Exact Formulas of the Transition Probabilities of the Multi-Species Asymmetric Simple Exclusion Process

Abstract

We find the formulas of the transition probabilities of the N-particle multi-species asymmetric simple exclusion processes (ASEP), and show that the transition probabilities are written as a determinant when the order of particles in the final state is the same as the order of particles in the initial state.

Integrability of the Multi-Species TASEP with Species-Dependent Rates

“…Remark 4. Theorem 1 partially extends (2.12) in [4]. In other words, (6) with b l = 1 is equal to (2.12) in [4] with p = 1.…”

“…Extending the technique used in [4], we will find the formulas of A σ in (19), which satisfy (20). Define an N × N diagonal matrix,…”

“…The transition probabilities and some determinantal formulas for the multi-species ASEP and its special cases were found in [1][2][3][4][5]. For certain special initial conditions with a single second class particle, some distributions and their asymptotics were studied in [6,7].…”

“…In this paper, we consider the multi-species totally asymmetric simple exclusion processes with N particles in which particles move to the right and each species l is allowed to have its own rate b l . Following the notations used in [4]…”

“…and that these labels are listed lexicographically, unless stated otherwise. The main result of this paper is that the multi-species TASEP with species-dependent rates is an integrable model, and we provide a formula analogous to (2.12) in [4] using the Bethe ansatz method.…”