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

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.