Parameter Permutation Symmetry in Particle Systems and Random Polymers
Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with the notable exception of ASEP) remain integrable when we equip each particle ᵢ with its own jump rate parameter νᵢ. It is a consequence...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211167 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Parameter Permutation Symmetry in Particle Systems and Random Polymers. Leonid Petrov. SIGMA 17 (2021), 021, 34 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862737406759796736 |
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| author | Petrov, Leonid |
| author_facet | Petrov, Leonid |
| citation_txt | Parameter Permutation Symmetry in Particle Systems and Random Polymers. Leonid Petrov. SIGMA 17 (2021), 021, 34 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with the notable exception of ASEP) remain integrable when we equip each particle ᵢ with its own jump rate parameter νᵢ. It is a consequence of integrability that the distribution of each particle ₙ() in a system started from the step initial configuration depends on the parameters ⱼ, j ≤ , symmetrically. A transposition ₙ ↔ ₙ₊₁ of the parameters thus affects only the distribution of ₙ(). For q-Hahn TASEP and its degenerations (q-TASEP and directed beta polymer), we realize the transposition ₙ ↔ ₙ₊₁ as an explicit Markov swap operator acting on the single particle ₙ(). For a beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process Q⁽ᵗ⁾ preserving the time t distribution of the -TASEP (with step initial configuration, where ∈ ℝ˃₀ is fixed). The dual system is a certain transient modification of the stochastic q-Boson system. We identify asymptotic survival probabilities of this transient process with q-moments of the -TASEP, and use this to show the convergence of the process Q⁽ᵗ⁾ with arbitrary initial data to its stationary distribution. Setting = 0, we recover the results about the usual TASEP established recently in [arXiv:1907.09155] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.
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| first_indexed | 2026-04-17T16:53:43Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211167 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T16:53:43Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Petrov, Leonid 2025-12-25T13:20:22Z 2021 Parameter Permutation Symmetry in Particle Systems and Random Polymers. Leonid Petrov. SIGMA 17 (2021), 021, 34 pages 1815-0659 2020 Mathematics Subject Classification: 82C22; 60C05; 60J27 arXiv:1912.06067 https://nasplib.isofts.kiev.ua/handle/123456789/211167 https://doi.org/10.3842/SIGMA.2021.021 Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with the notable exception of ASEP) remain integrable when we equip each particle ᵢ with its own jump rate parameter νᵢ. It is a consequence of integrability that the distribution of each particle ₙ() in a system started from the step initial configuration depends on the parameters ⱼ, j ≤ , symmetrically. A transposition ₙ ↔ ₙ₊₁ of the parameters thus affects only the distribution of ₙ(). For q-Hahn TASEP and its degenerations (q-TASEP and directed beta polymer), we realize the transposition ₙ ↔ ₙ₊₁ as an explicit Markov swap operator acting on the single particle ₙ(). For a beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process Q⁽ᵗ⁾ preserving the time t distribution of the -TASEP (with step initial configuration, where ∈ ℝ˃₀ is fixed). The dual system is a certain transient modification of the stochastic q-Boson system. We identify asymptotic survival probabilities of this transient process with q-moments of the -TASEP, and use this to show the convergence of the process Q⁽ᵗ⁾ with arbitrary initial data to its stationary distribution. Setting = 0, we recover the results about the usual TASEP established recently in [arXiv:1907.09155] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions. I am grateful to Vadim Gorin for helpful discussions, and to Matteo Mucciconi and Axel Saenz for remarks on the first version of the manuscript. I am grateful to the organizers of the workshop, Dimers, Ising Model, and their Interactions, and the support of the Ban International Research Station, where a part of this work was done. The work was partially supported by the NSF grant DMS-1664617. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Parameter Permutation Symmetry in Particle Systems and Random Polymers Article published earlier |
| spellingShingle | Parameter Permutation Symmetry in Particle Systems and Random Polymers Petrov, Leonid |
| title | Parameter Permutation Symmetry in Particle Systems and Random Polymers |
| title_full | Parameter Permutation Symmetry in Particle Systems and Random Polymers |
| title_fullStr | Parameter Permutation Symmetry in Particle Systems and Random Polymers |
| title_full_unstemmed | Parameter Permutation Symmetry in Particle Systems and Random Polymers |
| title_short | Parameter Permutation Symmetry in Particle Systems and Random Polymers |
| title_sort | parameter permutation symmetry in particle systems and random polymers |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211167 |
| work_keys_str_mv | AT petrovleonid parameterpermutationsymmetryinparticlesystemsandrandompolymers |