Some arithmetic properties of P\'olya's urn

Abstract

Following Hales (2018), the evolution of P\'olya's urn may be interpreted as a walk, a P\'olya walk, on the integer lattice $\mathbb{N}^2$. 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.