Random walks in the quarter plane with zero drift: an explicit criterion for the finiteness of the associated group

Abstract

In many recent studies on random walks with small jumps in the quarter plane, it has been noticed that the so-called "group" of the walk governs the behavior of a number of quantities, in particular through its "order". In this paper, when the "drift" of the random walk is equal to 0, we provide an effective criterion giving the order of this group. More generally, we also show that in all cases where the "genus" of the algebraic curve defined by the kernel is 0, the group is infinite, except precisely for the zero drift case, where finiteness is quite possible.

References in corpus (2)

• Walks with small steps in the quarter plane
• Two Non-holonomic Lattice Walks in the Quarter Plane

Cited by in corpus (11)

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• Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks
• $t$-Martin boundary of killed random walks in the quadrant
• On the nature of four models of symmetric walks avoiding a quadrant
• Constructing discrete harmonic functions in wedges
• Random Walks in the Quarter-Plane: Advances in Explicit Criterions for the Finiteness of the Associated Group in the Genus 1 Case
• Differential algebraic generating series of weighted walks in the quarter plane
• Discrete harmonic functions in the three-quarter plane