论文标题
高度功能的二分法理论
A dichotomy theory for height functions
论文作者
论文摘要
高度函数是给定图上的随机函数，在我们的情况下，在二维方晶格上的整数值函数。我们考虑（非正式地）位于离散高斯和固体固体模型（包含）之间的梯度电位。在1981年Fröhlich和Spencer的突破性工作中，严格建立了该模型中的相变，称为粗糙过渡，Berezinskii-Kosterlitz-无处不在的过渡或本地化 - 迁移转变。直到2005年，谢菲尔德才得出了相变的连续性。首先，我们建立了清晰度，从某种意义上说，协方差在本地化阶段呈指数衰减。其次，我们表明该模型是在批判性上被定位的，因为在自然拓扑中诱导定位的一组势能是开放的。第三，我们证明，高度函数的重点差异至少为$ c \ log n $在DELAICALISE机制中，其中$ n $是到边界的距离，其中$ c> 0 $表示通用常数。这意味着任何潜力的有效温度都不能在于间隔$（0，C）$（只要定义明确时），而在关键点，从$ 0 $上升到至少$ c $。我们将此范围称为有效温度差距。
Height functions are random functions on a given graph, in our case integer-valued functions on the two-dimensional square lattice. We consider gradient potentials which (informally) lie between the discrete Gaussian and solid-on-solid model (inclusive). The phase transition in this model, known as the roughening transition, Berezinskii-Kosterlitz-Thouless transition, or localisation-delocalisation transition, was established rigorously in the 1981 breakthrough work of Fröhlich and Spencer. It was not until 2005 that Sheffield derived continuity of the phase transition. First, we establish sharpness, in the sense that covariances decay exponentially in the localised phase. Second, we show that the model is delocalised at criticality, in the sense that the set of potentials inducing localisation is open in a natural topology. Third, we prove that the pointwise variance of the height function is at least $c\log n$ in the delocalised regime, where $n$ is the distance to the boundary, and where $c>0$ denotes a universal constant. This implies that the effective temperature of any potential cannot lie in the interval $(0,c)$ (whenever it is well-defined), and jumps from $0$ to at least $c$ at the critical point. We call this range of forbidden values the effective temperature gap.
