Interacting urn models with strong reinforcement

TL;DR

This paper investigates interacting urn models with polynomial reinforcement, disproving a conjecture that strong interaction always leads to monopoly, and providing new conditions for monopoly when the interaction is complete.

Contribution

It disproves the conjecture that any positive interaction parameter guarantees monopoly and offers improved conditions for monopoly in the case of full interaction.

Findings

Disproved the conjecture for $p>0$

Provided sufficient conditions for monopoly at $p=1$

Improved previous results by Launay

Abstract

For the interacting urn model with polynomial reinforcement, it has been conjectured that almost surely one color monopolizes all the urns if the interaction parameter $p>0$. We disprove the conjecture. For the case $p=1$, we give a sufficient condition for monopoly, which improves a previous result obtained by Launay.