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 independantly established that all infinitevolume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature β ≥ 0 are of the form αµ + β + (1 − α)µ − β , where µ + β and µ − β are the two pure phases and 0 ≤ α ≤ 1. We present here a new approach to this result, with a number of advantages:1. We obtain a finite-volume, quantitative analogue (implying the classical claim);2. the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon;3. this new approach seems substantially more robust. Known results for Gibbs measures and the 2d Ising model

Wetting and layering for Solid-on-Solid II: Layering transitions, Gibbs states, and regularity of the free energy

“…We believe that there are no infinite mean translation invariant Gibbs state for h > h w (β) and that the finite mean assumption is present only for technical reasons. We would also tend to believe that in analogy with low temperature two dimensional Ising model (see [2,13,22] for results and proofs) P n−1,h * n β and P n,h * n β are in fact the only ergodic Gibbs states when h = h * n , but proving such a statement is out of the scope of this paper. Finally we conclude the exposition with a result showing that our Gibbs states exhibit exponential decay of correlation.…”

Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions

“…Note that the behavior of the macroscopic interfaces induced by an arbitrary boundary condition was studied in [12]. We refer to [7] for a review on the microscopic theory of equilibrium crystal shapes.…”

Graphical Representations for Ising and Potts Models in General External Fields

“…Fortuin and Kasteleyn [22], marked the beginning of four decades of intense activity that produced a rather complete theory for translation invariant systems. These representations were successfully employed to obtain non-perturbative and deep results for Ising and Potts models on the hypercubic lattice using percolation-type methods, namely the discontinuity of the magnetization at the phase transition point for the one-dimensional Ising and Potts models with 1/r 2 interactions [3], the knowledge of the asymptotic behavior of the eigenvalues of the covariance matrix of the Potts model [10], the Aizenman-Higuchi Theorem on the Choquet decomposition of the two-dimensional Ising and Potts models [1,15,16,25,32] and the proof that the self-dual point on the square lattice p sd (q) = √ q/(1 + √ q) is the critical point for percolation in the randomcluster model (q 1) [5], see also the review [40]. For a detailed introduction to the random-cluster model we refer the reader to [18,24,27,30].…”