On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866915042668052480 |
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| author | Bryc, Wlodzimierz Swieca, Marcin |
| author_facet | Bryc, Wlodzimierz Swieca, Marcin |
| contents | We study a substitute for the matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the ``singular case'' $αβ=q^Nγδ$, when the standard form of the matrix product ansatz of Derrida, Evans, Hakim and Pasquier does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $φ_1$, is defined on the entire algebra, and determines stationary probabilities for large systems on $L\geq N+1$ sites. The other functional, $φ_0$, is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $L< N+1$ sites. Functional $φ_0$ vanishes on non-constant Askey-Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey-Wilson polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_09536 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case Bryc, Wlodzimierz Swieca, Marcin Mathematical Physics Probability 82C22, 60K35, 33D45, 33D15 We study a substitute for the matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the ``singular case'' $αβ=q^Nγδ$, when the standard form of the matrix product ansatz of Derrida, Evans, Hakim and Pasquier does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $φ_1$, is defined on the entire algebra, and determines stationary probabilities for large systems on $L\geq N+1$ sites. The other functional, $φ_0$, is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $L< N+1$ sites. Functional $φ_0$ vanishes on non-constant Askey-Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey-Wilson polynomials. |
| title | On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case |
| topic | Mathematical Physics Probability 82C22, 60K35, 33D45, 33D15 |
| url | https://arxiv.org/abs/1904.09536 |