Continuity of the Ising Phase Transition on Nonamenable Groups
Abstract
We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization ⟨σo⟩β,h+ is a locally Hölder-continuous function of the inverse temperature β and external field h throughout the non-negative quadrant (β,h)∈[0,∞)2. As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at pc on any transitive nonamenable graph.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- November 2023
- DOI:
- arXiv:
- arXiv:2007.15625
- Bibcode:
- 2023CMaPh.404..227H
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics