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

Entanglement in gapless resonating valence bond states

We study resonating-valence-bond (RVB) states on the square lattice of spins and of dimers, as well as SU(N)-invariant states that interpolate between the two. These states are ground states of gapless models, although the SU(2)-invariant spin RVB state is also believed to be a gapped liquid in its spinful sector. We show that the gapless behavior in spin and dimer RVB states is qualitatively similar by studying the R\'enyi entropy for splitting a torus into two cylinders, We compute this exactly for dimers, showing it behaves similarly to the familiar one-dimensional log term, although not identically. We extend the exact computation to an effective theory believed to interpolate among these states. By numerical calculations for the SU(2) RVB state and its SU(N)-invariant generalizations, we provide further support for this belief. We also show how the entanglement entropy behaves qualitatively differently for different values of the R\'enyi index $n$, with large values of $n$ proving a more sensitive probe here, by virtue of exhibiting a striking even/odd effect.Comment: 44 pages, 14 figures, published versio

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