发表机构
University of California, Berkeley（加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究分形渗流中的最后通行渗流量化问题，开发多尺度框架证明通行时间与线性增长有几乎多项式差距，提出不可矩形定向渗流模型以构造符合要求的定向路径。
AI 中文摘要
尽管具有快速衰减尾部的独立同分布无序下的无向和定向几何模型被认为表现相似，并呈现 Kardar–Parisi–Zhang 普适类的特征，但背景噪声为分形的情况预计会有显著差异。后者的一个典型例子是刘维尔量子引力，其中平面高斯自由场构成分形背景。捕捉部分关键特征的玩具模型是 Mandelbrot 分形渗流，即一种随机康托集。在最简单的设定中，不同尺度的二进立方体以概率 p（模型参数）独立保留，所有保留立方体的交集构成开位点集。这类集合的连通性已被深入研究，文献[CCD88, Cha95]证明了一个令人惊讶的结果：尽管分形渗流集的连通性存在相变（如同普通键渗流），但定向渗流从未发生。几何测度论也获得了类似结果。为理解由高斯自由场驱动的最后通行渗流（LPP，始于文献[GGN24]），我们研究分形渗流中 LPP 的量化问题。通过开发多尺度框架，我们证明通行时间与线性增长存在几乎多项式的差距。高 LPP 值的阻碍是定向路径的利普希茨性质，沿此思路，文献[Cha96]证明分形渗流的任何连通子集的豪斯多夫维数必须严格大于1。我们证明，只要定向路径在空间方向上指数级快速移动，就可以构造经过分形渗流开位点的定向路径——我们引入并将此模型命名为不可矩形定向渗流。
英文摘要
While models of undirected and directed geometry in random i.i.d. disorder with rapidly decaying tails are expected to behave similarly and exhibit features of the Kardar--Parisi--Zhang universality class, the scenario when the background noise is fractal is expected to be significantly different. A canonical example of the latter is Liouville Quantum Gravity, where the planar Gaussian free field forms the fractal background. A toy model capturing some of the essential features is given by Mandelbrot's fractal percolation, a random Cantor set. In the simplest setting, dyadic cubes of different scales are retained independently with probability $p$ (a parameter of the model), and the intersection of all retained cubes forms the set of open sites. Connectivity properties of such sets have been intensely studied. Across [CCD88, Cha95], a surprising result was proven. Namely, while there is a phase transition for connectivity of the fractal percolation set (like in usual bond percolation), directed percolation never occurs. Results of a similar spirit have been obtained in geometric measure theory as well. As a step towards understanding last passage percolation (LPP) driven by the Gaussian free field (initiated in [GGN24]), we study the problem of quantifying LPP in fractal percolation. Developing a multi-scale framework, we show an almost polynomial gap from linear growth of the passage time. The obstruction to high LPP values is the Lipschitz nature of directed paths. Along these lines, [Cha96] showed that any connected subset of fractal percolation must have Hausdorff dimension strictly larger than one. We show that one can construct directed paths passing through open sites in fractal percolation provided they move exponentially fast in the spatial direction---a model we introduce and term as unrectifiable directed percolation.
Comments49 pages, 8 figures
