1,248 research outputs found
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
Fluctuations of a one-dimensional polynuclear growth model in a half space
We consider the multi-point equal time height fluctuations of a
one-dimensional polynuclear growth model in a half space. For special values of
the nucleation rate at the origin, the multi-layer version of the model is
reduced to a determinantal process, for which the asymptotics can be analyzed.
In the scaling limit, the fluctuations near the origin are shown to be
equivalent to those of the largest eigenvalue of the orthogonal/symplectic to
unitary transition ensemble at soft edge in random matrix theory.Comment: 51 pages, 8 figure
Annihilation-diffusion processes: an exactly solvable model
A family of diffusion-annihilation processes is introduced, which is exactly
solvable. This family contains parameters that control the diffusion- and
annihilation- rates. The solution is based on the Bethe ansatz and using
special boundary conditions to represent the reaction. The processes are
investigated, both on the lattice and on the continuum. Special cases of this
family of processes are the simple exclusion process and the drop-push model.Comment: 11 pages, LaTe
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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