Generalized matrix Ansatz in the multispecies exclusion process - partially asymmetric case

TL;DR

This paper develops explicit representations of a generalized matrix Ansatz for the multispecies partially asymmetric exclusion process, revealing its complex spectral structure and extending previous totally asymmetric results.

Contribution

It constructs explicit representations of the quadratic algebra for the partially asymmetric case, advancing understanding of multispecies exclusion processes.

Findings

Explicit representations for the quadratic algebra are constructed.

The solution extends beyond simple deformations of the totally asymmetric case.

The process exhibits a rich combinatorial spectral structure.

Abstract

We investigate one of the simplest multispecies generalization of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and reported recently. The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In the present work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case.