壁面上方SOS曲面的标度极限和立方根涨落
Scaling limit and cube-root fluctuations in SOS surfaces above a wall
摘要：
考虑在$\bbZ^2$的$L\times L$盒上的硬壁上方的经典$(2+1)$维固体上固体模型。该模型通过将非负整数高度$η_x$分配给框中的每个位点$x$并将0高度分配给其边界来描述晶体表面。表面构型$η$的概率与$\exp(-β\mathcal{H}(η))$成比例，其中$β$是逆温度，而$\mathcal{H}(η)$是相邻位点之间高度差的绝对值之和。我们给出了一个完整的描述SOS表面的形状足够低的温度。首先，我们表明，几乎所有网站的高度是集中在两个级别，$H(L)=\lfloor (1/4β)\log L\rfloor$和$H(L)-1$的概率很高。此外，对于$L$的大多数值，高度集中在单个值$H(L)$上。接下来，我们研究与高度$(H(L),H(L)-1,...)$对应的能级线系综。我们证明了对于每一高度，存在唯一的宏观能级线。此外，当采取发散序列的系统尺寸$L_k$，在高度$H(L_k)-n$的重新缩放的宏观水平线具有限制的形状，如果$(1/4β)\log L_k$的小数部分收敛到一个非临界值。标度极限是单位正方形$Q$的显式凸子集，其边界在$Q$的边界上具有平坦分量。最后，最高宏观能级线沿其极限形状边界的平坦部分具有$L_k^{1/3+o(1)}$涨落。
英文摘要：
Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer height $η_x$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp(-β\mathcal{H}(η))$, where $β$ is the inverse-temperature and $\mathcal{H}(η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures. First we show that with high probability the height of almost all sites is concentrated on two levels, $H(L)=\lfloor (1/4β)\log L\rfloor$ and $H(L)-1$. Moreover, for most values of $L$ the height is concentrated on the single value $H(L)$. Next, we study the ensemble of level lines corresponding to the heights $(H(L),H(L)-1,...)$. We prove that w.h.p. there is a unique macroscopic level line for each height. Furthermore, when taking a diverging sequence of system sizes $L_k$, the rescaled macroscopic level line at height $H(L_k)-n$ has a limiting shape if the fractional parts of $(1/4β)\log L_k$ converge to a noncritical value. The scaling limit is an explicit convex subset of the unit square $Q$ and its boundary has a flat component on the boundary of $Q$. Finally, the highest macroscopic level line has $L_k^{1/3+o(1)}$ fluctuations along the flat part of the boundary of its limiting shape.
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