Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

Abstract

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is c x −(d−1)/2 . The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik [7].

Ornstein-Zernike behavior for Ising models with infinite-range interactions

“…This flaw was corrected more recently by the second and third authors [40], who showed how one could recover independence using techniques originating in the field of perfect simulations [17]; an alternative derivation, more combinatorial in nature, can be found in [26] (a version of which can also be found in Section 7). Finally, these methods were adapted to the random-current representation of the Ising model, allowing the first proof of OZ behavior for the Ising model in the presence of a magnetic field [38].…”

“…Sketch of the proof and organization of the paper. The proof follows the same key steps as the procedure developed in [11,13] with the technical and conceptual refinements of [15,40,38]: one uses a graphical representation of correlations, then a coarse-graining of the graphs appearing in this representation, followed by an analysis of the fine geometry of the graphs to construct a suitable renewal structure. The output of the construction is a coupling of the graphs with a random walk.…”

Failure of Ornstein-Zernike asymptotics for the pair correlation function at high temperature and small density

“…We consider the Ising model on Z with interactions as in (7); for simplicity, we assume that ψ(y) = ψ(|y|) for all y and that ψ is monotone in R >0 . Introduce the generating functions (Laplace transforms)…”

Asymptotics of even–even correlations in the Ising model

“…Again, there is a unique Gibbs measure µ β,h . The best nonperturbative result to date is due to Ott [30] and states that the 2-point function displays OZ decay: there exists an analytic function u → Ψ β,h (u) such that, uniformly in the unit vector u,…”