• DOI:10.1214/20-ejp471
• Corpus ID: 119319426

Attracting random walks

@article{Gaudio2019AttractingRW,
  title={Attracting random walks},
  author={Julia Gaudio},
  journal={arXiv: Probability},
  year={2019}
}

• Julia Gaudio
• Published 1 March 2019
• Mathematics
• arXiv: Probability

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.