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

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.