The dominating colour of an infinite Pólya urn model

Abstract

ABSTRACT. We study a Pólya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n ≥ 1, 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 Móri [4,5] and Thörnblad [17]. 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 Pólya urn model (which corresponds to p = 1).

Detecting Phase Transitions through Non-Equilibrium Work Fluctuations

“…The modification of the classical Ehrenfest urn model [41,44] that we analyze was originally introduced in [38] to understand billiard dynamics comprising a Maxwell demon [45][46][47], and is embedded in a long history of urn models studied in statistical mechanics [48][49][50][51][52][53][54][55][56][57][58][59]. It refers to N particles initially distributed inside two urns.…”

Why Stake When You Can Borrow?

“…If X(t) is constructed correctly, then the laws of {X i } i∈N and {X(τ i )} i∈N are equal in distribution. Using a construction for an embedding from [62], we are able get an explicit distribution for π stake (h) i under the assumptions of this section.…”

“…This follows directly from [62, Lemma 2.1], once we map the setup of §3 to their problem. In [62], one constructs a Pólya urn such that if a ball of color c is chosen, then in their notation, with probability p lemma , Algorithm 7 update markowitz(µ prev , π stake , π lend , , γ t , λ)…”

“…We modify the arguments made in [62] and sketch how they apply to our scenario. The Pólya urn process π stake (h) embeds into a continuous time Markov branching process X(t) where X(0) = π stake (0) [61,v.9 Theorem 1].…”

Strong convergence of infinite color balanced urns under uniform ergodicity

“…He introduced a new type of urn scheme [22], where at each step with some positive probability a new color can be introduced. Several authors studied this slightly different model in the context of the Griffiths-Engen-McCloskey (GEM) model, the Poisson-Dirichlet distribution, and the Ewens sampling formula [13,20,21,38]. It was also observed that Hoppe's urn scheme has a deep relation with certain combinatorial stochastic processes, known as the Chinese restaurant process [1,34].…”