The Current Distribution of the Multiparticle Hopping Asymmetric Diffusion Model

Lee, Eunghyun

doi:10.1007/s10955-012-0582-y

Cited by 8 publications

(18 citation statements)

References 25 publications

(87 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the work of Schütz [24], the coordinate Bethe ansatz has been successfully used to find the transition probabilities of some stochastic particle models with countable state spaces [7,9,14,15,20,26,29]. The time evolution of a stochastic particle model is governed by the master equation, which is a system of first-order differential equations, and for each state X the corresponding differential equation describes the time-evolution of the probability P (X; t) that the system is at a state X at time t. In a matrix form, the master equation is given by d dt P (X; t) = HP (X; t)…”

Section: Bethe Ansatz Applicabilitymentioning

confidence: 99%

See 1 more Smart Citation

The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process

Korhonen

Lee

2014

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We treat the N -particle ZRP whose jumping rates satisfy a certain condition. This condition is required to use the Bethe ansatz and the resulting model is the q-boson model that appeared in [J. Phys. A, 31 6057-6071 (1998)] by Sasamoto and Wadati or the q-TAZRP in Macdonald processes by Borodin and Corwin. We find the explicit formula of the transition probability of the q-TAZRP via the Bethe ansatz. By using the transition probability we find the probability distribution of the left-most particle's position at time t. To find the probability for the left-most particle's position we find a new identity corresponding to Tracy and Widom's identity for the ASEP in [Commun. Math. Phys., 279 815-844 (2008)]. For the initial state that all particles occupy a single site, the probability distribution of the left-most particle's position at time t is represented by the contour integral of a determinant.

show abstract

Section: Bethe Ansatz Applicabilitymentioning

confidence: 99%

“…These weights W X are known to give the stationary measure of the general ZRP up to a constant [12]. Compared to other models [15,26], a novel point of proving the transition probability of the q-TAZRP is that it needs an identity related to a [k] q . (See Lemma 2.4.)…”

Section: Bethe Ansatz Applicabilitymentioning

confidence: 99%

The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process

Korhonen

Lee

2014

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The rates are now given by R/[j] q −1 and L/[j] q . The zerorange model with N particles is exactly the "multi-particle asymmetric diffusion model" introduced by Sasamoto and Wadati 2 in [SW98] and further studied by Lee [Lee12] (see also [AKK99,AKK98]). For the corresponding exclusion process, we prove (by an asymptotic analysis of the Fredholm determinant in (35)) in Section 5 that the rescaled positions of particles converge to the TracyWidom GUE distribution (Theorem 5.2).…”

mentioning

confidence: 99%

The $q$-Hahn asymmetric exclusion process

Barraquand¹,

Corwin²

2016

Ann. Appl. Probab.

View full text Add to dashboard Cite

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.

show abstract

“…(ii) We prove by induction on N . When N = 2, the statement is obvious from (5). For the induction step, let us write…”

Section: Explicit Formulas Of Pmentioning

confidence: 99%

Some conditional probabilities in the TASEP with second class particles

Lee

2017

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

In this paper we consider the TASEP with second class particles which consists of k first class particles and N − k second class particles. We assume that all first class particles are initially located to the left of the leftmost second class particle. Under this assumption, we find the probability that the first class particles are at x, x + 1, · · · , x + k − 1 and these positions are still to the left of the leftmost second class particle at time t. If we additionally assume that the initial positions of the particles are 1, · · · , N , that is, step initial condition, then the formula of the probability does not depend on k and is very similar to a formula for the TASEP (without second class particles) with step initial condition.

show abstract

The Current Distribution of the Multiparticle Hopping Asymmetric Diffusion Model

Cited by 8 publications

References 25 publications

The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process

The transition probability and the probability for the left-most particle's position of the q-totally asymmetric zero range process

The $q$-Hahn asymmetric exclusion process

Some conditional probabilities in the TASEP with second class particles

Contact Info

Product

Resources

About