Lattice paths with catastrophes

Banderier, Cyril; Wallner, Michael

doi:10.1016/j.endm.2017.05.010

Cited by 12 publications

(26 citation statements)

References 30 publications

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“…where a E (z) and b E (z) are analytic functions around z = ρ with a E (ρ) = 1 and b E (ρ) = −2C/τ . From (8) we see that…”

Section: Sign Changes Of Motzkin Walksmentioning

confidence: 96%

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A half-normal distribution scheme for generating functions

Wallner

2020

European Journal of Combinatorics

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We present an extension of a theorem by Michael Drmota and Michèle Soria [Images and Preimages in Random Mappings, 1997] which can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytic and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of lattice paths.

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“…where a E (z) and b E (z) are analytic functions around z = ρ with a E (ρ) = 1 and b E (ρ) = −2C/τ . From (8) we see that…”

Section: Sign Changes Of Motzkin Walksmentioning

confidence: 96%

“…Finally, in the case δ > 0 the same reasoning as in [4] gives the result, as the perturbation by M <0 (z) does not pose any problems. Yet, an alternative proof can be given via the perturbed supercritical sequence scheme [8].…”

Section: Proposition 45 the Generating Functions Of Strictly Negatimentioning

confidence: 99%

A half-normal distribution scheme for generating functions

Wallner

2020

European Journal of Combinatorics

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“…Five consequences of our main results. Our first contribution is about methodology: we show that under the symmetry condition (H), three-quadrant walk models are exactly soluble, in the sense that their generating function admits an explicit (contour-integral) expression (2).…”

Section: Simple Walk Kreweras Reverse Kreweras Union Krewerasmentioning

confidence: 99%

“…with the same function w as in (2) but different functions f (rational) and g (algebraic). Though simpler, Equation (3) is quite similar to (2). This similarity opens the way to prove combinatorial formulas relating the two models.…”

Section: Simple Walk Kreweras Reverse Kreweras Union Krewerasmentioning

confidence: 99%

On Walks Avoiding a Quadrant

Raschel

Trotignon²

2019

Electron. J. Combin.

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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

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“…The standard reference for lattice paths with catastrophes is the paper [2]; the very recent paper [3] contains some bijective aspects.…”

Section: Introductionmentioning

confidence: 99%