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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 toits position. We allow the probability of choosing a given edge to depend onthe sum of, the number of times that particle traversed that edge, a quantitywhich depends on the behaviour of the other particles, and possibly externalfactors. We study localization properties of this system and our main tool is anew result we establish for a very general class of urn models. Morespecifically, we study attraction properties of urns composed of balls with twodistinct colors which evolve as follows. At each stage a ball is extracted. Theprobability of picking a ball of a certain color evolves in time. Thisevolution may depend not only on the composition of the urn but also onexternal factors or internal ones depending on the history of the urn. Aparticular example of the latter is when the reinforcement is a function of thecomposition of the urn and the biggest run of consecutive picks with the samecolor. The model that we introduce and study is very general, and we prove thatunder mild conditions, one of the colors in the urn is picked only finitelyoften.

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