Exact solution of the master equation for the asymmetric exclusion process

Schütz, Gunter M.

doi:10.1007/bf02508478

Cited by 241 publications

(352 citation statements)

References 32 publications

Supporting

Mentioning

337

Contrasting

Unclassified

Order By: Relevance

“…The result is obtained in greater generality with jump rates L and R being both time and particle-dependent (Proposition 3.1). The first part (the determinantal formula, see Proposition 2.1) is a generalization of similar results due to [2,19,20] obtained by the Bethe Ansatz techniques. Also, a closely related result have been obtained very recently in [10] using a version of the Robinson-Schensted-Knuth correspondence.…”

Section: Introductionsupporting

confidence: 54%

Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP

Borodin

Ferrari

Sasamoto

2008

Commun. Math. Phys.

152

View full text Add to dashboard Cite

We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way.We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy 1 and Airy 2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

show abstract

Section: Introductionsupporting

confidence: 54%

Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP

Borodin

Ferrari

Sasamoto

2008

Commun. Math. Phys.

152

View full text Add to dashboard Cite

show abstract

“…For p ≤ 0 and n ≥ 0: F p (n; t) can be written as a (finite or infinite) sum also in other regions of the (n, p) parameter space (see [17]) but those formulas are not used here.…”

Section: Discussionmentioning

confidence: 99%

“…Given the determinant, the proof that it is the solution of the master equation for the TASEP follows from standard relations for determinants, for details see [17]. In the next subsection we show how (3) can be derived directly from (6) without reference to combinatorial properties of the process.…”

Section: Bethe Ansatzmentioning

confidence: 98%

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

Rákos¹,

Schütz

2005

J Stat Phys

100

View full text Add to dashboard Cite

We calculate the time-evolution of a discrete-time fragmentation process in which clusters of particles break up and reassemble and move stochastically with size-dependent rates. In the continuous-time limit the process turns into the totally asymmetric simple exclusion process (only pieces of size 1 break off a given cluster). We express the exact solution of master equation for the process in terms of a determinant which can be derived using the Bethe ansatz. From this determinant we compute the distribution of the current across an arbitrary bond which after appropriate scaling is given by the distribution of the largest eigenvalue of the Gaussian unitary ensemble of random matrices. This result confirms universality of the scaling form of the current distribution in the KPZ universality class and suggests that there is a link between integrable particle systems and random matrix ensembles.

show abstract

“…Our approach is based on the determinantal formula for the Green function found by Schütz [13]. Suppose that N particles labelled 1, 2, · · · , N from the left start from y 1 , y 2 , .…”

mentioning

confidence: 99%

“…It is also remarked that F n (m, t) can be written in terms of the confluent hypergeometric function (or Laguerre function), though it is not explicitly stated in [13]. The Green function is already a nontrivial result.…”

mentioning

confidence: 99%

Asymmetric simple exclusion process and modified random matrix ensembles

Nagao

Sasamoto

2004

Nuclear Physics B

View full text Add to dashboard Cite

We study the fluctuation properties of the asymmetric simple exclusion process (ASEP) on an infinite one-dimensional lattice. When N particles are initially situated in the negative region with a uniform density ρ − = 1, Johansson showed the equivalence of the current fluctuation of ASEP and the largest eigenvalue distribution of random matrices. We extend Johansson's formula and derive modified ensembles of random matrices, corresponding to general ASEP initial conditions. Taking the scaling limit, we find that a phase change of the asymptotic current fluctuation occurs at a critical position.

show abstract

Exact solution of the master equation for the asymmetric exclusion process

Cited by 241 publications

References 32 publications

Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP

Large Time Asymptotics of Growth Models on Space-like Paths II: PNG and Parallel TASEP

Current Distribution and Random Matrix Ensembles for an Integrable Asymmetric Fragmentation Process

Asymmetric simple exclusion process and modified random matrix ensembles

Contact Info

Product

Resources

About