连通空间网络中任意点上路长的形状定理
The Shape Theorem for Route-lengths in Connected Spatial Networks on Random Points
摘要：
对于平面上Poisson点上的连通网络，考虑原点附近的点和极坐标附近的点之间的路线长度$D(r,θ) $ $(r,θ)$，并假设$E D(r,θ) = O(r)$为$r \to \infty$。通过与第一次通过渗流的形状定理类比，对于一个不变量和遍历网络，人们期望$r^{-1} D(r, θ)$收敛为$r \to \infty$到常数$ρ(θ)$。事实证明，在做出精确的公式和证明时，有一些微妙之处。我们给出了一个配方和证明通过一个变种的次加性遍历定理，其中随机变量有时是无限的。
英文摘要：
For a connected network on Poisson points in the plane, consider the route-length $D(r,θ) $ between a point near the origin and a point near polar coordinates $(r,θ)$, and suppose $E D(r,θ) = O(r)$ as $r \to \infty$. By analogy with the shape theorem for first-passage percolation, for a translation-invariant and ergodic network one expects $r^{-1} D(r, θ)$ to converge as $r \to \infty$ to a constant $ρ(θ)$. It turns out there are some subtleties in making a precise formulation and a proof. We give one formulation and proof via a variant of the subadditive ergodic theorem wherein random variables are sometimes infinite.
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