Attraction properties for general urn processes and applications to a class of interacting reinforced particle systems

TL;DR

This paper investigates the attraction properties of a broad class of urn models and applies these findings to analyze localization in interacting reinforced random walks on polygons, revealing conditions for eventual dominance of a single color.

Contribution

It introduces a new general result for urn models with complex reinforcement rules and applies it to interacting particle systems, advancing understanding of localization phenomena.

Findings

In certain conditions, one color in the urn is eventually chosen only finitely often.

The study establishes attraction properties for a wide class of urn models with external influences.

Application to interacting reinforced random walks shows localization behavior under mild assumptions.

Abstract

We study a system of interacting reinforced random walks defined on polygons. At each stage, each particle chooses an edge to traverse which is incident to its position. We allow the probability of choosing a given edge to depend on the sum of, the number of times that particle traversed that edge, a quantity which depends on the behaviour of the other particles, and possibly external factors. We study localization properties of this system and our main tool is a new result we establish for a very general class of urn models. More specifically, we study attraction properties of urns composed of balls with two distinct colors which evolve as follows. At each stage a ball is extracted. The probability of picking a ball of a certain color evolves in time. This evolution may depend not only on the composition of the urn but also on external factors or internal ones depending on the history of the urn. A particular example of the latter is when the reinforcement is a function of the composition of the urn and the biggest run of consecutive picks with the same color. The model that we introduce and study is very general, and we prove that under mild conditions, one of the colors in the urn is picked only finitely often.