A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model
By H. Duminil-Copin, Vincent Tassion. Communications in Mathematical Physics, 2015
207 citations recorded by the source corpus.
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<βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta < \beta_c}$$\end{document}, and the mean-field lower bound Pβ[0⟷∞]≥(β-βc)/β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}$$\end{document} for β>βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta > \beta_c}$$\end{document}. For finite-range models, we also prove that for any β<βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta < \beta_c}$$\end{document}, 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<βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta < \beta_c}$$\end{document}, and the mean-field lower bound ⟨σ0⟩β+≥(β2-βc2)/β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}$$\end{document} for β>βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta > \beta_c}$$\end{document}. For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for β<βc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta < \beta_c}$$\end{document}.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/587ec15e3ec1f12cd4537445bd3333748e25e5cc
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Condensed Matter Physics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Marginal triviality of the scaling limits of critical 4D Ising and 𝜑44 models (2019, methodology): “To prove the uniqueness of crossings, we employ the notion of the current’s backbone, on which more can be found in [1, 4, 14, 15, 17].”
- Sharp phase transition for the random-cluster and Potts models via decision trees (2017, methodology): “For the 2-state Potts model, better known as the Ising model, the result goes back to [ABF87] (see also [DT15]).”