Résumé : Lattice walks taking steps from a fixed step set, and staying within some region of the lattice (most prominently, a quarter plane) have occupied combinatorialists for the past few decades and produced a huge range of rich results, drawing on algebra, probability theory, differential equations, and computational methods. I will discuss a new twist on this idea -- walks on the edges of the square lattice which obey two-step rules. These allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a number of criteria, and show how these properties affect their generating functions, asymptotic enumerations and limiting shapes, on the full lattice as well as the upper half plane. The situation in the quarter plane is still very unclear, but I'll discuss some preliminary computational results.

 [arXiv]

Dernière modification : Monday 18 March 2024

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