A Class of Multi-particle Reinforced Interacting Random Walks

TL;DR

This paper studies multi-particle reinforced interacting random walks on graphs, showing that with strong reinforcement, particles tend to occupy a small set of vertices asymptotically, revealing their long-term behavior.

Contribution

It introduces a dynamical approach to analyze multi-particle reinforced random walks and characterizes their asymptotic occupation measures on finite graphs.

Findings

Particles' occupation measures have small joint support under strong reinforcement

The analysis applies to both finite and infinite graphs

The approach provides insights into long-term behavior of reinforced interacting particles

Abstract

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.