Mapping TASEP back in time

Abstract

We obtain a new relation between the distributions $$\\upmu _t$$\n \n μ\n t\n \n at different times $$t\\ge 0$$\n \n t\n ≥\n 0\n \n of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\\upmu _t$$\n \n μ\n t\n \n backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\\upmu _t$$\n \n μ\n t\n \n which in turn brings new identities for expectations with respect to $$\\upmu _t$$\n \n μ\n t\n \n . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.