The dominating colour of an infinite P\'olya urn model

TL;DR

This paper analyzes a supercritical Pólya urn model, proving that eventually a single dominating colour emerges with probability one, extending classical results to a broader parameter range.

Contribution

It establishes the almost sure dominance of a single colour in a supercritical Pólya urn, generalizing classical results to the case where 1/2<p<1.

Findings

Existence of a dominating colour with probability 1

Dominance occurs after a finite random time

Generalization of classical Pólya urn results

Abstract

We study a P\'olya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with another ball of the same colour. With probability 1-p, recolour the ball to a new colour and then return it to the urn. This is equivalent to the supercritical case of a random graph model studied by Backhausz and M\'ori, and Th\"ornblad. We prove that, with probability 1, there is a dominating colour, in the sense that, after some random but finite time, there is a colour that always has the most number of balls. A crucial part of the proof is the analysis of an urn model with two colours, in which the observed ball is returned to the urn along with another ball of the same colour with probability p, and removed with probability 1-p. Our results here generalise a classical result about the P\'olya urn model (which corresponds to p=1).