Distributions of a particle’s position and their asymptotics in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml14" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>q</mml:mi></mml:math>-deformed totally asymmetric zero range process with site dependent jumping rates

Lee, Eunghyun; Wang, Dong

doi:10.1016/j.spa.2018.06.005

Cited by 10 publications

(10 citation statements)

References 23 publications

(68 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…strength of the asymmetry (see also [LW17] where the same distribution arises for the first particle's position in a certain zero-range process). In [BC16], another partially asymmetric process called the MADM exclusion process was studied.…”

Section: Introductionmentioning

confidence: 83%

Facilitated Exclusion Process

Baik

Barraquand

Corwin

et al. 2018

Computation and Combinatorics in Dynamics, Stochastics and Control

View full text Add to dashboard Cite

We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant which is exactly solvable through Pfaffian Schur processes.Our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE).We also discuss a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of the first particles when the rate of the first one varies. In terms of half-space last passage percolation, this corresponds to last passage times close to the boundary when the size of the boundary weights is simultaneously scaled close to the critical point.

show abstract

Section: Introductionmentioning

confidence: 83%

Facilitated Exclusion Process

Baik

Barraquand

Corwin

et al. 2018

Computation and Combinatorics in Dynamics, Stochastics and Control

View full text Add to dashboard Cite

show abstract

“…In this section we consider the q-Whittaker processes, which are obtained by a specialization of Macdonald processes [6,Section 3], and the Asymmetric Simple Exclusion Process (ASEP) as primary examples. We also consider the q-deformed Totally Asymmetric Simple Exclusion Process (q-TASEP), which is a continuous limit of the q-Whittaker process [6, Section 3.3], [7] and the q-deformed Totally Asymmetric Zero Range Process (q-TAZRP), which is the dual process of q-TASEP [20], [22].…”

Section: Relation To Interacting Particle Systemsmentioning

confidence: 99%

“…q-TAZRP The q-deformed Totally Asymmetric Zero Range process (q-TAZRP) is a dual process to the q-TASEP, see [20], [34] and [22] for a detailed definition of the model and the duality. The q-TAZRP was originally defined in [29] with the name q-boson process.…”

Section: Relation To Interacting Particle Systemsmentioning

confidence: 99%

“…The q-TAZRP was originally defined in [29] with the name q-boson process. Let the (inhomogeneous) q-TAZRP be defined as in [22], with the conductance of the sites given by a k = (1 − q) −1 b k (k ∈ Z), and assume that the particle number is N and the initial condition is the so-called step initial condition that x 1 (0) = · · · = x N (0) = 0. Then the distribution of the leftmost particle x N at time t > 0 is by [22, Proposition 2.1, Formulas (117) and (118)]…”

Section: Relation To Interacting Particle Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe–Neuberger–Shapiro random matrix model

Liechty¹,

Wang²

2020

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

We derive the local statistics of the canonical ensemble of free fermions in a quadratic potential well at finite temperature, as the particle number approaches infinity. This free fermion model is equivalent to a random matrix model proposed by Moshe, Neuberger and Shapiro. Limiting behaviors obtained before for the grand canonical ensemble are observed in the canonical ensemble: We have at the edge the phase transition from the Tracy-Widom distribution to the Gumbel distribution via the Kardar-Parisi-Zhang (KPZ) crossover distribution, and in the bulk the phase transition from the sine point process to the Poisson point process. A similarity between this model and a class of models in the KPZ universality class is explained. We also derive the multi-time correlation functions and the multi-time gap probability formulas for the free fermions along the imaginary time.

show abstract

“…Ferrari and Vető [16] and Barraquand [9] proved Tracy-Widom limits for current fluctuations in q-TASEP, while recently Imamura and Sasamoto [18] proved KPZ-related scaling limits of the tagged particle distribution. Finally, we mention Barraquand [9] and Lee, Wang [21] as investigations for the non-homogeneous version of the dynamics.…”

Section: Introductionmentioning

confidence: 99%

q-Zero Range has Random Walking Shocks

2019

View full text Add to dashboard Cite

but no other surprises in the zero range world. We check all nearest neighbour 1-dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature. We find the totally asymmetric version of the celebrated q-zero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates. We also examine the interaction of shocks, which appears somewhat more involved for q-zero range than in the already known cases.

show abstract

Distributions of a particle’s position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates

Cited by 10 publications

References 23 publications

Facilitated Exclusion Process

Facilitated Exclusion Process

Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe–Neuberger–Shapiro random matrix model

q-Zero Range has Random Walking Shocks

Contact Info

Product

Resources

About