A Class of Multi-particle Reinforced Interacting Random Walks

We consider a class of multi-particle reinforced interacting random walks. In this model, there are some (finite or infinite) particles performing random walks on a given (finite or infinite) connected graph, so that each particle has higher probability to visit neighboring vertices or edges which have been seldom visited by the other particles. Specifically we investigate two particles' vertex-reinforced interacting random walks on finite complete graphs. By a dynamical approach we prove that the two particles' occupation measure asymptotically has small joint support almost surely if the reinforcement is strong.