Some arithmetic properties of P\'olya's urn

TL;DR

This paper investigates the divisibility and visibility properties of Pólya's urn modeled as a walk on the integer lattice, analyzing asymptotic behaviors related to coprimality and drawing connections to standard random walks.

Contribution

It extends previous work by studying the divisibility properties of Pólya's urn as a walk, revealing asymptotic visibility and coprimality proportions using de Finetti's theorem.

Findings

Asymptotic proportion of coprime compositions analyzed

Visibility from the origin characterized in the limit

Connections established between Pólya's urn and standard random walks

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.