FK–Ising coupling applied to near-critical planar models

Federico Camia, Jianping Jiang, Charles M. Newman

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

Fingerprint

Reflection Positivity
Ising model
Continuum Limit
Correlation Length
Critical Temperature
Magnetization
Ising Model
External Field
Tail
Magnetic Field
Exponent
Magnetic fields
Upper bound
Estimate
Temperature
Model

Keywords

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

Cite this

Camia, Federico ; Jiang, Jianping ; Newman, Charles M. / FK–Ising coupling applied to near-critical planar models. In: Stochastic Processes and Their Applications. 2020 ; Vol. 130, No. 2. pp. 560-583.
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FK–Ising coupling applied to near-critical planar models. / Camia, Federico; Jiang, Jianping; Newman, Charles M.

In: Stochastic Processes and Their Applications, Vol. 130, No. 2, 01.02.2020, p. 560-583.

Research output: Contribution to JournalArticleAcademicpeer-review

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AU - Jiang, Jianping

AU - Newman, Charles M.

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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.

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