Orthogonal Dualities of Markov Processes and Unitary Symmetries

Abstract

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.

q−Orthogonal dualities for asymmetric particle systems

“…The families of orthogonal polynomials dualities for these processes were found for the first time in [17] by explicit computations relying on the hypergeometric structure of the polynomials. The same dualities were found in [37] via generating functions, while an algebraic approach is followed in [26] and [8], relying, respectively, on the use of unitary intertwiners and unitary symmetries. In [8] yet another approach to (orthogonal) duality is described, based on scalar products of classical duality functions.…”

“…The same dualities were found in [37] via generating functions, while an algebraic approach is followed in [26] and [8], relying, respectively, on the use of unitary intertwiners and unitary symmetries. In [8] yet another approach to (orthogonal) duality is described, based on scalar products of classical duality functions.…”

“…We refer to duality functions of this type also as classical duality functions. Orthogonal polynomial duality functions are, on the other hand, a very recent discovery and were found, up to date, only for symmetric processes (SEP(α), SIP(α) and IRW) in a series of papers [8,17,18,37]. The duality functions for these processes are products of univariate orthogonal polynomials, where the orthogonality is with respect to the reversible measures of the process itself.…”

“…In this paper we use this latter approach to extend the results obtained in [8] to the case of asymmetric processes. Differently from [8], the q-orthogonal duality functions for asymmetric processes are not yet known in the literature.…”

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

“…Finally, we may also expect that orthogonal duality functions arise when considering unitary equivalent representations of the sl(2) algebra, as it happens in the case of the spin chain related to the KMP model [50][51][52].…”