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A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. (English) Zbl 1342.82026
Commun. Math. Phys. 343, No. 2, 725-745 (2016); correction ibid. 359, No. 2, 821-822 (2018).
Authors’ abstract: We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime \(\beta<\beta_c\), and the mean-field lower bound \(\mathbb P_\beta[0\longleftrightarrow\infty]\geq(\beta-\beta_c)/\beta\) for \(\beta>\beta_c\). For finite-range models, we also prove that for any \(\beta<\beta_c\), the probability of an open path from the origin to distance \(n\) decays exponentially fast in \(n\). For the Ising model, we prove finiteness of the susceptibility for \(\beta<\beta_c\), and the mean-field lower bound \(\langle\sigma_0\rangle_\beta^+\geq\sqrt{(\beta^2-\beta_c^2)/\beta^2}\) for \(\beta>\beta_c\). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \(\beta<\beta_c\).

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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