AI 中文总结
该研究针对对应3D伊辛模型低温的大倾斜强度，证明面积倾斜线系的无限版本相关性指数衰减、有限版本有一致正谱间隙，采用嵌入超临界分支过程的概率方法。
AI 中文摘要
硬基底上的随机表面常表现出熵排斥，即表面被向上推动以允许熵更优的向下波动，这类例子中一类特别丰富的情况来自低温3D伊辛模型。研究此类表面的一种有力方法是通过其水平曲线，这些曲线构成一族不相交的随机曲线。在[CIW18, CIW19]中，提出了具有几何递增面积倾斜的布朗线系，作为该情况下的候选极限模型。该模型超出了可积结构或SDE结构相关技术的适用范围，而这些技术曾是研究Airy线系的关键要素。关于此类线系的一个特别引人关注的问题是，当将其视为马尔可夫过程时的混合性质，尤其是时间上的相关性衰减速率。对于Airy线系，已知这种衰减是逆二次的；而在[CG25]中，针对面积倾斜模型的相关性衰减的首个定量界慢于时间的多项式。更早的结果[DLZ24]已确立了有限线版本线系的谱间隙为正，但未给出定量界，这留下了无限线系相关性真实衰减速率的重要问题。针对足够大的面积倾斜强度（对应3D伊辛模型足够低的温度），我们证明了无限线系的相关性呈指数衰减，且有限线系具有与线数无关的一致正谱间隙。我们的证明基于确立不同指标曲线之间尺度分离的精确形式，采用了一种新颖的概率方法，涉及在该线系中嵌入超临界分支过程。
英文摘要
Random surfaces on a hard substrate often exhibit entropic repulsion, wherein the surface is propelled upwards to allow entropically preferable downward fluctuations. A particularly rich class of examples arises from the low-temperature 3D Ising model. A powerful approach to studying such surfaces is through their level curves, which form a family of non-intersecting random curves. In [CIW18, CIW19], an ensemble of Brownian lines with geometrically increasing area tilts was proposed as a putative limiting model in this case. This model falls outside the scope of techniques based on integrable or SDE structures, which have been key ingredients in the study of the Airy line ensemble. A particularly intriguing question about such line ensembles concerns their mixing properties when viewed as a Markov process, and in particular the rate of decay of correlations in time. For the Airy line ensemble, this decay is known to be inverse quadratic. The first quantitative bound on the decay of correlations in the area-tilted model, established in [CG25], was slower than polynomial in time. An earlier result [DLZ24] had established positivity of the spectral gap for the finite-line version of the ensemble, without quantitative bounds. This left open the important question of the true decay rate of correlations for the infinite ensemble. Settling this question for sufficiently large area-tilt strength, corresponding to sufficiently low temperature for the 3D Ising model, we prove exponential decay of correlations for the infinite ensemble and a uniform (in the number of lines) positive spectral gap for the finite ensemble. Our proof is based on establishing a precise form of separation of scales between curves of different indices, using a novel probabilistic approach involving embedding supercritical branching processes in the line ensemble.
Comments42 pages, 6 figures
