# Attracting Random Walks

## TL;DR

This paper introduces the Attracting Random Walks model, analyzing its phase transition in mixing times on graphs, revealing a rich get richer dynamic with a critical temperature affecting convergence speed.

## Contribution

The paper defines a new non-reversible Markov chain model on graphs, demonstrating a phase transition in mixing times without relying on Gibbsian stationary distributions.

## Key findings

- Mixing time is $O(n ext{log} n)$ at high temperature.
- Mixing time is exponential in $n$ at low temperature.
- The model exhibits a dynamic phase transition independent of stationary distribution decomposition.

## Abstract

This paper introduces the Attracting Random Walks model, which describes the dynamics of a system of particles on a graph with $n$ vertices. At each step, a single particle moves to an adjacent vertex (or stays at the current one) with probability proportional to the exponent of the number of other particles at a vertex. From an applied standpoint, the model captures the rich get richer phenomenon. We show that the Markov chain exhibits a phase transition in mixing time, as the parameter governing the attraction is varied. Namely, mixing time is $O(n\log n)$ when the temperature is sufficiently high and $\exp(\Omega(n))$ when temperature is sufficiently low. When $\mathcal{G}$ is the complete graph, the model is a projection of the Potts model, whose mixing properties and the critical temperature have been known previously. However, for any other graph our model is non-reversible and does not seem to admit a simple Gibbsian description of a stationary distribution. Notably, we demonstrate existence of the dynamic phase transition without decomposing the stationary distribution into phases.