带白噪声体积项的平面等周问题的渐近性
Asymptotics of a planar isoperimetric problem with a white-noise volume term
中文总结 AI 辅助
该论文研究平面随机等周问题中白噪声积分与周长最大比值的渐近行为，通过几何线性化和粗粒化方法，建立了期望值的前导阶渐近及超集中现象。
中文摘要 AI 辅助
在这项工作中，我们研究集合上白噪声积分与集合周长之间的最大比值 $I$，这可以看作一个随机等周问题，并出现在随机场伊辛模型（Ding-Wirth）和极小极大最优匹配（Leighton-Shor）中。在本文考虑的平面情形中，该比值是尺度不变的，因此问题具有临界性。为此，它需要紫外截断，我们通过限制在球 $B_L$ 内且边长至少为1的多边形集合来施加截断。我们的主要结果建立了前导阶渐近 $\u200b\u200b\mathbb{E}I\approx i\ln^{3/4}L$，其中 $i\in(0,\infty)$，以及在尺度 $O(\ln^{-1/4}L)$ 上的超集中现象。这是通过联系到一个更简单的 $(1+1)$ 维作用量实现的，该作用量由最后两位作者和 C. Wagner 研究过。这种联系是通过将周长经典几何线性化为狄利克雷能量而发现的，并由 Ried-Wagner 大尺度正则性理论证明其合理性，从而允许进行粗粒化和逐尺度迭代论证。
英文摘要
In this work we study the maximal ratio $I$ between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball $B_L$. Our main result establishes the leading-order asymptotics $\mathbb{E}I\approx i\ln^{3/4}L$ for some $i\in(0,\infty)$ and superconcentration at scale $O(\ln^{-1/4}L)$. This is done by relating to a simpler $(1+1)$-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.
