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Continuity of the Ising phase transition on nonamenable groups

Hutchcroft, Tom (2020) Continuity of the Ising phase transition on nonamenable groups. . (Unpublished)

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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.

Item Type:Report or Paper (Discussion Paper)
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Hutchcroft, Tom0000-0003-0061-593X
Additional Information:We thank Jonathan Hermon for making us aware of Freedman’s work on maximal inequalities for martingales [33], which inspired Lemmas 3.4 and 3.5. We also thank Hugo Duminil-Copin, Geoffrey Grimmett, and Russ Lyons for helpful comments on an earlier version of the manuscript.
Record Number:CaltechAUTHORS:20210924-202122977
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111033
Deposited By: George Porter
Deposited On:27 Sep 2021 14:46
Last Modified:27 Sep 2021 14:46

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