论文标题
刚性iSing界面的近似域Markov属性
Approximate domain Markov property for rigid Ising interfaces
论文作者
论文摘要
考虑一下$ \ mathbb z^d $的侧面长度$ n $的中心盒子，带有$ \ mp $ bundary条件，这些条件在上半个空间中，在下半个空间中。多布鲁什（Dobrushin）著名地表明，在尺寸$ d \ ge 3 $中，在低温下，伊辛接口（将加号/减阶段分开的双面表面）是刚性的，即，它具有$ O（1）$ $高度波动。最近，作者将这些振荡分解为支柱，并确定了它们的典型形状，从而导致了大量和最大程度的紧密度。 假设我们在界面的高度-H $级别曲线上进行条件，将$ s \ subset \ Mathbb z^{d-1} $以及圆柱体外部的整个接口$ s \ times \ times \ mathbb z $：$ s \ times s \ times \ times \ mathbb z $看起来像什么？许多随机表面（例如SOS和DGFF）的许多模型从根本上满足了Markov的域名，因此它们在$ s $上的高度仅取决于$ s^c $通过$ \ partial s $的高度。 Ising接口重要的是不满足该属性；接口的定律取决于$ s \ times \ mathbb z $之外的完整旋转配置。 在这里，我们在Ising接口的级别曲线内建立了近似域Markov属性。我们首先将Dobrushin的结果扩展到此设置，显示$ S \ Times \ Mathbb Z $中的接口刚好在高度$ h $上，其高度振荡上有指数式的尾巴。然后，我们证明，对于无条件的Ising界面的高柱，$ S \ Times \ Mathbb Z $中的典型高柱绝对连续。使用此功能，我们确定了$ S \ times \ Mathbb z $在高度$ h $的最大振荡上的大量，紧密度和牙龈尾部的定律，这仅取决于$ s $的基数。
Consider the Ising model on a centered box of side length $n$ in $\mathbb Z^d$ with $\mp$-boundary conditions that are minus in the upper half-space and plus in the lower half-space. Dobrushin famously showed that in dimensions $d\ge 3$, at low-temperatures the Ising interface (dual-surface separating the plus/minus phases) is rigid, i.e., it has $O(1)$ height fluctuations. Recently, the authors decomposed these oscillations into pillars and identified their typical shape, leading to a law of large numbers and tightness of their maximum. Suppose we condition on a height-$h$ level curve of the interface, bounding a set $S \subset \mathbb Z^{d-1}$, along with the entire interface outside the cylinder $S\times \mathbb Z$: what does the interface in $S\times \mathbb Z$ look like? Many models of random surfaces (e.g., SOS and DGFF) fundamentally satisfy the domain Markov property, whereby their heights on $S$ only depend on the heights on $S^c$ through the heights on $\partial S$. The Ising interface importantly does not satisfy this property; the law of the interface depends on the full spin configuration outside $S\times \mathbb Z$. Here we establish an approximate domain Markov property inside the level curves of the Ising interface. We first extend Dobrushin's result to this setting, showing the interface in $S\times \mathbb Z$ is rigid about height $h$, with exponential tails on its height oscillations. Then we show that the typical tall pillars in $S\times \mathbb Z$ are uniformly absolutely continuous with respect to tall pillars of the unconditional Ising interface. Using this we identify the law of large numbers, tightness, and Gumbel tail bounds on the maximum oscillations in $S\times \mathbb Z$ about height $h$, showing that these only depend on the conditioning through the cardinality of $S$.
