标题： 面积约束随机行走模型中的精确立方根波动 显示英文标题

标题： Exact cube-root fluctuations in an area-constrained random walk model

摘要： 本文致力于研究一个(1+1)维的随机游走模型，该模型被条件限制以包围一个阶为$N^2$的面积。 这种条件强制了一个全局凹的轨迹。 我们研究了游走相对于其凸包的局部偏离。 为此，我们引入了两个量——平均面长度$\mathsf{MeanFL}$和平均局部粗糙度$\mathsf{MeanLR}$——用来测量随机游走在凸包边界周围的典型纵向和横向波动。 我们的主要结果是$\mathsf{MeanFL}$的阶为$N^{2/3}$且$\mathsf{MeanLR}$的阶为$N^{1/3}$。 此外，遵循 Hammond (Ann. 概率，2012)，我们确定了最大面长度和最大局部粗糙度的标度中的多对数修正，表明前者标度为$N^{2/3}(\log N)^{1/3}$，而后者标度为$N^{1/3}(\log N)^{2/3}$。研究对象旨在作为二维统计力学模型（如伊辛模型）在相分离区域的界面的玩具模型——我们在本文结尾讨论了这个问题。 显示英文摘要

摘要： This article is devoted to the study of the behaviour of a (1+1)-dimensional model of random walk conditioned to enclose an area of order $N^2$. Such a conditioning enforces a globally concave trajectory. We study the local deviations of the walk from its convex hull. To this end, we introduce two quantities -- the mean facet length $\mathsf{MeanFL}$ and the mean local roughness $\mathsf{MeanLR}$ -- measuring the typical longitudinal and transversal fluctuations around the boundary of the convex hull of the random walk. Our main result is that $\mathsf{MeanFL}$ is of order $N^{2/3}$ and $\mathsf{MeanLR}$ is of order $N^{1/3}$. Moreover, following the strategy of Hammond (Ann. Prob., 2012), we identify the polylogarithmic corrections in the scaling of the maximal facet length and of the maximal local roughness, showing that the former one scales as $N^{2/3}(\log N)^{1/3}$, while the latter scales as $N^{1/3}(\log N)^{2/3}$. The object of study is intended to be a toy model for the interface of a two-dimensional statistical mechanics model (such as the Ising model) in the phase separation regime -- we discuss this issue at the end of this work.