Continuity of the Ising phase transition on nonamenable groups

2020/07/30 by Tom Hutchcroft, Hutchcroft, Tom · 1 citation
Physics and Astronomy · Mathematics · #Theoretical and Computational Physics #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2007.15625

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 ⟨ σ_o ⟩_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 p_c on any transitive nonamenable graph.

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