19 research outputs found
Combinatorial mappings of exclusion processes
We review various combinatorial interpretations and mappings of
stationary-state probabilities of the totally asymmetric, partially asymmetric
and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In
these steady states, the statistical weight of a configuration is determined
from a matrix product, which can be written explicitly in terms of generalised
ladder operators. This lends a natural association to the enumeration of random
walks with certain properties.
Specifically, there is a one-to-many mapping of steady-state configurations
to a larger state space of discrete paths, which themselves map to an even
larger state space of number permutations. It is often the case that the
configuration weights in the extended space are of a relatively simple form
(e.g., a Boltzmann-like distribution). Meanwhile, various physical properties
of the nonequilibrium steady state - such as the entropy - can be interpreted
in terms of how this larger state space has been partitioned.
These mappings sometimes allow physical results to be derived very simply,
and conversely the physical approach allows some new combinatorial problems to
be solved. This work brings together results and observations scattered in the
combinatorics and statistical physics literature, and also presents new
results. The review is pitched at statistical physicists who, though not
professional combinatorialists, are competent and enthusiastic amateurs.Comment: 56 pages, 21 figure
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