非凸势的表面张力的严格凸性
Strict Convexity of the Surface Tension for Non-convex Potentials
摘要：
研究了具有非凸相互作用的格点$\mathbb{Z}^d$上的梯度模型。这些吉布斯场（具有连续自旋的晶格模型）出现在物理学和数学的各个分支中。在量子场论中，它们表现为无质量的场论。尽管我们的动机源于考虑向量值场作为晶体结构原子的位移和这些模型的柯西-玻恩规则的研究，但我们的注意力主要集中在界面上，梯度场作为\emph{effective}界面相互作用。在这种情况下，我们证明了严格的凸性的表面张力（界面自由能）的低温和足够小的界面倾斜使用多尺度（重正化组分析）技术的Brydges和同事\cite{B07}的方法。这是对\cite{CDM09}中高温区研究的补充，是Funaki和Spohn对严格凸相互作用有效的结果\cite{FS97}的推广.
英文摘要：
We study gradient models on the lattice $\mathbb{Z}^d$ with non-convex interactions. These Gibbs fields (lattice models with continuous spin) emerge in various branches of physics and mathematics. In quantum field theory they appear as massless field theories. Even though our motivation stems from considering vector valued fields as displacements for atoms of crystal structures and the study of the Cauchy-Born rule for these models, our attention here is mostly devoted to interfaces, with the gradient field as an \emph{effective} interface interaction. In this case we prove the strict convexity of the surface tension (interface free energy) for low temperatures and sufficiently small interface tilts using muli-scale (renormalisation group analysis) techniques following the approach of Brydges and coworkers \cite{B07}. This is a complement to the study of the high temperature regime in \cite{CDM09} and it is an extension of Funaki and Spohn's result \cite{FS97} valid for strictly convex interactions.
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