Related papers: Counting walks in a quadrant: a unified approach v…
In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use…
We address the enumeration of walks with small steps confined to a two-dimensional cone, for example the quarter plane, three-quarter plane or the slit plane. In the quarter plane case, the solutions for unweighted step-sets are already…
This paper is the first application of the compensation approach to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of…
Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…
In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of…
In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on…
The set of random walks with different step sets (of short steps) in the quarter plane has provided a rich set of models that have profoundly different integrability properties. In particular, 23 of the 79 effectively different models can…
We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set…
In the 1970s, William Tutte developed a clever algebraic approach, based on certain "invariants", to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks…
We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x,…
In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and…
In the 1970s, Tutte developed a clever algebraic approach, based on certain "invariants" , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to…
In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies…
Gessel's walks are the planar walks that move within the positive quadrant $\mathbb{Z}_{+}^{2}$ by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the…
In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the one-dimensional case. We compute these…
Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion formula to arbitrary powers of roots of the characteristic equation, that being a…
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich…
In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many…
Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some…
We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these…