A simple rank-based Markov chain with self-organized criticality

2014/05/14 by Jan M. Swart, Swart, Jan M.
Mathematics · Physics and Astronomy · #60J05 (Secondary) #60K35 #82C26 #82C27 (Primary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1405.3609

openalex publication_date 2014/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a self-reinforced point processes on the unit interval that appears to exhibit self-organized criticality, somewhat reminiscent of the well-known Bak-Sneppen model. The process takes values in the finite subsets of the unit interval and evolves according to the following rules. In each time step, a particle is added at a uniformly chosen position, independent of the particles that are already present. If there are any particles to the left of the newly arrived particle, then the left-most of these is removed. We show that all particles arriving to the left of p_{\rm c}=1-e^-1 are a.s. eventually removed, while for large enough time, particles arriving to the right of p_{\rm c} stay in the system forever.

Citations

• A stochastic model of evolution
• A simple model of a limit order book

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