Three Attractive Osculating Walkers and a Polymer Collapse Transition (original) (raw)

Staircase polygons and recurrent lattice walks

Physical Review E, 1993

In this paper we derive a direct relationship between the staircase-polygon-generating function Zd of Guttmann and Prellberg [Phys. Rev. E 47, R2233 (1993)] and the generating function for recurrent lattice walks Pd for the simple (hyper-) cubic lattice in all dimensions d. A recursion formula is obtained for the Zd with respect to dimension, which leads to a simplified derivation of Guttmann and Prellberg s result for d = 3, avoiding the use of the Heun function, and a derivation of their formula for d =4 from an integral representation is given in the Appendix.

Catalan States of Lattice Crossing: An Application of Plucking Polynomial

Topology and its Applications, 2018

For a Catalan state C of a lattice crossing L (m, n) with no returns on one side, we find its coefficient C (A) in the Relative Kauffman Bracket Skein Module expansion of L (m, n). We show, in particular, that C (A) can be found using the plucking polynomial of a rooted tree with a delay function associated to C. Furthermore, for C with returns on one side only, we prove that C (A) is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

European Journal of Combinatorics, 2017

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N 2 , counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1-39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers [FPSAC 2009, Discrete Math. Theor. Comput. Sci. Proc., pp. 201-215, 2009]. We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions.

Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

Journal of Statistical Physics, 1979

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.

The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths

J. Integer Seq., 2019

We study the central coefficients of a family of Pascal-like triangles defined by Riordan arrays. These central coefficients count left-factors of colored Schröder paths. We give various forms of the generating function, including continued fraction forms, and we calculate their Hankel transform. By using the A and Z sequences of the defining Riordan arrays, we obtain a matrix whose row sums are equal to the central coefficients under study. We explore the row polynomials of this matrix. We give alternative formulas for the coefficient array of the sequence of central coefficients.

Some applications arising from the interactions between the theory of Catalan-like numbers and the ECO method

In [FP] the ECO method and Aigner's theory of Catalan-like numbers are compared, showing that it is often possible to translate a combinatorial situation from one theory into the other by means of a standard change of basis in a suitable vector space. In the present work we emphasize the soundness of such an approach by finding some applications suggested by the above mentioned translation. More precisely, we describe a presumably new bijection between two classes of lattice paths and we give a combinatorial interpretation to an integer sequence not appearing in [Sl]. * This work was partially supported by MIUR project: Linguaggi formali e automi: metodi, modelli e applicazioni.