AI 中文总结
该研究证明了一维和二维非均匀长程渗流的重发性准则，推导了二维平面随机几何图满足重发性的条件，并得出带插值核的二维权重依赖随机连接模型在特定强衰减区域内连通分量为重发的结论。
AI 中文摘要
我们证明了一维和二维非均匀长程渗流的重发性准则。在一维情形下，重发性源于纯粹的几何稀缺条件：长边在指数尺度上最终消失。只要满足标准强衰减长边估计，该准则适用于权重依赖随机连接模型及相关一维空间无标度图。在二维情形下，我们结合Lüchtrath的线性化学距离估计与度测度的面积序界，指数分离带中的图距离层为满足多项式混合和长边估计的平面随机几何图提供了所需的Nash-Williams割集[J. Theoret. Probab. 39 (2026), Paper No. 12]。作为具体推论，二维带插值核的权重依赖随机连接模型的每个连通分量，在强衰减区域δ>2、γ<1−1/δ且α<1−γ中均为重发的。
英文摘要
We prove recurrence criteria for inhomogeneous long-range percolation in dimensions one and two. In dimension one, recurrence follows from a purely geometric scarcity condition: long edges eventually disappear on exponential scales. This applies to weight-dependent random connection models and related one-dimensional spatial scale-free graphs whenever the standard strong-decay long-edge estimate holds. In dimension two, we combine the linear chemical-distance estimate of Lüchtrath with an area-order bound on the degree measure. Graph-distance layers in exponentially separated bands then give the required Nash-Williams cutsets for planar random geometric graphs satisfying the polynomial mixing and long-edge estimates [J. Theoret. Probab. 39 (2026), Paper No. 12]. As a concrete consequence, every connected component of the two-dimensional weight-dependent random connection model with interpolation kernel is recurrent throughout the strong-decay region $δ>2$, $γ<1-\frac{1}δ$, and $α<1-γ$.
Comments9 pages, 1 figure; this paper extends the results of arxiv:2408.06918 and supersedes it
