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

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 is an element of]0, 1[, we provide the conditions on the jump probabilities {p(i,j)'s} to decide whether walks are singular or regular, as defined in [3, Section 2.3]. These conditions are independent of t E]0, 1[ and given in terms of step set configurations. We also find the configurations for the kernel to be of genus 0, knowing that the genus is always <= 1. All these conditions are very similar to that of the stationary case considered in [3]. Our results extend the work [2], which considers only the special situation where t is an element of]0, 1[ is a transcendental number over the field Q(p(i,j)). In addition, when p(0,0) = 0, our classification holds for all t is an element of[0, -infinity].