February 2024 Long-range models in 1D revisited
Hugo Duminil-Copin, Christophe Garban, Vincent Tassion
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 232-241 (February 2024). DOI: 10.1214/22-AIHP1355


In this short note, we revisit a number of classical results on long-range 1D percolation, Ising model and Potts models (Comm. Math. Phys. 84 (1982) 87–101; Comm. Math. Phys. 104 (1986) 547–571; J. Stat. Phys. 50 (1988) 1–40; Comm. Math. Phys. 118 (1988) 303–336). More precisely, we show that for Bernoulli percolation, FK percolation and Potts models, there is symmetry breaking for the 1/r2-interaction at large β, and that the phase transition is necessarily discontinuous. We also show, following the notation of (J. Stat. Phys. 50 (1988) 1–40) that β(q)=1 for all q1.

Dans cette courte note, nous revisitons un certain nombre de résultats classiques sur la percolation unidimensionnelle à longue portée, le modèle d’Ising et les modèles de Potts (Comm. Math. Phys. 84 (1982) 87–101 ; Comm. Math. Phys. 104 (1986) 547–571 ; J. Stat. Phys. 50 (1988) 1–40 ; Comm. Math. Phys. 118 (1988) 303–336). Plus précisément, nous montrons que pour la percolation de Bernoulli, la percolation FK et les modèles de Potts, il y a une rupture de symétrie pour l’interaction en 1/r2 lorsque β est suffisamment grand, et la transition de phase est nécessairement discontinue. Nous montrons également, en suivant la notation de (J. Stat. Phys. 50 (1988) 1–40), que β(q)=1 pour tous les q1.


The first author is funded by the ERC CriBLaM (grant agreement No 757296), the Swiss FNS and the NCCR SwissMap. The research of the second author is supported by the ERC grant LiKo 676999. The third author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 851565) and by the NCCR SwissMap.


Download Citation

Hugo Duminil-Copin. Christophe Garban. Vincent Tassion. "Long-range models in 1D revisited." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 232 - 241, February 2024. https://doi.org/10.1214/22-AIHP1355


Received: 5 July 2021; Revised: 29 September 2022; Accepted: 16 December 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718380
Digital Object Identifier: 10.1214/22-AIHP1355

Primary: 60K35 , 82B20 , 82B43

Keywords: critical , Long-range , One-dimension , percolation , renormalization

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


This article is only available to subscribers.
It is not available for individual sale.

Vol.60 • No. 1 • February 2024
Back to Top