倾斜固一固是液体:具有斜坡势能的 SOS 的缩放极限

Benoît Laslier, Eyal Lubetzky
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引用次数: 0

摘要

$(2+1)$D固-固(SOS)模型表现出著名的粗糙化转变:在一个原点高度为$0的$N\times N$环上,$h(x)$的方差,即$x$处的高度,在大逆温$\beta$时为$O(1)$,而在小$\beta$时为$\asymp \log ||x|$(如在高斯自由场(GFF)中)。前者--大$\beta$时的刚性--对于一大类$|\nabla\phi|^p$模型($p=1$为SOS)是已知的,然而一旦表面处于斜坡上(倾斜边界条件),就会失效。据推测,斜坡会破坏刚度的稳定性,并在较小的 $\beta$ 时诱发表面的 GFF 型行为。关于这一点的唯一严格结果是 Sheffield(2005)提出的:对于这些整数高度函数模型,如果斜率 $\theta$ 是无理的,那么 Var$(h(x))\to\infty$ 与 $|x|$ 有关(没有已知的定量约束)。我们研究了在一个足够大的固定$\beta$下的SOS曲面族,它位于一个具有非零边界条件斜率$theta$的$N\times N$环上,并受到一个每个位点强度为$\epsilon_\beta$(任意小)的势$V$的扰动。我们的主要结果是:(a)高度梯度的度量 $\nabla h$ 随着 $N\to\infty$ 的增大而具有敬畏极限 $\mu_\infty$;(b)从 $\mu_\infty$ 开始的采样的缩放极限收敛到全平面 GFF。特别是,我们恢复了 Var$(h(x))\sim c\log|x|$ 的渐近线。据我们所知,这是倾斜$|\nabla\phi|^p$模型或其扰动的第一个实例,它在大\beta$时恢复了极限。证明着眼于近似 SOS 曲面的随机单调曲面,并表明:(i) 这些曲面构成了一个弱相互作用的二聚体模型;(ii) Giuliani、Mastropietro 和 Toninelli(2017)的重正化框架导致了 GFF 极限。这两部分都需要新的成分,包括将[GMT17]从有限相互作用非难扩展到任何长程可求和相互作用。
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Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope
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.
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