Exact large deviation bounds up to \(T_ c\) for the Ising model in two dimensions
type: Item
instance of: scholarly article
title: Exact large deviation bounds up to \(T_ c\) for the Ising model in two dimensions
zbMATH Open document ID: 0830.60018
DOI: 10.1007/BF01192464
publication date: 1996-01-15T00:00:00Z
review text: Consider the Ising model on \(Z^2\). Let \(\Lambda\) be a square box of volume \(L^2\). Let \(P^\beta_{\Lambda, +}\) be the corresponding Gibbs measure on \(\Lambda\) with boundary conditions and inverse temperature \(\beta\). Define the block-spin magnetization \(X_\Lambda\) by \(X_\Lambda = {1 \over |\Lambda |} \sum_{x \in \Lambda} \sigma_x\), where \(\sigma_x = \pm 1\). Since \(P^\beta_+\) is ergodic, \(X_\Lambda\) converges to \(m^*\) \(P^\beta_+\)-a.s. The main result of the present paper is the following Theorem. Let \(m \in [-m^*, m^*]\) and \(\beta > \beta_c\). Then \[ \limsup_{L \to \infty} {1 \over L} \log P^\beta_{\Lambda, +} (X_\Lambda \leq m) \leq -2 \bigl( \lambda_F \alpha (m) \bigr)^{1/2}, \] where \(\alpha (m) = (m^*- m)/2 m^*\) and \(\lambda_F\) is the volume of the Wulff shape, generated by the surface tension function \(F\) of the model. This result combined with the lower large deviation bounds for \(X_\Lambda\), obtained by the author [J. Stat. Phys. 74, 411-432 (1994)], gives exact large deviation bounds for the \(2D\) Ising model.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B24
zbMATH DE Number: 784420
zbMATH Keywords: Ising model
zbMATH Keywords: large deviations
zbMATH Keywords: Wulff construction
zbMATH Keywords: FK percolation
author: Dmitry Ioffe
reviewed by: S. Poghosyan
MaRDI profile type: Publication
cites work: Discontinuity of the magnetization in one-dimensional \(1/| x-y| ^ 2\) Ising and Potts models.
cites work: The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation
cites work: Exponential decay of connectivities in the two-dimensional Ising model
cites work: Q3744974
cites work: Large deviations for the empirical field of a Gibbs measure
cites work: Large deviations for the 2D Ising model: A lower bound without cluster expansions
cites work: Q3976835
cites work: Large deviations for Gibbs random fields
cites work: Random fields
cites work: Complete analyticity for 2D Ising completed
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Recommended article: Large deviations for the 2D Ising model: A lower bound without cluster expansions
Recommended article: Large deviations and continuum limit in the 2D Ising model
Recommended article: A CLT for the total energy of the two-dimensional critical Ising model
Recommended article: A power-law upper bound on the correlations in the 2D random field Ising model
Recommended article: Scaling limit of the energy variable for the two-dimensional Ising ferromagnet
Recommended article: Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model
Recommended article: On the ground-state threshold in random two-dimensional Ising \(\pm J\) models
description: scientific article; zbMATH DE number 784420