Integrability of the Multi-Species Asymmetric Simple Exclusion Processes with Long-Range Jumps on Z

Let us consider a two-sided multi-species stochastic particle model with finitely many particles on Z, defined as follows. Suppose that each particle is labelled by a positive integer l, and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p, or the left direction with probability q=1−p. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle l′<l (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it jumps to the next site on the left only if that site is either empty or occupied by a particle l′<l, and in the latter case, particles l and l′ swap their positions. We show that this model is integrable, and provide the exact formula of the transition probability using the Bethe ansatz.