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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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