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2020 Attracting random walks
Julia Gaudio, Yury Polyanskiy
Electron. J. Probab. 25: 1-31 (2020). DOI: 10.1214/20-EJP471

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.

Citation

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Julia Gaudio. Yury Polyanskiy. "Attracting random walks." Electron. J. Probab. 25 1 - 31, 2020. https://doi.org/10.1214/20-EJP471

Information

Received: 16 August 2019; Accepted: 26 May 2020; Published: 2020
First available in Project Euclid: 1 July 2020

zbMATH: 07225527
MathSciNet: MR4119119
Digital Object Identifier: 10.1214/20-EJP471

Subjects:
Primary: 60J10

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