A Simple Rank-Based Markov Chain with Self-Organized Criticality

#### J.M. Swart

2017, v.23, Issue 1, 87-102

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\tire 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.

Keywords: self-reinforcement, self-organized criticality, canyon