A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model

2015/02/28 by Hugo Duminil-Copin, Vincent Tassion · 1 citation
Mathematics · Physics and Astronomy · #Bernoulli's principle #Directed percolation #Exponential function #Ising model #Markov Chains and Monte Carlo Methods #Percolation (cognitive psychology) #Percolation threshold #Phase transition #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Upper and lower bounds #math-ph #math.MP #math.PR

paper · pdf · doi:10.1007/s00220-015-2480-z

25 pages, 1 figure, correction in Lemma 2.6

crossref issued 2015/11/05 · crossref published 2015/11/05 · crossref published-online 2015/11/05 · openalex publication_date 2015/11/05 · crossref created 2015/11/05 · crossref published-print 2016/04/01 · openalex created_date 2016/06/24 · arxiv created 2018/01/21 · arxiv updated 2018/01/23 · crossref deposited 2019/09/01 · crossref indexed 2026/07/13 · openalex updated_date 2026/08/05

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 β<β_c, and the mean-field lower bound ℙ_β[0\longleftrightarrow∞]≥ (β-β_c)/β for β>β_c. For finite-range models, we also prove that for any β<β_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 β<β_c, and the mean-field lower bound ⟨ σ₀⟩_β⁺≥ √((β²-β_c²)/β²) for β>β_c. For finite-range models, we also prove that the two-point correlations functions decay exponentially fast in the distance for β<β_c.

Citations

• Sharpness of the phase transition in percolation models
• On the critical behavior of the magnetization in high-dimensional Ising models
• Correlation inequalities and the decay of correlations in ferromagnets
• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition
• Percolation

Cited by

• Ising model [wikipedia]

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