Asymptotics for the number of walks in a Weyl chamber of type <i>B</i>

Feierl, Thomas

doi:10.1002/rsa.20467

Cited by 6 publications

(3 citation statements)

References 18 publications

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“…Grabiner and Magyar [28] give the complete list of all random walks to which the reflection principle can be applied. Recently, the reflection principle of Gessel and Zeilberger has been slightly generalized by Feierl [21]. He derived a new version of the reflection principle for random walks with steps which are at most finite combinations of steps from the list of Grabiner and Magyar.…”

mentioning

confidence: 99%

Random walks in cones

Denisov¹,

Wachtel²

2015

Ann. Probab.

143

368

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We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.Comment: Published at http://dx.doi.org/10.1214/13-AOP867 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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mentioning

confidence: 99%

Random walks in cones

Denisov¹,

Wachtel²

2015

Ann. Probab.

143

368

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“…The reflection argument in Weyl chambers involves a signed sum over permutations of terms, which leads to the presence of determinants. The recent work of Feierl develops the related asymptotic formulas for type A walks [8] using a theorem of Hörmander to estimate a Fourier-Laplace integral, numbered in this paper as Theorem 7. The results of Melczer and Mishna [14] and Melczer and Wilson [15], and their formulas are made similarly explicit by using the formalism articulated in the book by Pemantle and Wilson [16] to apply some of these integral formulas in a more systematic setting.…”

Section: Reflectable Walksmentioning

confidence: 99%

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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“…These results have focused largely on unweighted quadrant walks with small steps, and a wide variety of techniques have been developed and adapted to treat them [10,18,5,37]. Ongoing works consider generalizations of this model, such as walks in higher dimensions [6], walks with longer steps [19,4], and walks with weights [20,29]. A lattice path model is comprised of a finite set of vectors, or steps, denoted S, and a cone in which the walks are contained, very often R 2 0 .…”

Section: Enumeration Of Walks In Conesmentioning

confidence: 99%

Weighted lattice walks and universality classes

Courtiel¹,

Melczer²,

Mishna³

et al. 2017

Journal of Combinatorial Theory, Series A

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In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

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Asymptotics for the number of walks in a Weyl chamber of type B

Cited by 6 publications

References 18 publications

Random walks in cones

Random walks in cones

The asymptotics of reflectable weighted walks in arbitrary dimension

Weighted lattice walks and universality classes

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