Some toy models of self-organized criticality in percolation

Quelques modèles jouets de criticalité auto-organisée en percolation

Raphaël Cerf (1, 2) , Nicolas Forien (3, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 LMO - Laboratoire de Mathématiques d'Orsay

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 pc 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. Along the way, we prove a general geometric inequality for subgraphs of Z d , which is of independent interest.

Domaines: Probabilités [math.PR]

Raphaël Cerf, Nicolas Forien. Some toy models of self-organized criticality in percolation. 2019. ⟨hal-02417506v1⟩