标题： 倾斜固-固模型是液体：斜面上SOS模型的尺度极限 显示英文标题

标题： Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope

摘要： $(2+1)$D固体-固体（SOS）模型以表现出粗糙化相变而著称：在一个$N\times N$环面上，原点处的高度被固定在$0$，$h(x)$即$x$处的高度的方差在大的反温度$\beta$下为$O(1)$，而在小的$\beta$下为$\asymp \log |x|$（如高斯自由场（GFF）所示）。 前者——大尺度刚性$\beta$——对于一大类$|\nabla\phi|^p$模型（$p=1$为SOS）是已知的，但被认为一旦表面处于斜坡上（倾斜边界条件）就会失效。 人们推测斜率会破坏刚性并引发表面在小$\beta$时的GFF型行为。 对此唯一的严格结果是Sheffield（2005年）提出的：对于这些整数高度函数的模型，如果斜率$\theta$是无理数，那么Var$(h(x))\to\infty$随$|x|$而变化（没有已知的定量界）。 我们研究一个在足够大的固定$\beta$上的 SOS 表面，在一个具有非零边界条件斜率$\theta$的$N\times N$扭曲环面上，受到每个格点强度为$\epsilon_\beta$的势能$V$的扰动（任意小）。 我们的主要结果是(a)高度梯度的测度$\nabla h$在$N\to\infty$时具有弱极限$\mu_\infty$; 以及(b)来自$\mu_\infty$的样本的缩放极限收敛到整个平面的GFF。特别地，我们恢复了渐近性 Var$(h(x))\sim c\log|x|$。据我们所知，这是第一个倾斜的$|\nabla\phi|^p$模型或其扰动模型，其中在大的$\beta$时得到了极限。证明研究了近似SOS表面的随机单调表面，并表明(i)这些形成了弱相互作用的偶素模型，以及(ii)Giuliani、Mastropietro和Toninelli (2017)的重整化框架导致GFF极限。两部分都需要新的要素，包括将[GMT17]从有限相互作用非平凡地扩展到任何长程可求和相互作用。 显示英文摘要

摘要： The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an $N\times N$ torus with the height at the origin rooted at $0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large inverse-temperature $\beta$, vs. $\asymp \log |x|$ at small $\beta$ (as in the Gaussian free field (GFF)). The former--rigidity at large $\beta$--is known for a wide class of $|\nabla\phi|^p$ models ($p=1$ being SOS) yet is believed to fail once the surface is on a slope (tilted boundary conditions). It is conjectured that the slope would destabilize the rigidity and induce the GFF-type behavior of the surface at small $\beta$. The only rigorous result on this is by Sheffield (2005): for these models of integer height functions, if the slope $\theta$ is irrational, then Var$(h(x))\to\infty$ with $|x|$ (with no known quantitative bound). We study a family of SOS surfaces at a large enough fixed $\beta$, on an $N\times N$ torus with a nonzero boundary condition slope $\theta$, perturbed by a potential $V$ of strength $\epsilon_\beta$ per site (arbitrarily small). Our main result is (a) the measure on the height gradients $\nabla h$ has a weak limit $\mu_\infty$ as $N\to\infty$; and (b) the scaling limit of a sample from $\mu_\infty$ converges to a full plane GFF. In particular, we recover the asymptotics Var$(h(x))\sim c\log|x|$. To our knowledge, this is the first example of a tilted $|\nabla\phi|^p$ model, or a perturbation thereof, where the limit is recovered at large $\beta$. The proof looks at random monotone surfaces that approximate the SOS surface, and shows that (i) these form a weakly interacting dimer model, and (ii) the renormalization framework of Giuliani, Mastropietro and Toninelli (2017) leads to the GFF limit. New ingredients are needed in both parts, including a nontrivial extension of [GMT17] from finite interactions to any long range summable interactions.