The Ising magnetization exponent on Z2 is 1/15

Federico Camia, Christophe Garban, Charles Newman

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)175-187
JournalProbability Theory and Related Fields
DOIs
StatePublished - 2014

Fingerprint

Ising
Magnetization
Exponent
Triangular Lattice
Critical Exponents
Ising Model
External Field
Upper and Lower Bounds
Magnetic Field
Theorem

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 Z2 is 1/15. / Camia, Federico; Garban, Christophe; Newman, Charles.

In: Probability Theory and Related Fields, 2014, p. 175-187.

Research output: Contribution to journalArticle

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