A planar Ising model of self-organized criticality

Abstract

We consider the planar Ising model in a finite square box and we replace the temperature parameter with a function depending on the magnetization. This creates a feedback from the spin configuration onto the parameter, which drives the system towards the critical point. Using the finite-size scaling results of [CM11], we show that, when the size of the box grows to infinity, the temperature concentrates around the critical temperature of the planar Ising model on the square lattice.

Some toy models of self-organized criticality in percolation

“…As a consequence, our construction of the fixed point using the geometric surgery procedure does not work any more in FK percolation, because it would require to control this phenomenon of simultaneous openings of edges, which is not yet well understood. Yet, in the article Forien (2021), we have managed to bypass this problem in the particular setting of the planar FK-Ising model, using the estimates about the near-critical regime proved by Cerf and Messikh (2011) in this context.…”