论文标题
关键前润湿中2D ISING接口的本地和全局几何形状
Local and global geometry of the 2D Ising interface in critical pre-wetting
论文作者
论文摘要
考虑在$ n \ times n $ box上的低温和正外田$λ$的伊辛模型，其dobrushin边界条件在北，东，西边界和南部边界的北部，东和西边界以及减去。如果$λ= 0 $，则分开加号和减去阶段的接口是扩散的，具有$ O（\ sqrt n）$高度波动，并且模型已完全润湿。在一个订单的一个字段下，接口波动为$ o（1）$，界面仅部分润湿，被固定在其南部边界上。我们研究了$λ_n\ downarrow 0 $的关键预润湿状态，其中预计高度波动将扩展为$λ^{-1/3} $，并且预计重新验证的界面将融合到法拉利 - 苏普兰扩散。 Velenik（2004）将接口下的区域的顺序确定为对数校正。从那时起，仅在区域倾斜下的更简单的随机步行模型中发现了此类接口的更精致的特征。 在本文中，我们解决了Velenik的几种猜想，内容涉及关键预润湿制度中Ising界面的精致特征。我们的主要结果是在单点高度波动上急剧结束，证明了$ e^{ - θ（x^{3/2}）} $上尾巴，让人联想到tracy- widom分布，从而捕获了本地布朗尼振荡与全球场效应之间的折衷。我们进一步证明了该界面高度高的点数的浓度估计值。这些用于推断界面的各种几何特性，包括其限制区域的顺序和尾巴，以及控制其最大高度波动的多同源术前因素。我们的参数结合了从ISING界面的随机线表示的经典输入，以及新型的本地重采样和耦合方案。
Consider the Ising model at low-temperatures and positive external field $λ$ on an $N\times N$ box with Dobrushin boundary conditions that are plus on the north, east, and west boundaries and minus on the south boundary. If $λ= 0$, the interface separating the plus and minus phases is diffusive, having $O(\sqrt N)$ height fluctuations, and the model is fully wetted. Under an order one field, the interface fluctuations are $O(1)$ and the interface is only partially wetted, being pinned to its southern boundary. We study the critical pre-wetting regime of $λ_N \downarrow 0$, where the height fluctuations are expected to scale as $λ^{ -1/3}$ and the rescaled interface is predicted to converge to the Ferrari--Spohn diffusion. Velenik (2004) identified the order of the area under the interface up to logarithmic corrections. Since then, more refined features of such interfaces have only been identified in simpler models of random walks under area tilts. In this paper, we resolve several conjectures of Velenik regarding the refined features of the Ising interface in the critical pre-wetting regime. Our main result is a sharp bound on the one-point height fluctuation, proving $e^{ - Θ(x^{3/2})}$ upper tails reminiscent of the Tracy--Widom distribution, capturing a tradeoff between the locally Brownian oscillations and the global field effect. We further prove a concentration estimate for the number of points above which the interface attains a large height. These are used to deduce various geometric properties of the interface, including the order and tails of the area it confines, and the poly-logarithmic pre-factor governing its maximum height fluctuation. Our arguments combine classical inputs from the random-line representation of the Ising interface, with novel local resampling and coupling schemes.
