On Walks Avoiding a Quadrant

Abstract

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 the first quadrant have been studied a lot, the case of planar, non-convex cones-equivalent to the three-quarter plane after a linear transform-has been approached only recently. In this article we develop an analytic approach to the case of walks in three quadrants. The advantage of this method is to provide uniform treatment in the study of models corresponding to different step sets. After splitting the three quadrants in two symmetric convex cones, the method is composed of three main steps: write a system of functional equations satisfied by the counting generating function, which may be simplified into one single equation under symmetry conditions; transform the functional equation into a boundary value problem; and finally solve this problem, using a concept of anti-Tutte's invariant. The result is a contour-integral expression for the generating function. Such systems of functional equations also appear in queueing theory with the famous Join-the-Shortest-Queue model, which is still an open problem in the non-symmetric case. Figure 1: Walks in the quarter plane and in three-quarter plane

Discrete Harmonic Functions in the Three-Quarter Plane

“…In [6], Bousquet-Mélou sees the three-quarter plane as the union of three quadrants and obtains some results for the simple and diagonals walks avoiding a quadrant. Integral expressions for the generating function of walks avoiding a quadrant with symmetric step sets for walks are derived in [20], where the three-quarter plane is seen as the union of two symmetric convex cones of opening angle 3 4. Asymptotics of the number of excursions of walks with small steps in the three-quadrant is computed in [17] by Mustapha. In this article, following [4,11], Mustapha expresses the critical exponent of harmonic functions in three quadrants as a function of the critical exponent of harmonic functions in a quadrant.…”

“…In this article, we find an explicit expression for generating functions of discrete harmonic functions associated to random walks avoiding a quadrant with a mixed approach of [19] and [20]. We focus on the analytic approach developed in [19], which consists in writing a functional equation for the generating function for a fixed harmonic function, transforming this functional equation into a boundary value problem and finally solving this problem, which results in an explicit expression for the generating function.…”

“…Moreover, with assumption (H2), we suppose the walks to be symmetric with no anti-diagonal jumps. If we do not exclude non-zero probabilities 1 1 and 1 1 , as noted in [20,Sec. 2], after the change of variable we would end up with random walks with non-zero probabilities 2 1 and 2 1 respectively for big jumps of vector 2 1 and 2 1 .…”

“…Note that the analytic theory for walks with big steps is still incomplete [9,10]. A related consequence would be the presence of more unknowns in the functional (20) (namely 1 2 and 1 2), making the problem not solvable with our techniques. The third hypothesis (H3) is not essential in the study, but automatically excludes degenerate cases which could have been studied with easier methods.…”

“…However, even if this functional (13) seems close to the quarter plane case (17) solved in [19], it is in fact fundamentally different: the three quadrants case involves negative powers of and making the series not convergent anymore. To remedy this difficulty, we follow the same strategy as in [20]: we divide the three quadrants into two symmetric convex cones and the diagonal (see Fig. 5) and write a system of two functional equations (one for each cone).…”