向中间层的熵排斥

中文摘要 AI 辅助

我们考虑在格点 $\mathbb{Z}^d$ 上具有偶且凸的相互作用能量 $W$ 的 $\nabla\varphi$ 高度函数，这些函数被限制取整数值 $\{-S, \ldots, S\}$，其中 $S \ge 1$ 为整数。我们研究熵排斥的效应，它倾向于将自旋值推向中间层。我们证明该模型具有唯一的吉布斯测度，且关联函数指数衰减，在以下情形成立：(a) 维度 $d=2$，在所有温度下。(b) 维度 $d\ge3$，在所有温度下，对于一大类具有非增二阶导数的 $W$，包括族 $W(x)=|x|^p$（$p\in[1,2]$）。(c) 维度 $d\ge 3$，在低温与高温下，即 $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$，并采用归一化 $W(0)=0$。在低温下，我们的证明提供了 Pirogov--Sinai 方法的一种替代方案。相反，我们展示了一类偶且凸的相互作用能量 $W$，它们在高维和合适的温度范围内具有多个吉布斯测度。尽管唯一性可能失效，我们证明每个吉布斯测度的磁化强度都位于 $(-\frac{1}{2},\frac{1}{2})$ 内。这意味着限制取值于 $\{0,1,\ldots\}$（即条件位于地板之上）的模型在所有维度、任意偶且凸的 $W$ 以及所有温度下都是去局域化的。我们的方法扩展到其他设置：我们证明取值于实数区间 $[-1,1]$ 的高度函数总是具有唯一的吉布斯测度，这一结果此前仅对二次相互作用得到证明。对于取值于 $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$（$S \ge 1$ 为整数）的高度函数，我们证明每个吉布斯测度的磁化强度都位于 $(-1,1)$ 内。我们结果的特殊情形 $W(x)=x^2$ 解决了 Bricmont--El Mellouki--Fröhlich (1986) 工作中留下的问题。

英文摘要

We consider $\nablaφ$ height functions with even and convex interaction energy $W$ on the lattice $\mathbb{Z}^d$, which are restricted to take values in the set $\{-S, \ldots, S\}$ for some integer $S \ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\in[1,2]$. (c) Dimensions $d\ge 3$ at both low and high temperatures, $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\frac{1}{2},\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\{0,1,\ldots\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont--El Mellouki--Fröhlich (1986).

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57 pages