hal-03008556
https://inria.hal.science/hal-03008556
https://inria.hal.science/hal-03008556v2/document
https://inria.hal.science/hal-03008556v2/file/noyau.pdf
Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0
Fayolle, Guy
Iasnogorodski, Roudol
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
UNDEFINED
We analyze the kernel K(x, y, t) of the basic functional equation associated with the tri-variate counting generating function (CGF) of walks in the quarter plane. In this short paper, taking $t ∈]0, 1[$, we provide the conditions on the step set {p i,j } to decide whether the walks are singular or regular, as defined in [3, Section 2.3]. These conditions are independent of $t ∈]0, 1[$ and given in terms of step set configurations. We also give the configurations for the kernel to be of genus 0, knowing that the genus is always ≤ 1. All these conditions are very similar to the case t = 1 considered in [3]. Our results extend the work [2], which considers only very special situations, namely when $t ∈]0, 1[$ is a transcendental number and the $p i,j s$ are rational.
2020-11-19
2020-11-19
en