Stationary distributions of the multi-type ASEP

Martin, James B.

doi:10.1214/20-ejp421

Cited by 20 publications

(17 citation statements)

References 40 publications

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“…As before, one would like to find an expression for each steady state probability as a manifestly positive sum over some set of combinatorial objects. One may give such a formula in terms of Ferrari-Martin's multiline queues shown in Figure 8, see [Mar20,CMW22].…”

Section: The (Multispecies) Asep On a Ringmentioning

confidence: 99%

The combinatorics of hopping particles and positivity in Markov chains

Williams¹

2022

Preprint

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The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. In this article I give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics (tableaux and multiline queues) and special functions (Askey-Wilson polynomials, Macdonald polynomials, and Schubert polynomials). I also make some general observations about positivity in Markov chains.

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Section: The (Multispecies) Asep On a Ringmentioning

confidence: 99%

The combinatorics of hopping particles and positivity in Markov chains

Williams¹

2022

Preprint

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“…Algebraic properties have been used to generate alternative matrix representations of the Ferrari-Martin construction [72,73]. Very recently [74] a generalised queueing construction, in which each potential service is unused with probability q k when the queue-length is k was shown to give a recursive construction of the stationary distribution of multispecies process with asymmetry parameter q.…”

Section: Multispecies Generalisationmentioning

confidence: 99%

Combinatorial mappings of exclusion processes

Wood¹,

Blythe²,

Evans³

2020

J. Phys. A: Math. Theor.

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We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties.Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state-such as the entropy-can be interpreted in terms of how this larger state space has been partitioned.These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.

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“…Reversible measures of n-species ASEP1,q on a finite interval were found in [BS18,Kua17]. (See also [Mar18] for stationary measures on an infinite line or on a ring). The expression is…”

Section: Asepmentioning

confidence: 99%

Stochastic Fusion of Interacting Particle Systems and Duality Functions

Kuan

2019

Preprint

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We introduce a new method, which we call stochastic fusion, which takes an exclusion process and constructs an interacting particle systems in which more than one particle may occupy a lattice site. The construction only requires the existence of stationary measures of the original exclusion process on a finite lattice. If the original exclusion process satisfies Markov duality on a finite lattice, then the construction produces Markov duality functions (for some initial conditions) for the fused exclusion process. The stochastic fusion construction is based off of the Rogers-Pitman intertwining.In particular, we have results for three types of models: 1. For symmetric exclusion processes, the fused process and duality functions are inhomogeneous generalizations of those in [GKRV09]. The construction also allows a general class of open boundary conditions: as an application of the duality, we find the hydrodynamic limit and stationary measures of the generalized symmetric simple exclusion process SSEP(m/2) on Z+ for open boundary conditions.2. For the asymmetric simple exclusion process, the fused process and duality functions are inhomogeneous generalizations of those found in [CGRS16] for the ASEP(q, j). As a by-product of the construction, we show that the multi-species ASEP(q, j) preserves q-exchangeable measures, and use this to find new duality functions for the ASEP, ASEP(q, j) and q-Boson.3. For dynamic models, we fuse the dynamic ASEP from [Bor17], and produce a dynamic and inhomogeneous version of ASEP(q, j). We also apply stochastic fusion to IRF models and compare them to previously found models.We include an appendix, co-authored with Amol Aggarwal, which explains the algebraic roots of the model for the interested reader, as well as the relationship between fusion for interacting particle systems and fusion for stochastic vertex models. We compare the fused dynamical vertex weights with previously found dynamical vertex weights.

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Stationary distributions of the multi-type ASEP

Cited by 20 publications

References 40 publications

The combinatorics of hopping particles and positivity in Markov chains

The combinatorics of hopping particles and positivity in Markov chains

Combinatorial mappings of exclusion processes

Stochastic Fusion of Interacting Particle Systems and Duality Functions

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