Abstract
We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP(q,θ), asymmetric exclusion process, with a repulsive interaction, allowing up to θ∈N particles in each site, and the ASIP(q,θ), θ∈R+, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate q−analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP(q,θ).
Highlights
In this paper we study two models of interacting particle systems with asymmetric jump rates exhibiting a self-duality property
We show that well-known families of q−hypergeometric orthogonal polynomials, the q−Krawtchouk polynomials and q−Meixner polynomials, occur as 1-site duality functions for corresponding stochastic models
We conjecture that the orthogonal self-duality polynomials complete the picture of nested-product duality functions for ASEP(q, θ) and ASIP(q, θ), summing up to the classical or triangular ones, already known for these processes from [10, 11]
Summary
In this paper we study two models of interacting particle systems with asymmetric jump rates exhibiting a self-duality property. The asymmetric processes ASEP(α, θ) and ASIP(α, θ) were introduced in [10, 11] where self-duality properties are proved These are due to the algebraic structure of the generator that is constructed passing through the (α + 1)-dimensional representation of a quantum Hamiltonian with Uq(sl2) invariance. We conjecture that the orthogonal self-duality polynomials complete the picture of nested-product duality functions for ASEP(q, θ) and ASIP(q, θ), summing up to the classical or triangular ones, already known for these processes from [10, 11]. The strategy followed in [10, 11] to construct the so-called classical dualities relies on an algebraic approach based on the study of the symmetries of the generator. In the last part of the paper we will follow this algebraic approach to write (in terms of elements of Uq(su(2))) the symmetries of the generator yielding the q−polynomial dualities obtained via the scalar-product method
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