Conditions for Some Non Stationary Random Walks in the Quarter Plane to Be Singular or of Genus 0

#### G. Fayolle, R. Iasnogorodski

2021, v.27, Issue 1, 111-122

ABSTRACT

We analyze the \emph{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 $t\in]0,1[$, we provide the conditions on the jump probabilities \{$p_{i,j}$'s\} to decide whether walks are \emph{singular} or \emph{regular}, as defined in \cite[Section~2.3]{FIM2017}. These conditions are independent of $t\in]0,1[$ and given in terms of \emph{step set configurations}. We also find the configurations for the kernel to be of genus~$0$, knowing that the genus is always~$\leq1$. All these conditions are very similar to that of the stationary case considered in \cite{FIM2017}. Our results extend the work \cite{DHRS2020}, which considers only the special situation where~$t\in]0,1[$ is a transcendental number over the field
$\Qb(p_{i,j})$. In addition, when $p_{0,0}=0$, our classification holds for all~$t\in]0,+\infty]$.

Keywords: algebraic curve, functional equation, generating function, genus, quarter-plane, Riemann surface, singular random walk.