AI 中文总结
本文研究低维瞬态加权图上临界高斯自由场符号分形的宏观等价性，证明其尺度极限的豪斯多夫维数为 $1+d/2$，并给出二点穿越估计。
AI 中文摘要
我们研究了一类足够低维（即低于平均场机制）的瞬态加权图 $G$ 上高斯自由场的度量图符号分形，并证明了宏观等价性质：几种自然的定义宏观分形的方式——涉及直径、体积或容量泛函中的任意一种——实际上是相同的。作为推论，在低维中，宏观分形被刻画为在相应的环汤图像中包含大环的那些分形，这与高维中的典型行为形成对比。然后，我们将这些结果应用于 $\varepsilon\mathbb{Z}^d$（$d=3,4,5$）上临界分形的小网格尺度极限，其中 $\varepsilon \downarrow 0$。我们证明了每个子序列尺度极限由具有正布朗容量的集合组成，其豪斯多夫维数和闵可夫斯基维数均等于 $1+\tfrac d2$，正如 Werner 所猜想的那样。我们还在极限中推导了二点穿越估计，并表明每个极限环汤分形是其所包含的宏观环的尺度极限的并集的闭包。值得注意的是，与例如平面伯努利渗流不同，豪斯多夫维数的确定并非源于涉及极限对象（如 $\text{SLE}_6$）的直接计算，而是源于离散模型本身的独特特征。
英文摘要
We study metric graph sign clusters of the Gaussian free field on a large class of transient weighted graphs $G$ that are sufficiently low-dimensional, i.e. below the mean-field regime, and prove a macroscopic equivalence property: several natural ways to define macroscopic clusters - involving any of diameter, volume, or capacity functionals - are in fact the same. As a corollary, macroscopic clusters are characterized in low dimensions as those containing large loops in the corresponding loop soup picture, in contrast with the typical behavior in high dimensions. We then apply these results to the small-mesh scaling limit of critical clusters on $\varepsilon\mathbb{Z}^d$, $d=3,4,5$, as $\varepsilon \downarrow 0$. We prove that every subsequential scaling limit consists of sets with positive Brownian capacity whose Hausdorff and Minkowski dimensions are both equal to $1+\tfrac d2$, as conjectured by Werner. We also derive two-point crossing estimates in the limit, and show that each limiting loop soup cluster is the closure of the union of the scaling limits of the macroscopic loops it contains. Remarkably, unlike with planar Bernoulli percolation for instance, the determination of the Hausdorff dimension does not emerge from direct calculations involving a limiting object (such as $\text{SLE}_6$), but rather from the distinctive features of the discrete model itself.
Comments63 pages
