一类具有非凸位势的梯度场的标度极限
Scaling limit for a class of gradient fields with nonconvex potentials
摘要：
我们考虑梯度场$(ϕ_x:x\in \mathbb{Z}^d)$，其定律采用吉布斯-玻尔兹曼形式$Z^{-1}\exp\{-\sum_{< x,y>}V(ϕ_y-ϕ_x)\}$，其中和在最近的邻居上运行。我们假设潜在的$V$允许表示\[V(η):=-\log\int\varrho({d}κ)\exp\biggl[-{1/2}κ\et a^2\biggr],\]，其中$\varrho$是$(0,\infty)$中具有紧支撑的正测度。因此，势$V$是对称的，但一般是非凸的。而对于严格凸的$V$，不变的遍历梯度Gibbs测度完全由它们的倾斜度来刻画，一个非凸的势函数可能导致几个零倾斜度的遍历梯度Gibbs测度.尽管如此，上面的势$V$的每一个遍历的、零倾斜梯度的吉布斯测度都可以标度到高斯自由场。
英文摘要：
We consider gradient fields $(ϕ_x:x\in \mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}\exp\{-\sum_{< x,y>}V(ϕ_y-ϕ_x)\}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation \[V(η):=-\log\int\varrho({d}κ)\exp\biggl[-{1/2}κ\et a^2\biggr],\] where $\varrho$ is a positive measure with compact support in $(0,\infty)$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$'s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.
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