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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 Cerf and Messikh (Theory Relat Fields 150(1–2):193–217, 2011. https://doi.org/10.1007/s00440-010-0272-0), 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.

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Acknowledgements

I wish to thank Raphaël Cerf for suggesting this problem, and for fruitful discussions.

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Correspondence to Nicolas Forien.

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Forien, N. A planar Ising model of self-organized criticality. Probab. Theory Relat. Fields 180, 163–198 (2021). https://doi.org/10.1007/s00440-021-01025-9

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  • DOI: https://doi.org/10.1007/s00440-021-01025-9

Keywords

  • Ising
  • Self-organized criticality
  • Random-cluster model
  • Percolation

Mathematics Subject Classification

  • 82B20
  • 82B27
  • 82B43
  • 60K35