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arXiv · 1904.09536

On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case

Wlodzimierz Bryc·Marcin Swieca
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Abstract

We study a substitute for the matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the ``singular case'' $\alpha\beta=q^N\gamma\delta$, 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, $\varphi_1$, is defined on the entire algebra, and determines stationary probabilities for large systems on $L\geq N+1$ sites. The other functional, $\varphi_0$, 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 $\varphi_0$ vanishes on non-constant Askey-Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey-Wilson polynomials.

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Wlodzimierz Bryc, Marcin Swieca. 2019-04-21. On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case. https://doi.org/10.1007/s10955-019-02367-4

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