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

Abstract

We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions. Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk. We aimed at a pedagogical and self-contained presentation.

Figures (20)

• Figure 1: The results in this paper apply to the regime $\beta\in(\beta_{\rm sat}(s),\beta_{\mathrm c})$. The asymptotic behavior of the Green function $G$ is actually not of Ornstein--Zernike type in the regime $\beta\in (0,\beta_{\rm sat}(s))$.
• Figure 2: The unit ball and the Wulff shape associated to the $L^4$-norm $\rho(x) = \|x\|_4$. A pair of $\rho$-dual vectors $x$ and $t$ are also represented. We have used the notation $\hat{x} = x/\|x\|$ and $\hat{t} = t/\|t\|$.
• Figure 3: The forward cone $\mathcal{Y}^\blacktriangleleft_{\rho,t,\delta}$ for the same $t$ as in Fig. \ref{['fig:UnitBallWulff']}.
• Figure 4: $v$ is a $(t,\delta)$-cone-point of the path $\gamma$.
• Figure 5: The Diamond $D_{t,\delta}(x,y)$.

Theorems & Definitions (85)

• Remark 1.1
• Definition 2.1: Path models
• Definition 2.2: Conditional weights
• Definition 2.3: Two-point function
• Definition 2.4: Inverse correlation length
• Definition 2.5
• Lemma 2.1
• Definition 2.6: Unit ball, Wulff shape
• Definition 2.7: Extremal radius
• Definition 2.8: Dual vectors; see Fig. \ref{['fig:UnitBallWulff']}