Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones

Price, Andrew Elvey

doi:10.48550/arxiv.2204.06847

Cited by 2 publications

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“…This is however not sufficient, as we need to continue meromorphically the Laplace transform onto an open domain of C 2 containing (0, 0). For that purpose, we introduce a continuation procedure related to the one used in [11], and close to those in [10,13] in the discrete setting. This is done in Section 4 and in the Appendix; the main idea is to split the three-quarter plane into two convex cones, and to state two intermediate functional equations for Laplace transforms whose convergence is clear.…”

Section: Main Results and Scheme Of The Proofsmentioning

confidence: 99%

“…The literature contains a few examples of discrete walks in a three-quarter plane, which, starting from Fourier transforms, might also be solved by means of a single functional equation, through subtle manipulations on the various generating functions (see e.g., [3,4,10,21,24]).…”

Section: Main Results and Scheme Of The Proofsmentioning

confidence: 99%

“…1.1 for an example). Such random walks have an intrinsic interest and have already been studied in a number of cases: see [3,4,10,21] in the combinatorial literature and [20,24] for more probabilistic works.…”

Section: Reflected Brownian Motion In a Three-quarter Planementioning

confidence: 99%

See 2 more Smart Citations

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Fayolle

Franceschi

Raschel

2023

Indagationes Mathematicae

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Section: Main Results and Scheme Of The Proofsmentioning

confidence: 99%

Section: Main Results and Scheme Of The Proofsmentioning

confidence: 99%

See 1 more Smart Citation

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Fayolle

Franceschi

Raschel

2023

Indagationes Mathematicae

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“…The rest of this subsection simply rephrases Section 2.2 of [31], with a slightly different presentation. (Another strategy, namely splitting in three quadrants, is applied in [9,11,12,20].) We write 12…”

Section: Functional Equations For Walks Avoiding a Quadrantmentioning

confidence: 99%

“…In a very recent preprint[20], Elvey Price proves this conjecture for the dependence in x.2022/09/25 12:58 https://doi.org/10.4153/S0008414X22000487 Published online by Cambridge University Press…”

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confidence: 92%

Enumeration of three-quadrant walks via invariants: some diagonally symmetric models

Bousquet-Mélou

2022

Can. J. Math.-J. Can. Math.

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In the past 20 years, the enumeration of plane lattice walks confined to a convex conenormalized into the first quadrant -has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of them deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps. More recently, similar questions have been raised for non-convex cones, typically the three-quadrant cone C = {(i, j) : i ≥ 0 or j ≥ 0}. They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in C, which can be seen as the first level of difficulty beyond quadrant problems. This collection consists of diagonally symmetric models in {−1, 0, 1} 2 \ {(−1, 1), (1, −1)}. Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte's notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model { , , , }, which is D-finite. The three algebraic models are those of the Kreweras trilogy, S = { , ←, ↓}, S * = {→, ↑, }, and S ∪ S * . Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in S is an explicit rational function in the quadrant generating function with steps in S := {(j − i, j) : (i, j) ∈ S }. We derive various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in C for the (reverses of the) five models that are at least D-finite.

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Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones

Cited by 2 publications

References 0 publications

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann–Hilbert problem via Fourier transforms

Enumeration of three-quadrant walks via invariants: some diagonally symmetric models

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