发表机构
University of Maryland; Northwestern University（马里兰大学; 西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了d≥3时斜高度函数的局域化猜想，确立了无限体积极限的唯一性，还研究了其结构性质并推导了非交叉曲面的体吉布斯测度的存在性。
AI 中文摘要
我们考虑带有一般凸相互作用的、取值为整数的∇φ高度函数，这些函数被置于一个斜面上。Sheffield（2003）猜想，对于所有斜率而言，这类高度函数在维度d≥3时是局域化的，具体表现为在有限体积内具有紧密的涨落，并能存在无限体积极限。我们在如下条件下证明了该猜想：存在两个坐标，在这两个坐标上斜率向量为零，且相互作用是偶的、处于低温状态。在该设定下，我们还证明了无限体积极限在适当意义下的唯一性，并对其极值分量进行了分类。我们进一步研究了无限体积极限的结构性质，确立了宏观畴壁（水平集的边界）之间的熵排斥，表明它们在由圆旋转动力系统描述的精确意义上被最大程度地分离。最后，我们证明了极值分量中相关性的指数衰减。我们的设定包含一个特例：无限多个零斜率的、取值为整数的曲面（维度为2或更高）被条件限制不得交叉。我们由此推导出，对于这类非交叉曲面，存在任意给定平均间距的体吉布斯测度，且该测度具有相应的结构性质。
英文摘要
We consider integer-valued $\nablaϕ$ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions $d\ge3$, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.
CommentsFixed typos and added references to related works. 44 pages, 3 figures
