laplace (or saddle-point) method which has been implemented in similar contexts in [BCF14,FV13,Bar15,Vet15].…”

Section: Introductionmentioning

confidence: 99%

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The $q$-Hahn asymmetric exclusion process

Barraquand¹,

Corwin²

2016

Ann. Appl. Probab.

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Abstract. We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the q-Hahn TASEP and the q-Hahn Boson (zero-range) process introduced in [Pov13] and further studied in [Cor14], by allowing jumps in both directions. Owing to a Markov duality, we prove moment formulas for the locations of particles in the exclusion process. This leads to a Fredholm determinant formula that characterizes the distribution of the location of any particle. We show that the model-dependent constants that arise in the limit theorems predicted by the KPZ scaling theory are recovered by a steepest descent analysis of the Fredholm determinant. For some choice of the parameters, our model specializes to the multi-particle-asymmetric diffusion model introduced in [SW98]. In that case, we make a precise asymptotic analysis that confirms KPZ universality predictions. Surprisingly, we also prove that in the partially asymmetric case, the location of the first particle also enjoys cube-root fluctuations which follow Tracy-Widom GUE statistics.

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Section: Asymptotic Analysismentioning

confidence: 99%

Section: Asymptotic Analysismentioning

confidence: 99%

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The $q$-Hahn asymmetric exclusion process

Barraquand¹,

Corwin²

2016

Ann. Appl. Probab.

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“…In [15] it is proved that the totally asymmetric zero range process (2.8) is in the KPZ universality class. It is an interesting open problem to prove or disprove that the same conclusion holds true for (2.7) [26].…”

Section: Limiting Casesmentioning

confidence: 99%

Asymmetric Stochastic Transport Models with $${\mathscr {U}}_q(\mathfrak {su}(1,1))$$ U q ( su ( 1 , 1 ) ) Symmetry

et al. 2016

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By using the algebraic construction outlined in Carinci et al. (arXiv:1407.3367, 2014, we introduce several Markov processes related to the U q (su(1, 1)) quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the and which turns out to have the Symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

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“…> r Ã for˛< 0 still holds with exp replaced by exp . Indeed, this is the case with e (this fact has been used successfully in recent years to derive the asymptotics of some related models, although not for the half-flat or flat initial conditions; see, for instance, [5,6,19]). For x > .1 / 1 it is the case that exp .´/ looks quite like exp.´/, and in fact it converges to it uniformly on OE a; 1/ for any a > 0 as % 1.…”

Section: Fredholm Pfaffian Formulamentioning

confidence: 99%