### Abstract

We consider the Ising model at its critical temperature with external magnetic field ha
^{15∕8}
on aZ
^{2}
. 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 Z
^{2}
satisfies M(h)∕h
^{1∕15}
→ some B∈(0,∞) as h↓0.

Original language | English (US) |
---|---|

Journal | Stochastic Processes and their Applications |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### 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

### Cite this

**FK–Ising coupling applied to near-critical planar models.** / Camia, Federico; Jiang, Jianping; Newman, Charles.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - FK–Ising coupling applied to near-critical planar models

AU - Camia, Federico

AU - Jiang, Jianping

AU - Newman, Charles

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider the Ising model at its critical temperature with external magnetic field ha 15∕8 on aZ 2 . 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 Z 2 satisfies M(h)∕h 1∕15 → some B∈(0,∞) as h↓0.

AB - We consider the Ising model at its critical temperature with external magnetic field ha 15∕8 on aZ 2 . 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 Z 2 satisfies M(h)∕h 1∕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

UR - http://www.scopus.com/inward/record.url?scp=85062019738&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062019738&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2019.02.003

DO - 10.1016/j.spa.2019.02.003

M3 - Article

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

ER -