105 research outputs found
Combinatorics of the asymmetric exclusion process on a semi-infinite lattice
We study two versions of the asymmetric exclusion process (ASEP) -- an ASEP
on a semi-infinite lattice with an open left boundary, and an ASEP on a finite
lattice with open left and right boundaries -- and we demonstrate a surprising
relationship between their stationary measures. The semi-infinite ASEP was
first studied by Liggett and then Grosskinsky, while the finite ASEP had been
introduced earlier by Spitzer and Macdonald-Gibbs-Pipkin. We show that the
finite correlation functions involving the first L sites for the stationary
measures on the semi-infinite ASEP can be obtained as a nonphysical
specialization of the stationary distribution of an ASEP on a finite
one-dimensional lattice with L sites. Namely, if the output and input rates of
particles at the right boundary of the finite ASEP are beta and delta,
respectively, and we set delta=-beta, then this specialization corresponds to
sending the right boundary of the lattice to infinity. Combining this
observation with work of the second author and Corteel, we obtain a
combinatorial formula for finite correlation functions of the ASEP on a
semi-infinite lattice
A systematic way to find and construct exact finite dimensional matrix product stationary states
We explain how to construct matrix product stationary states which are
composed of finite-dimensional matrices. Our construction explained in this
article was first presented in a part of [Hieida and Sasamoto:J. Phys. A: Math.
Gen. 37 (2004) 9873] for general models. In this article, we give more details
on the treatment than in the above-mentioned reference, for one-dimensional
asymmetric simple exclusion process(ASEP).Comment: This article will appear in the proceedings of "Workshop on Matrix
Product State Formulation and Density Matrix Renormalization Group
Simulations (MPS&DMRG)" to be published by World Scientifi
From duality to determinants for q-TASEP and ASEP
We prove duality relations for two interacting particle systems: the
-deformed totally asymmetric simple exclusion process (-TASEP) and the
asymmetric simple exclusion process (ASEP). Expectations of the duality
functionals correspond to certain joint moments of particle locations or
integrated currents, respectively. Duality implies that they solve systems of
ODEs. These systems are integrable and for particular step and half-stationary
initial data we use a nested contour integral ansatz to provide explicit
formulas for the systems' solutions, and hence also the moments. We form
Laplace transform-like generating functions of these moments and via residue
calculus we compute two different types of Fredholm determinant formulas for
such generating functions. For ASEP, the first type of formula is new and
readily lends itself to asymptotic analysis (as necessary to reprove GUE
Tracy--Widom distribution fluctuations for ASEP), while the second type of
formula is recognizable as closely related to Tracy and Widom's ASEP formula
[Comm. Math. Phys. 279 (2008) 815--844, J. Stat. Phys. 132 (2008) 291--300,
Comm. Math. Phys. 290 (2009) 129--154, J. Stat. Phys. 140 (2010) 619--634]. For
-TASEP, both formulas coincide with those computed via Borodin and Corwin's
Macdonald processes [Probab. Theory Related Fields (2014) 158 225--400]. Both
-TASEP and ASEP have limit transitions to the free energy of the continuum
directed polymer, the logarithm of the solution of the stochastic heat equation
or the Hopf--Cole solution to the Kardar--Parisi--Zhang equation. Thus,
-TASEP and ASEP are integrable discretizations of these continuum objects;
the systems of ODEs associated to their dualities are deformed discrete quantum
delta Bose gases; and the procedure through which we pass from expectations of
their duality functionals to characterizing generating functions is a rigorous
version of the replica trick in physics.Comment: Published in at http://dx.doi.org/10.1214/13-AOP868 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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