Some arithmetic properties of P?olya?s urn

Following Hales (2018), the evolution of P ' olya's urn may be interpreted as a walk, a P ' olya walk, on the integer lattice N2. We study the visibility properties of P ' olya's walk or, equivalently, the divisibility properties of the composition of the urn. In particular, we are interested in the asymptotic average time that a P ' olya walk is visible from the origin, or, alternatively, in the asymptotic proportion of draws so that the resulting composition of the urn is coprime. Via de Finetti's exchangeability theorem, P ' olya's walk appears as a mixture of standard random walks. This paper is a follow-up of Cilleruelo-Fern ' andez-Fern ' andez (2019), where similar questions were studied for standard random walks.