3d positive lattice walks and spherical triangles

Bogosel, Beniamin; Perrollaz, Vincent; Raschel, Kilian; Trotignon, Amélie

doi:10.48550/arxiv.1804.06245

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Comparison To Earlier Formulas1

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2020

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“…We show how to prove this property in the concluding remarks. The excursion enumeration formulas of Denisov and Wachtel [6] agree with ours for the known 2D and 3D cases [4,2]. The 1D results of Banderier and Flajolet [1] also agree with our formulas.…”

Section: Comparison To Earlier Formulassupporting

confidence: 85%

The asymptotics of reflectable weighted walks in arbitrary dimension

Mishna

Simon

2020

Advances in Applied Mathematics

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Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For some models that are invariant under the Weyl group action, they express the counting function for the walks with fixed starting and endpoint as a constant term in the Taylor series expansion of a rational function. Here, we focus on the simplest case, the Weyl groups A d 1 , which correspond to walks in the first orthant N d taking steps from a subset of {±1, 0} d which is invariant under reflection across any axis. The principle novelty here is the incorporation of weights on the steps and the main result is a very general theorem giving asymptotic enumeration formulas for walks that end anywhere in the orthant. The formulas are determined by singularity analysis of multivariable rational functions, an approach that has already been successfully applied in numerous related cases.

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