  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.
