FK–Ising coupling applied to near-critical planar models

Federico Camia; Jianping Jiang; Charles M. Newman

doi:10.1016/j.spa.2019.02.003

FK–Ising coupling applied to near-critical planar models

Federico Camia, Jianping Jiang, Charles M. Newman

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the Ising model at its critical temperature with external magnetic field ha^15∕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)∕h^1∕15→ some B∈(0,∞) as h↓0.

Original language	English (US)
Pages (from-to)	560-583
Number of pages	24
Journal	Stochastic Processes and their Applications
Volume	130
Issue number	2
DOIs	https://doi.org/10.1016/j.spa.2019.02.003
State	Published - Feb 2020

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2019.02.003

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title = "FK–Ising coupling applied to near-critical planar models",

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.",

keywords = "Correlation length, Exponential decay, FK-Ising coupling, Ising model, Magnetization exponent, Magnetization field, Near-critical",

author = "Federico Camia and Jianping Jiang and Newman, {Charles M.}",

note = "Funding Information: 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. Publisher Copyright: {\textcopyright} 2019 Elsevier B.V.",

year = "2020",

month = feb,

doi = "10.1016/j.spa.2019.02.003",

language = "English (US)",

volume = "130",

pages = "560--583",

journal = "Stochastic Processes and their Applications",

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AU - Camia, Federico

AU - Jiang, Jianping

AU - Newman, Charles M.

N1 - Funding Information: 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. Publisher Copyright: © 2019 Elsevier B.V.

PY - 2020/2

Y1 - 2020/2

N2 - 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.

AB - 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.

KW - Correlation length

KW - Exponential decay

KW - FK-Ising coupling

KW - Ising model

KW - Magnetization exponent

KW - Magnetization field

KW - Near-critical

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ER -

FK–Ising coupling applied to near-critical planar models

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Keywords

ASJC Scopus subject areas

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