5 research outputs found
Matrix product solution to an inhomogeneous multi-species TASEP
We study a multi-species exclusion process with inhomogeneous hopping rates.
This model is equivalent to a Markov chain on the symmetric group that
corresponds to a random walk in the affine braid arrangement. We find a matrix
product representation for the stationary state of this model. We also show
that it is equivalent to a graphical construction proposed by Ayyer and
Linusson, which generalizes Ferrari and Martin's construction
Remarks on the multi-species exclusion process with reflective boundaries
We investigate one of the simplest multi-species generalizations of the one
dimensional exclusion process with reflective boundaries. The Markov matrix
governing the dynamics of the system splits into blocks (sectors) specified by
the number of particles of each kind. We find matrices connecting the blocks in
a matrix product form. The procedure (generalized matrix ansatz) to verify that
a matrix intertwines blocks of the Markov matrix was introduced in the periodic
boundary condition, which starts with a local relation [Arita et al, J. Phys. A
44, 335004 (2011)]. The solution to this relation for the reflective boundary
condition is much simpler than that for the periodic boundary condition
Transfer matrices for the totally asymmetric exclusion process
We consider the totally asymmetric simple exclusion process (TASEP) on a
finite lattice with open boundaries. We show, using the recursive structure of
the Markov matrix that encodes the dynamics, that there exist two transfer
matrices and that intertwine the Markov
matrices of consecutive system sizes:
. This semi-conjugation property of
the dynamics provides an algebraic counterpart for the matrix-product
representation of the steady state of the process.Comment: 7 page
Generalized matrix Ansatz in the multispecies exclusion process - partially asymmetric case
We investigate one of the simplest multispecies generalization of the
asymmetric simple exclusion process on a ring. This process has a rich
combinatorial spectral structure and a matrix product form for the stationary
state. In the totally asymmetric case operators that conjugate the dynamics of
systems with different numbers of species were obtained by the authors and
reported recently. The existence of such nontrivial operators was reformulated
as a representation problem for a specific quadratic algebra (generalized
matrix Ansatz). In the present work, we construct the family of representations
explicitly for the partially asymmetric case. This solution cannot be obtained
by a simple deformation of the totally asymmetric case