  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.

Probability