265 research outputs found

### Simple Asymmetric Exclusion Model and Lattice Paths: Bijections and Involutions

We study the combinatorics of the change of basis of three representations of
the stationary state algebra of the two parameter simple asymmetric exclusion
process. Each of the representations considered correspond to a different set
of weighted lattice paths which, when summed over, give the stationary state
probability distribution. We show that all three sets of paths are
combinatorially related via sequences of bijections and sign reversing
involutions.Comment: 28 page

### Bi-orthogonal Polynomials and the Five parameter Asymmetric Simple Exclusion Process

We apply the bi-moment determinant method to compute a representation of the
matrix product algebra -- a quadratic algebra satisfied by the operators
$\mathbf{d}$ and $\mathbf{e}$ -- for the five parameter ($\alpha$, $\beta$,
$\gamma$, $\delta$ and $q$) Asymmetric Simple Exclusion Process. This method
requires an $LDU$ decomposition of the ``bi-moment matrix''. The decomposition
defines a new pair of basis vectors sets, the `boundary basis'. This basis is
defined by the action of polynomials $\{P_n\}$ and $\{Q_n\}$ on the quantum
oscillator basis (and its dual). Theses polynomials are orthogonal to
themselves (ie.\ each satisfy a three term recurrence relation) and are
orthogonal to each other (with respect to the same linear functional defining
the stationary state). Hence termed `bi-orthogonal'. With respect to the
boundary basis the bi-moment matrix is diagonal and the representation of the
operator $\mathbf{d}+\mathbf{e}$ is tri-diagonal. This tri-diagonal matrix
defines another set of orthogonal polynomials very closely related to the the
Askey-Wilson polynomials (they have the same moments)

### Chebyshev type lattice path weight polynomials by a constant term method

We prove a constant term theorem which is useful for finding weight
polynomials for Ballot/Motzkin paths in a strip with a fixed number of
arbitrary `decorated' weights as well as an arbitrary `background' weight. Our
CT theorem, like Viennot's lattice path theorem from which it is derived
primarily by a change of variable lemma, is expressed in terms of orthogonal
polynomials which in our applications of interest often turn out to be
non-classical. Hence we also present an efficient method for finding explicit
closed form polynomial expressions for these non-classical orthogonal
polynomials. Our method for finding the closed form polynomial expressions
relies on simple combinatorial manipulations of Viennot's diagrammatic
representation for orthogonal polynomials. In the course of the paper we also
provide a new proof of Viennot's original orthogonal polynomial lattice path
theorem. The new proof is of interest because it uses diagonalization of the
transfer matrix, but gets around difficulties that have arisen in past attempts
to use this approach. In particular we show how to sum over a set of implicitly
defined zeros of a given orthogonal polynomial, either by using properties of
residues or by using partial fractions. We conclude by applying the method to
two lattice path problems important in the study of polymer physics as models
of steric stabilization and sensitized flocculation.Comment: 27 pages, 14 figure

### A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model

The free fermion condition of the six-vertex model provides a 5 parameter
sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter
into the eigenfunctions of the transfer matrices of the model decouple, hence
allowing explicit solutions. Such conditions arose originally in early
field-theoretic S-matrix approaches. Here we provide a combinatorial
explanation for the condition in terms of a generalised Gessel-Viennot
involution. By doing so we extend the use of the Gessel-Viennot theorem,
originally devised for non-intersecting walks only, to a special weighted type
of \emph{intersecting} walk, and hence express the partition function of $N$
such walks starting and finishing at fixed endpoints in terms of the single
walk partition functions

### One-transit paths and steady-state of a non-equilibrium process in a discrete-time update

We have shown that the partition function of the Asymmetric Simple Exclusion
Process with open boundaries in a sublattice-parallel updating scheme is equal
to that of a two-dimensional one-transit walk model defined on a diagonally
rotated square lattice. It has been also shown that the physical quantities
defined in these systems are related through a similarity transformation.Comment: 8 pages, 2 figure

### Exact Solution of the Discrete (1+1)-dimensional RSOS Model in a Slit with Field and Wall Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model
confined to a slit. We include a field-like energy, which equivalently weights
the area under the interface, and also include independent interaction terms
with both walls. This model can also be mapped to a lattice polymer model of
Motzkin paths in a slit interacting with both walls and including an osmotic
pressure. This work generalises previous work on the RSOS model in the
half-plane which has a solution that was shown recently to exhibit a novel
mathematical structure involving basic hypergeometric functions ${}_3\phi_2$.
Because of the mathematical relationship between half-plane and slit this work
hence effectively explores the underlying $q$-orthogonal polynomial structure
to that solution. It also generalises two other recent works: one on Dyck paths
weighted with an osmotic pressure in a slit and another concerning Motzkin
paths without an osmotic pressure term in a slit

### Nonequilibrium stationary states and equilibrium models with long range interactions

It was recently suggested by Blythe and Evans that a properly defined steady
state normalisation factor can be seen as a partition function of a fictitious
statistical ensemble in which the transition rates of the stochastic process
play the role of fugacities. In analogy with the Lee-Yang description of phase
transition of equilibrium systems, they studied the zeroes in the complex plane
of the normalisation factor in order to find phase transitions in
nonequilibrium steady states. We show that like for equilibrium systems, the
``densities'' associated to the rates are non-decreasing functions of the rates
and therefore one can obtain the location and nature of phase transitions
directly from the analytical properties of the ``densities''. We illustrate
this phenomenon for the asymmetric exclusion process. We actually show that its
normalisation factor coincides with an equilibrium partition function of a walk
model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure

### A simple model of a vesicle drop in a confined geometry

We present the exact solution of a two-dimensional directed walk model of a
drop, or half vesicle, confined between two walls, and attached to one wall.
This model is also a generalisation of a polymer model of steric stabilisation
recently investigated. We explore the competition between a sticky potential on
the two walls and the effect of a pressure-like term in the system. We show
that a negative pressure ensures the drop/polymer is unaffected by confinement
when the walls are a macroscopic distance apart

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