Attracting Random Walks
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2019
- DOI:
- 10.48550/arXiv.1903.00427
- arXiv:
- arXiv:1903.00427
- Bibcode:
- 2019arXiv190300427G
- Keywords:
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- Mathematics - Probability;
- 60J10
- E-Print:
- 32 pages, 7 figures