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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 jump probabilities {pi,j ’s} to decide whether 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 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 ∈]0, 1[ is a transcendental number over the field Q(pi,j). In addition, when p(0,0) = 0, our classification holds for all t ∈]0, +∞].
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https://hal.inria.fr/hal-03008556
Contributor : Guy Fayolle Connect in order to contact the contributor
Submitted on : Sunday, January 10, 2021 - 11:51:13 AM
Last modification on : Friday, January 21, 2022 - 3:16:41 AM

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  • HAL Id : hal-03008556, version 3

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Guy Fayolle, Roudolf Iasnogorodski. Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0. Markov Processes And Related Fields, Polymat Publishing Company, 2021, 27 (1), pp.12. ⟨hal-03008556v3⟩

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