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

Guy Fayolle
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• PersonId : 832458

#### 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. 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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-03008556 , version 1 (16-11-2020)
hal-03008556 , version 2 (19-11-2020)
hal-03008556 , version 3 (10-01-2021)

### Identifiers

• HAL Id : hal-03008556 , version 1

### Cite

Guy Fayolle, Roudol Iasnogorodski. Conditions for some non stationary random walks in the quarter plane to be singular or of genus 0. 2020. ⟨hal-03008556v1⟩

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