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

Bicoloured Dyck paths and the contact polynomial for n non-intersecting paths in a half plane.

In this paper configurations of n non-intersecting lattice paths which begin and end on the line y = 0 and are excluded from the region below this line are considered. Such configurations are called Hankel n-paths which make c intersections with the line y = 0 the lowest of which has length 2r. These configurations may also be described as parallel Dyck paths. It is found that replacing by the length generating function for Dyck paths, (!) P1 r=0 Cr!r, where C_r is the rth Catalan number, results in a remarkable simplification of the coefficients of the contact polynomial. In particular it is shown that the polynomial for configurations of a single Dyck path has the expansion ^ ZH 2r(1; (!)) = P1 b=0 Cr+b!b. This result is derived using a bijection between bi-coloured Dyck paths and plain Dyck paths. A bi-coloured Dyck path is a Dyck path in which each edge is coloured either red or blue with the constraint that the colour can only change at a contact with the line y = 0. For n > 1, the coefficient of !b in ^ ZW 2r (n; (!)) is expressed as a determinant of Catalan numbers which has a combinatorial interpretation in terms of a modified class of n non-intersecting Dyck paths. The determinant satisfies a recurrence relation which leads to the proof of a product form for the cofficients in the ! expansion of the contact polynomial

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

The Grand-Canonical Asymmetric Exclusion Process and the One-Transit Walk

The one-dimensional Asymmetric Exclusion Process (ASEP) is a paradigm for nonequilibrium dynamics, in particular driven diffusive processes. It is usually considered in a canonical ensemble in which the number of sites is fixed. We observe that the grand-canonical partition function for the ASEP is remarkably simple. It allows a simple direct derivation of the asymptotics of the canonical normalization in various phases and of the correspondence with One-Transit Walks recently observed by Brak et.al.Comment: Published versio

On directed interacting animals and directed percolation

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.Comment: 23 pages, 3 figures; J. Phys. A 35 (2002) 272

Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon

The raise and peel model of a one-dimensional fluctuating interface (model A) is extended by considering one source (model B) or two sources (model C) at the boundaries. The Hamiltonians describing the three processes have, in the thermodynamic limit, spectra given by conformal field theory. The probability of the different configurations in the stationary states of the three models are not only related but have interesting combinatorial properties. We show that by extending Pascal's triangle (which gives solutions to linear relations in terms of integer numbers), to an hexagon, one obtains integer solutions of bilinear relations. These solutions give not only the weights of the various configurations in the three models but also give an insight to the connections between the probability distributions in the stationary states of the three models. Interestingly enough, Pascal's hexagon also gives solutions to a Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few references are adde

Directed compact percolation near a wall: III. Exact results for the mean length and number of contacts

Existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall. The results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below p_c. The results for the cluster length satisfy previously conjectured differential equations. A closed expression is conjectured for the mean size above p_c in terms of a hypergeometric function

Asymmetric Exclusion Model and Weighted Lattice Paths

We show that the known matrix representations of the stationary state algebra of the Asymmetric Simple Exclusion Process (ASEP) can be interpreted combinatorially as various weighted lattice paths. This interpretation enables us to use the constant term method (CTM) and bijective combinatorial methods to express many forms of the ASEP normalisation factor in terms of Ballot numbers. One particular lattice path representation shows that the coefficients in the recurrence relation for the ASEP correlation functions are also Ballot numbers. Additionally, the CTM has a strong combinatorial connection which leads to a new 'canonical' lattice path representation and to the 'W-expansion' which provides a uniform approach to computing the asymptotic behaviour in the various phases of the ASEP. The path representations enable the ASEP normalization factor to be seen as the partition function of a more general polymer chain model having a two-parameter interaction with a surface. We show, in the case alpha = beta = 1, that the probability of finding a given number of particles in the stationary state can be expressed via non-intersecting lattice paths and hence as a simple determinant

On directed compact percolation near a damp wall

A percolation probability for directed, compact percolation near a damp wall, which interpolates between the previously examined cases, is derived exactly. We find that the critical exponent ß = 2 in common with the dry wall, rather than the value previously found in the wet wall and bulk cases. The solution is found via a mapping to a particular model of directed walks. We evaluate the exact generating function for this walk model which is also related to the ASEP model of traffic flow. We compare the underlying mathematical structure of the various cases previously considered and this one by reviewing the common framework of solution via the mapping to different directed walk models. © 2009 IOP Publishing Ltd. © 2009 IOP Publishing Ltd