A toy model of self-critical percolation

Forien Nicolas, University Paris-Dauphine

We present the construction of a simple toy model presenting a phenomenon of self-organized criticality.

The model is defined as follows. We consider the Bernoulli percolation model in a finite box and we introduce an automatic control of the percolation parameter, which we take as a function of the percolation configuration.
For a suitable choice of this automatic control, the model is self-critical, i.e., the percolation parameter converges to the critical point of Bernoulli percolation, when the size of the box tends to infinity.
The proof is elementary and is related to the behaviour of percolation near the critical point.

This is joint work with Raphaël Cerf [1], with a generalization to the Ising model [2] that might be briefly mentioned, if time allows.

Area: CS37 - Percolation (Matteo Quattropani)

Keywords: percolation, self-organized criticality

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