149 research outputs found
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
Noncommutative integrability, paths and quasi-determinants
In previous work, we showed that the solution of certain systems of discrete
integrable equations, notably and -systems, is given in terms of
partition functions of positively weighted paths, thereby proving the positive
Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of
solution is amenable to generalization to non-commutative weighted paths. Under
certain circumstances, these describe solutions of discrete evolution equations
in non-commutative variables: Examples are the corresponding quantum cluster
algebras [BZ], the Kontsevich evolution [DFK09b] and the -systems themselves
[DFK09a]. In this paper, we formulate certain non-commutative integrable
evolutions by considering paths with non-commutative weights, together with an
evolution of the weights that reduces to cluster algebra mutations in the
commutative limit. The general weights are expressed as Laurent monomials of
quasi-determinants of path partition functions, allowing for a non-commutative
version of the positive Laurent phenomenon. We apply this construction to the
known systems, and obtain Laurent positivity results for their solutions in
terms of initial data.Comment: 46 pages, minor typos correcte
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