Bijective counting of Kreweras walks and loopless triangulations

Bernardi, Olivier

doi:10.1016/j.jcta.2006.09.009

Cited by 34 publications

(80 citation statements)

References 6 publications

(13 reference statements)

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“…A combinatorial explanation has been found when i = 0, in connection with the enumeration of planar triangulations [4]. To our knowledge, the generic case remains open.…”

Section: Explain Closed Form Expressionsmentioning

confidence: 89%

“…Given the connection between Kreweras' walks and planar triangulations [4], this could be of the same complexity as proving directly that families of planar maps have an algebraic generating function. (Much progress has been made recently on this problem by Schaeffer, Di Francesco and their co-authors.)…”

Section: Explain Algebraic Seriesmentioning

confidence: 99%

“…It was then proved by Gessel that the associated 3-variable generating function is algebraic [26]. Since then, simpler derivations of this series have been obtained [42,13], including an automated proof [36], and a purely bijective one for walks ending at the origin [4]. See also [23,21,13], where the stationary distribution of a related Markov chain in the quarter plane is obtained, and found to be algebraic.…”

Section: Introductionmentioning

confidence: 98%

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Walks with small steps in the quarter plane

Bousquet-Mélou¹,

Mishna²

2010

Algorithmic Probability and Combinatorics

159

569

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Abstract. Let S ⊂ {−1, 0, 1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a half-plane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study.To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases.We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be D-finite. The 23rd model, known as Gessel's walks, has recently been proved by Bostan et al. to have an algebraic (and hence D-finite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a non-D-finite generating function.Our approach allows us to recover and refine some known results, and also to obtain new results. For instance, we prove that walks with N, E, W, S, SW and NE steps have an algebraic generating function.

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“…A combinatorial explanation has been found when i = 0, in connection with the enumeration of planar triangulations [4]. To our knowledge, the generic case remains open.…”

Section: Explain Closed Form Expressionsmentioning

confidence: 89%

Section: Explain Algebraic Seriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Walks with small steps in the quarter plane

Bousquet-Mélou¹,

Mishna²

2010

Algorithmic Probability and Combinatorics

159

569

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“…The simplest example of such a bijection is the Mullin bijection [Mul67,Ber07b], which encodes a planar map decorated by a spanning tree by a nearest-neighbor random walk on Z 2 . There are other such bijections, with different walks, that encode planar maps decorated by e.g., percolation [Ber07a,BHS18], bipolar orientations [KMSW19], or the Fortuin-Kasteleyn model [She16b]. These bijections are called mating-of-trees bijections since the planar map is constructed from the walk by gluing together, or mating, the discrete random trees associated with the two coordinates of the walk.…”

Section: 3mentioning

confidence: 99%

Random Surfaces and Liouville Quantum Gravity

Gwynne¹

2020

Notices Amer. Math. Soc.

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Liouville quantum gravity (LQG) surfaces are a family of random fractal surfaces which can be thought of as the canonical models of random two-dimensional Riemannian manifolds, in the same sense that Brownian motion is the canonical model of a random path. LQG surfaces are the continuum limits of discrete random surfaces called random planar maps. In this expository article, we discuss the definition of random planar maps and LQG, the sense in which random planar maps converge to LQG, and the motivations for studying these objects. We also mention several open problems. We do not assume any background knowledge beyond that of a second-year mathematics graduate student.What is the most natural way of choosing a random surface (two-dimensional Riemannian manifold)? If we are given a finite set X, the easiest way to choose a random element of X is uniformly, i.e., by assigning equal probability to each element of X. More generally, if we are given a set X ⊂ R n with finite, positive Lebesgue measure, the simplest way of choosing a random element of X is by sampling from Lebesgue measure normalized to have total mass one. However, the space of all surfaces is infinite dimensional for any reasonable notion of dimension, so it is not immediately obvious whether there is a canonical way of choosing a random surface.Nevertheless, there is a class of canonical models of "random surfaces" called Liouville quantum gravity (LQG) surfaces. The reason for the quotations is that LQG surfaces are not Riemannian manifolds in the literal sense since they are too singular to admit a smooth structure. Instead, LQG surfaces are defined as random topological surfaces equipped with a measure, a metric, and a conformal structure. These surfaces are fractal in the sense that the Hausdorff dimension of an LQG surface, viewed as a metric space, is strictly bigger than 2.LQG surfaces have a rich geometric structure which is still not fully understood (see Section 4). Furthermore, such surfaces are important in statistical mechanics, string theory, and conformal field theory and have deep connections to other mathematical objects such as Schramm-Loewner evolution [Sch00] (see, e.g., [She16a]), random matrix theory (see, e.g., [Web15]), and random planar maps. Discrete random surfacesBefore we discuss random surfaces, let us first, by way of analogy, consider the simpler problem of finding a canonical way to choose a random curve in the plane. As in the case of surfaces, the space of all planar curves is infinite-dimensional and does not admit a canonical probability measure in an obvious way. To get around this, we discretize the problem. Let us consider for each n ∈ N the set of all nearest-neighbor paths in the integer lattice Z 2 with n steps. This is a finite set (of cardinality 4 n ), so we can choose a uniformly random element S n from it. This discrete random path S n is called the simple random walk.By linearly interpolating, we may view S n as a curve from [0, n] to R 2 . There is a classical theorem in probability due to Donsker wh...

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“…As we mentioned above, it has been studied in several different contexts, and a direct (i.e. combinatorial) explanation of the algebraicity of the excursions was offered by Bernardi [2]. He defines a bijection with a family of planar maps.…”

Section: Algebraic Walks 231 Kreweras' Walksmentioning

confidence: 99%

Classifying lattice walks restricted to the quarter plane

Mishna

2009

Journal of Combinatorial Theory, Series A

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This work considers the nature of generating functions of random lattice walks restricted to the first quadrant. In particular, we find combinatorial criteria to decide if related series are algebraic, transcendental holonomic or otherwise. Complete results for walks taking their steps in a maximum of three directions of restricted amplitude are given, as is a well-supported conjecture for all walks with steps taken from a subset of {0, ±1} 2 . New enumerative results are presented for several classes, each obtained with a variant of the kernel method.Lattice walks are a combinatorial object with a long history, however lately there has been a rise in interest in their generating functions (for example, [5,7,9,15]) and global approaches and results. A few such studies have been performed. For example, Banderier and Flajolet [1] examine the nature of their generating functions, and provide general results of asymptotic analysis for one-dimensional lattice walks. Two-dimensional walks in the quarter plane have been treated by a collection of case analyzes [7,9,14,23]. The goal of the present work is to determine general characterizations of the generating functions, based strictly on combinatorial properties of the allowable steps. For example, can we characterize the walks whose generating function is algebraic, transcendental holonomic (that is, which satisfies a linear differential equations with polynomial coefficients) or neither? The answer has important repercussions for sequence generation, asymptotics, amongst other consequences.We focus our attention to those walks in the quarter plane which take steps from some subset of {0, ±1} 2 . These can be either viewed as the eight allowable moves for a king in chess, or one of the eight standard compass directions. These are a superset of the nearest neighbor walks, and a subset of the next nearest neighbor walks. We give explicit complete generating function expressions and classifications when this subset is of cardinality three, and we are able to formulate well-supported

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Bijective counting of Kreweras walks and loopless triangulations

Cited by 34 publications

References 6 publications

Walks with small steps in the quarter plane

Walks with small steps in the quarter plane

Random Surfaces and Liouville Quantum Gravity

Classifying lattice walks restricted to the quarter plane

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