A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model

Abstract

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model are convex combinations of the two pure phases. We present here a new approach to this result, with a number of advantages: (i) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach might be applicable to systems for which the classical method fails.

A couple of typos corrected. To appear in Probab. Theory Relat. Fields

References in corpus (3)

• Ornstein-Zernike Theory for the finite range Ising models above T_c
• Fluctuation theory of connectivities for subcritical random cluster models
• On the Ising model with random boundary condition

Cited by in corpus (10)

• Conformal invariance of lattice models
• Graphical Representations for Ising and Potts Models in General External Fields
• On the Gibbs states of the noncritical Potts model on Z^2
• Existence of gradient Gibbs measures on regular trees which are not translation invariant
• Absence of Dobrushin states for $2d$ long-range Ising models
• Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions
• Phase Transitions in Nonlinear Filtering
• Structure of Gibbs measures for planar FK-percolation and Potts models
• Decimations for Two-dimensional Ising and Rotator Models I
• Ornstein-Zernike behavior for Ising models with infinite-range interactions