Ornstein-Zernike theory for finite range Ising models above T c

Abstract

Abstract. We derive a precise Ornstein-Zernike asymptotic formula for the decay of the two-point function σ 0 σ x β in the general context of finite range Ising type models on Z d . The proof relies in an essential way on the a-priori knowledge of the strict exponential decay of the two-point function and, by the sharp characterization of phase transition due to Aizenman, Barsky and Fernández, goes through in the whole of the high temperature region β < β c . As a byproduct we obtain that for every β < β c , the inverse correlation length ξ β is an analytic and strictly convex function of direction.

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

“…The purpose was to show polynomial correction to the exponential decay of the correlation, see [12] for details. Similar results have been proved for the finite range Ising model above the critical temperature [8], as well as the random cluster model [9]. Furthermore, similar graphical expansions have been proved to satisfy the Ornstein-Zernike equation also in the context of point processes and the random connection model of percolation [24].…”

Diffraction Spectrum of Lattice Gas Models above Tc

“…However, we invoked the stronger result of [4] here because it might help to analyze some finer details of the diffraction in the future.…”

Rigorous nonperturbative Ornstein-Zernike theory for Ising ferromagnets

“…It remains to check that up to exponentially negligible weights typical paths γ : 0 → x contain a density of break points. However, coarsegraining procedures similar to those developed above for SAW, and based on the properties (7), (8) and (9), show that there exist M < ∞ and ν > 0 such that λ: 0→y irreducible q β (λ) ≤ M e −ν|x| e −ξ β (x) , This is the analogue of the mass separation result (12). Both the local limit asymptotics of Theorem 1 and the geometry of K β can be now read from the analytic dependence of the leading eigenvalue ρ(z) of L z on the perturbation z ∈ C d in L z f (λ) = L e (z,V (λ1)) f .…”