A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model
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 ฮฒ<ฮฒc, and the mean-field lower bound Pฮฒ[0โทโ]โฅ(ฮฒ-ฮฒc)/ฮฒ for ฮฒ>ฮฒc. For finite-range models, we also prove that for any ฮฒ<ฮฒc, 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 ฮฒ<ฮฒc, and the mean-field lower bound โจฯ0โฉฮฒ+โฅ(ฮฒ2-ฮฒc2)/ฮฒ2 for ฮฒ>ฮฒc. For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for ฮฒ<ฮฒc.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- April 2016
- DOI:
- arXiv:
- arXiv:1502.03050
- Bibcode:
- 2016CMaPh.343..725D
- Keywords:
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- Mathematics - Probability;
- Mathematical Physics
- E-Print:
- 25 pages, 1 figure, correction in Lemma 2.6