FK–Ising coupling applied to near-critical planar models

Federico Camia, Jianping Jiang*, Charles M. Newman

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We consider the Ising model at its critical temperature with external magnetic field ha15∕8 on aZ2. We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for a=1, the correlation length is ≥const.h−8∕15 as h↓0. We extend to the a↓0 continuum limit the FK–Ising coupling for all h>0, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent 1∕8 for the one-arm event. Finally, we show that for a=1, the average magnetization, M(h), in Z2 satisfies M(h)∕h1∕15→ some B∈(0,∞) as h↓0.

Original languageEnglish
Pages (from-to)560-583
Number of pages24
JournalStochastic Processes and Their Applications
Volume130
Issue number2
Early online date11 Feb 2019
DOIs
Publication statusPublished - 1 Feb 2020

Funding

The research was supported in part by STCSM grant 17YF1413300 to JJ and U.S. NSF grant DMS-1507019 to CMN. The authors thank Rob van den Berg, Francesco Caravenna, Gesualdo Delfino, Roberto Fernandez, Alberto Gandolfi, Christophe Garban, Barry McCoy, Tom Spencer, Rongfeng Sun and Nikos Zygouras for useful comments and discussions related to this work. The authors benefited from the hospitality of several units of NYU during their work on this paper: the Courant Institute and CCPP at NYU-New York, NYU-Abu Dhabi, and NYU-Shanghai. The authors also thank the referee for valuable comments and suggestions. The research was supported in part by STCSM grant 17YF1413300 to JJ and U.S. NSF grant DMS-1507019 to CMN. The authors thank Rob van den Berg, Francesco Caravenna, Gesualdo Delfino, Roberto Fernandez, Alberto Gandolfi, Christophe Garban, Barry McCoy, Tom Spencer, Rongfeng Sun and Nikos Zygouras for useful comments and discussions related to this work. The authors benefited from the hospitality of several units of NYU during their work on this paper: the Courant Institute and CCPP at NYU-New York, NYU-Abu Dhabi, and NYU-Shanghai. The authors also thank the referee for valuable comments and suggestions.

FundersFunder number
NYU-Shanghai
National Science FoundationDMS-1507019
Science and Technology Commission of Shanghai Municipality17YF1413300

    Keywords

    • Correlation length
    • Exponential decay
    • FK-Ising coupling
    • Ising model
    • Magnetization exponent
    • Magnetization field
    • Near-critical

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