Some arithmetic properties of Pólya's urn (Q6042102)

Summary: Following \textit{A. W. Hales} [IEEE Trans. Inf. Theory 64, No. 4, Part 2, 3150--3152 (2018; Zbl 1391.60099)], the evolution of Pólya's urn may be interpreted as a walk, a Pólya walk, on the integer lattice \(\mathbb{N}^2\). We study the visibility properties of Pólya'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 Pólya 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, Pólya's walk appears as a mixture of standard random walks. This paper is a follow-up of \textit{J. Cilleruelo} et al. [Eur. J. Comb. 75, 92--112 (2019; Zbl 1400.05217)], where similar questions were studied for standard random walks.