### Abstract

We prove that for the Ising model defined on the plane {Mathematical expression} at {Mathematical expression} the average magnetization under an external magnetic field {Mathematical expression} behaves exactly like {Mathematical expression} The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that {Mathematical expression} on the triangular lattice (Kesten in Commun Math Phys 109(1):109-156, 1987; Smirnov and Werner in Math Res Lett 8(5-6):729-744, 2001)] relies on the GHS inequality as well as the RSW theorem for FK percolation from Duminil-Copin et al. (Commun Pure Appl Math 64:1165-1198, 2011). The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

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

Pages (from-to) | 175-187 |

Journal | Probability Theory and Related Fields |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Mathematics Subject Classification: 82B20, 82B27, 60K35

### ASJC Scopus subject areas

- Statistics and Probability
- Analysis
- Statistics, Probability and Uncertainty

### Cite this

**The Ising magnetization exponent on Z ^{2} is 1/15.** / Camia, Federico; Garban, Christophe; Newman, Charles.

Research output: Contribution to journal › Article

^{2}is 1/15',

*Probability Theory and Related Fields*, pp. 175-187. https://doi.org/10.1007/s00440-013-0526-8

}

TY - JOUR

T1 - The Ising magnetization exponent on Z2 is 1/15

AU - Camia, Federico

AU - Garban, Christophe

AU - Newman, Charles

PY - 2014

Y1 - 2014

N2 - We prove that for the Ising model defined on the plane {Mathematical expression} at {Mathematical expression} the average magnetization under an external magnetic field {Mathematical expression} behaves exactly like {Mathematical expression} The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that {Mathematical expression} on the triangular lattice (Kesten in Commun Math Phys 109(1):109-156, 1987; Smirnov and Werner in Math Res Lett 8(5-6):729-744, 2001)] relies on the GHS inequality as well as the RSW theorem for FK percolation from Duminil-Copin et al. (Commun Pure Appl Math 64:1165-1198, 2011). The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

AB - We prove that for the Ising model defined on the plane {Mathematical expression} at {Mathematical expression} the average magnetization under an external magnetic field {Mathematical expression} behaves exactly like {Mathematical expression} The proof, which is surprisingly simple compared to an analogous result for percolation [i.e. that {Mathematical expression} on the triangular lattice (Kesten in Commun Math Phys 109(1):109-156, 1987; Smirnov and Werner in Math Res Lett 8(5-6):729-744, 2001)] relies on the GHS inequality as well as the RSW theorem for FK percolation from Duminil-Copin et al. (Commun Pure Appl Math 64:1165-1198, 2011). The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

KW - Mathematics Subject Classification: 82B20, 82B27, 60K35

UR - http://www.scopus.com/inward/record.url?scp=84883524207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883524207&partnerID=8YFLogxK

U2 - 10.1007/s00440-013-0526-8

DO - 10.1007/s00440-013-0526-8

M3 - Article

AN - SCOPUS:84883524207

SP - 175

EP - 187

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

ER -