# Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0

Abstract : 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.
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https://hal.inria.fr/hal-03008556
Contributor : Guy Fayolle <>
Submitted on : Thursday, November 19, 2020 - 11:15:52 AM
Last modification on : Friday, November 20, 2020 - 3:31:57 AM

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• HAL Id : hal-03008556, version 2

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Guy Fayolle, Roudol Iasnogorodski. Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0. 2020. ⟨hal-03008556v2⟩

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