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A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

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A Correction to this article was published on 23 March 2018

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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 \({\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}\) 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 \({\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}\) for \({\beta > \beta_c}\). For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for \({\beta < \beta_c}\).

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  • 23 March 2018

    The differential inequality (2.5) should be replaced by

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Authors and Affiliations

  1. Département de Mathématiques, Université de Genève, Geneva, Switzerland

    Hugo Duminil-Copin & Vincent Tassion

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  1. Hugo Duminil-Copin
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  2. Vincent Tassion
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Correspondence to Hugo Duminil-Copin.

Additional information

Communicated by F. Toninelli

A correction to this article is available online at https://doi.org/10.1007/s00220-018-3118-8.

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Duminil-Copin, H., Tassion, V. A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model. Commun. Math. Phys. 343, 725–745 (2016). https://doi.org/10.1007/s00220-015-2480-z

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