Walks in the quarter plane: Kreweras’ algebraic model

By Mireille Bousquet-M'elou. 2004

99 citations recorded by the source corpus.

We consider planar lattice walks that start from (0, 0), remain in the first quadrant i, j ≥ 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: – they are counted by nice numbers (Kreweras 1965), – the generating function of these numbers is algebraic (Gessel 1986), – the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function (Flatto and Hahn 1984). These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, which is more elementary that those previously published. We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations, which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc3a4fc19583081f0df06993a7542e007fad8810

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

ON PARTITIONS AVOIDING 3-CROSSINGS (2005, unknown): “First of all, a number of equations with a finite diagram have been solved, and shown to have a D-finite solution [3, 4, 5, 18].”
On 3-Dimensional Lattice Walks Confined to the Positive Octant (2014, unknown): “, AND S. MELCZER In the past few years, the “next” natural case, namely walks conﬁned to the intersectionoftwohalf-spaces,hasbeenextensivelystudiedintheformofwalkson Z2 conﬁnedtothepositivequadrantN2 [5,8,10,11,17,18,26,31]. Wecallsuch walks quadrant walks. In the early days of this study, it was so”
Walks with small steps in the quarter plane (2008, methodology): “This was done in [13] for a probabilistic version of Kreweras’ walks, using a variant of the method presented in this paper.”
Counting walks in a quadrant: a unified approach via boundary value problems (2010, unknown): “, [5, 12, 13], [4, 16, 17] and [8, 23], respectively).”
Random walks in cones (2011, background): “Here, we mention papers of Bousquet-Melou [6] and of BousquetMelou and Mishna [7], where the authors obtained exact results for some random walks with bounded steps in the quarter plane.”