Some toy models of self-organized criticality in percolation

TL;DR

This paper introduces and analyzes simple models of self-organized criticality in percolation, where the percolation probability adapts based on the configuration to approach the critical point as the system size grows.

Contribution

It presents three toy models of self-organized criticality in percolation, demonstrating automatic control mechanisms that drive the system to criticality.

Findings

Models achieve convergence of percolation probability to critical point

Largest cluster size influences the self-organization process

Distribution of cluster sizes stabilizes at criticality

Abstract

We consider the Bernoulli percolation model in a finite box and we introduce an automatic control of the percolation probability, which is a function of the percolation configuration. For a suitable choice of this automatic control, the model is self-critical, i.e., the percolation probability converges to the critical point $p_c$ when the size of the box tends to infinity. We study here three simple examples of such models, involving the size of the largest cluster, the number of vertices connected to the boundary of the box, or the distribution of the cluster sizes.