Recursive structures in the multispecies TASEP

Abstract

We consider a multi-species generalization of the totally asymmetric simple\nexclusion process (TASEP) with the simple hopping rule: for x and yth-class\nparticles (x<y), the transition xy -> yx occurs with a rate independent from\nthe values x and y. P. A. Ferrari and J. Martin (2007) obtained the stationary\nstate of this model thanks to a combinatorial algorithm, which was subsequently\ninterpreted as a matrix product representation by Evans et al. (2009). This\n`matrix ansatz' shows that the stationary state of the multi-species TASEP with\nN classes of particles (N-TASEP) can be constructed algebraically by the action\nof an operator on the (N-1)-TASEP stationary state. Besides, Arita et al.\n(2009) analyzed the spectral structure of the Markov matrix: they showed that\nthe set of eigenvalues of the N-TASEP contains those of the (N-1)-TASEP and\nthat the various spectral inclusions can be encoded in a hierarchical\nset-theoretic structure known as the Hasse diagram. Inspired by these works, we\ndefine nontrivial operators that allow us to construct eigenvectors of the\nN-TASEP by lifting the eigenvectors of the (N-1)-TASEP. This goal is achieved\nby generalizing the matrix product representation and the Ferrari-Martin\nalgorithm. In particular, we show that the matrix ansatz is not only a\nconvenient tool to write the stationary state but in fact intertwines Markov\nmatrices of different values of N.\n