论文标题
对于具有非凸电势的梯度模型的自由能的缩放限制和严格的凸度
Scaling limit and strict convexity of free energy for gradient models with non-convex potential
论文作者
论文摘要
我们考虑了晶格$ \ mathbb {z}^d $上的梯度模型。这些模型是接口的有效模型，也称为连续的ISING模型。界面的高度是由具有二次相互作用的非凸透扰动的能量的随机场建模的。我们对此模型的反向温度$β$在倾斜边界条件$ u $的吉布斯度量感兴趣。在[AKM16]，[HIL16]和[ABKM19]中，作者表明，对于小倾斜$ u $和较大的反向温度$β$，表面张力严格凸出，其中限制在子序列中采取限制。此外，表明缩放极限（再次在子序列上）是连续圆环上的高斯自由场。证明的方法是根据Brydges和同事制定的一般策略的重新分析组方法的严格实施。 在本文中，重量级化组分析从有限体积流量扩展到无限 - 体积版本，以消除[AKM16]，[HIL16]和[ABKM19]结果中子序列的必要性。
We consider gradient models on the lattice $\mathbb{Z}^d$. These models serve as effective models for interfaces and are also known as continuous Ising models. The height of the interface is modelled by a random field with an energy which is a non-convex perturbation of the quadratic interaction. We are interested in the Gibbs measure with tilted boundary condition $u$ at inverse temperature $β$ of this model. In [AKM16], [Hil16] and [ABKM19] the authors show that for small tilt $u$ and large inverse temperature $β$ the surface tension is strictly convex, where the limit is taken on a subsequence. Moreover, it is shown that the scaling limit (again on a subsequence) is the Gaussian free field on the continuum torus. The method of the proof is a rigorous implementation of the renormalisation group method following a general strategy developed by Brydges and coworkers. In this paper the renormalisation group analysis is extended from the finite-volume flow to an infinite-volume version to eliminate the necessity of the subsequence in the results in [AKM16], [Hil16] and [ABKM19].
