彩虹渗流

中文摘要 AI 辅助

我们考虑在ℝ×(0,1)上强度为λ的泊松点过程上的权重依赖随机连接模型，其中顶点(x,t)和(y,s)恰好当(t∨s)|x−y|≤β时相连。距离为d的点以概率min(1,β/d)²相连，这是一维长程渗流的临界衰减，且共享一个顶点的边通过共同标记相关联。我们证明该模型存在真实的相变：当λβ<1时，几乎所有连通分量都是有限的；当λβ≥31时，存在无限分量，因此在强度为1时，临界值满足β_c∈[1,31]；附录中的数值研究将其置于约2附近。下界通过模型的离散骨架证明，该骨架通过将顶点固定到ℤ得到，具有独立意义：它完全没有超临界相，从完全碎裂到平凡连通性发生退化跃迁，尽管几乎肯定有无限多条边穿过每个固定位点。

英文摘要

We consider the weight-dependent random connection model on a Poisson point process of intensity $λ$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\leβ$. Points at distance $d$ are joined with probability $\min(1,β/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $λβ<1$ almost surely all connected components are finite, while for $λβ\ge31$ an infinite component exists, so at intensity one the critical value satisfies $β_c\in[1,31]$; a numerical study included as an appendix places it near $2$. By kernel and profile comparisons the supercritical bound extends to the age-dependent random connection model on the line, which with indicator profile has a non-degenerate phase transition at every value of its parameter, closing a case of the one-dimensional phase diagram left open in earlier work. The lower bound is proved by disconnecting nested pairs of long edges ("rainbows") with cut-point certificates, an argument developed first in a discrete skeleton of the model with the vertices pinned to $\mathbb{Z}$. The skeleton is of independent interest: it has no supercritical phase at all, jumping from total fragmentation to trivial connectivity even though almost surely infinitely many edges cross every fixed site. The supercritical argument is a Peierls argument on the binary tiling of the hyperbolic half-plane.

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48 pages, 15 figures