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

Abstract

We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ∈ (0, 1) and where at most 2 j ∈ N particles per site are allowed. The process is constructed from a (2 j + 1)-dimensional representation of a quantum Hamiltonian with U q (sl 2 ) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure. Mathematics Subject Classification

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

“…The main results are the following (a) Self-duality of ASIP (q, k). We proceed via the same construction as in [13] for the algebra U q (su (1,1)) to find the ASIP(q, k) which is the "correct" asymmetric analogue of the SIP(k). The parameter q tunes the asymmetry: q → 1 gives back the SIP(k).…”

“…and for this measure Proof The proof of item (2) is similar to the proof of Theorem 3.1, item (d) in [13] and we refer the reader to that paper for full details. The main idea is that if we have the rate c + (η i , η i+1 ) (resp.…”

“…Note that the rates are unbounded for y i+1 → ∞, nevertheless the process is well defined even in the infinite volume, as it belongs to the class considered in [4]. This is to be compared to the case of the deformed algebra U q (sl 2 ) [13] whose scaling limit with infinite spin is given by [6] …”

“…Recently Markov processes with the U q (su (2)) algebraic structure for higher spin value have been introduced and studied in [13]. This lead to a family of non-integrable asymmetric generalization of the partial exclusion process (see also [25]).…”

“…In [13] we constructed a generalization of the asymmetric exclusion process, allowing 2 j particles per site with self-duality properties reminiscent of the self-duality of the standard ASEP found initially by Schütz [29]. This construction followed a general scheme where one starts from the Casimir operator C of the quantum Lie algebra U q (su (2)), and applies a coproduct to obtain an Hamiltonian H i,i+1 working on the occupation number variables at sites i and i + 1.…”