Discrete Harmonic Functions in the Three-Quarter Plane

Trotignon, Amélie

doi:10.1007/s11118-020-09884-y

Cited by 5 publications

(9 citation statements)

References 14 publications

(43 reference statements)

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“…Method and main result. As mentioned in [RT19,Tro19], the study of walks avoiding a quadrant gives rise to convergence problems. Indeed, although we can easily write a functional equation for the generating function C(x, y; t), unlike the quadrant case, see [BMM10, Sec.…”

Section: Introductionmentioning

confidence: 99%

“…4.1], this functional equation involves infinitely many negative and positive power of x and y making the series not convergent anymore. In this article, we follow the same strategy as [RT19,Tro19] and divide the three quadrants into two symmetric convex cones of opening angle 3π/4 and the diagonal in between, see Figure 4. After making some assumptions on the step set of the walk (symmetry and no anti-diagonal jumps in (H1)), we derive a functional equation which can be solved.…”

Section: Introductionmentioning

confidence: 99%

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On the nature of four models of symmetric walks avoiding a quadrant

Dreyfus,

Trotignon

2020

Preprint

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We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel have genus one, and the step set is symmetric, with no anti-diagonal directions. In that situation, a functional equation can be solved. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the nature of four models of symmetric walks avoiding a quadrant

Dreyfus,

Trotignon

2020

Preprint

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“…As discussed in [30,Sec. 6], no simple expression is known for H (x, y) (the results of [30] only hold for walks with no drift; the same is true for the results of [32] on harmonic functions in C). algebraic, because quadrant walks with steps in S 1 (or S 2 ) are algebraic.…”

Section: Harmonic Functionsmentioning

confidence: 99%

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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“…Indeed, in the same way as in the quarter plane, reflected Brownian motion has been introduced to study scaling limits of large queueing networks (see Figure 1.1), a Brownian model in a non-convex cone could approximate discrete random walks on a wedge having obtuse angle (see Figure 1.2 for a concrete example). Such random walks have an independent interest and have already been studied in a number of cases: see [2,29,8] in the combinatorial literature and [33,27] for more probability inclined works.…”

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confidence: 99%

“…The main idea is to state a functional equation satisfied by the associated generating functions and to reduce it to certain boundary value problems, which after analysis happen to be solvable in closed form. This approach has been applied to the framework of Brownian diffusions in a quadrant [14,13,1], to symmetric random walks in a three-quarter plane [29,33], but never to the present setting of diffusions in non-convex wedges. From this technical point of view, the present work will bring the following novelty: we will prove that our problem is generically reducible to a system of two…”

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confidence: 99%

On the stationary distribution of reflected Brownian motion in a non-convex wedge

Fayolle¹,

Franceschi²,

Raschel³

2021

Preprint

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We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.

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Discrete Harmonic Functions in the Three-Quarter Plane

Cited by 5 publications

References 14 publications

On the nature of four models of symmetric walks avoiding a quadrant

On the nature of four models of symmetric walks avoiding a quadrant

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

On the stationary distribution of reflected Brownian motion in a non-convex wedge

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