Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two

Abstract

We study two generalizations of the asymmetric simple exclusion process with two types of particles. Particles of type 1 can jump over particles of type 2, while particles of type 2 can only influence the jump rates of particles of type 1. We prove that the processes are self-dual and explicitly write the duality function. As an application, an expression for the r-th moment of the exponentiated current is written in terms of r-particle evolution.The construction and proofs of duality are accomplished using symmetry of the quantum groups Uq(gl 3 ) and Uq(sp 4 ), with each node in the Dynkin diagram corresponding to a particle type, and the number of edges corresponding to the jump rates.

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

“…The method introduced in this paper is a fairly general way to construct particle systems with dualities from quantum algebras. Recently it has been applied to algebras with higher rank, such as U q (gl(3)) [2,15] or U q (sp (4)) [15], yielding two-component asymmetric exclusion process with multiple conserved species of particles. In [7] we also applied the method to non-compact algebras such as U q (su (1, 1)), finding new diffusion processes of heat transport with duality.…”

Asymmetric Stochastic Transport Models with $${\mathscr {U}}_q(\mathfrak {su}(1,1))$$ U q ( su ( 1 , 1 ) ) Symmetry

“…This depends on the chosen representation of the generators of the algebra, and the choice of the co-product. Recently the construction has been applied to algebras with higher rank, such as U q (gl (3)) [5,23] or U q (sp (4)) [23], yielding two-component asymmetric exclusion process with multiple conserved species of particles.…”

Integrable Stochastic Dualities and the Deformed Knizhnik–Zamolodchikov Equation

“…where the sum on the right hand side is over compositions ν which are strictly less than µ with respect to the ordering ≺, for some coefficients e µ,ν (q, t; w) which are polynomial in q. The polynomiality of the coefficients is ensured by (33) and (34). This equation therefore extends to specializations q = t −m .…”

“…For other recent progress related to higher-rank duality functions, making use of quantum group symmetries, we refer the reader to [10,11,33,34].…”

“…In many cases treated in the literature, duality functionals have been constructed in a more or less ad hoc fashion and only a few attempts have been made to systematically derive dualities in integrable stochastic models using quantum group symmetries [1,10,11,33,34,45]. In this paper we propose a new approach for methodically constructing integrable dualities by exploiting the algebraic structure provided by the t-deformed Knizhnik-Zamolodchikov (KZ) equations [20,31], which are consistency equations expressed in terms of the R-matrix of a quantum group, or alternatively, in terms of the Hecke algebra.…”