Geodesics and spanning trees for Euclidean first-passage percolation
Geodesics and spanning trees for Euclidean first-passage percolation
复制标题
欧几里得第一通道渗透的测地线和生成树
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
C. Newman
中科院分区：
文献类型：
--
作者：
C. D. Howard;C. Newman
R d 上齐次泊松过程的粒子位置集合 Q 上的度量 D α (q, q') 定义为 Q 中以 q 开始并以 q' 结束的序列上 (Σ i |q i -q i+1 | α ) 1/α 的下确界（其中 |.| 表示欧几里得距离），当 a > 1 时，具有非平凡的测地线。情况 1 < α < ∞ 是 作者之前介绍过欧几里得第一通道渗滤（FPP）模型，而 α = ∞ 情况下的测地线正是来自 Aldous 和 Steele 的欧几里得最小生成树/森林的路径。我们比较和对比这两种情况的结果和猜想。 1 < α < ∞（和任何 d）的新结果包括度量 (X ≤ 1/2) 和测地线 (xi ≤ 3/4) 的波动指数的不等式，其强度足够强，足以得出格子 FPP 尚未获得的结论：几乎可以肯定，每个半无限测地线都有一个渐近方向，并且每个方向都有一个半无限测地线（从每个 q）。对于 d = 2 和 2 < α < ∞，进一步的结果涉及半无限测地线和相关随机表面的生成树。
The metric D α (q, q') on the set Q of particle locations of a homogeneous Poisson process on R d , defined as the infimum of (Σ i |q i -q i+1 | α ) 1/α over sequences in Q starting with q and ending with q' (where |.| denotes Euclidean distance) has nontrivial geodesics when a > 1. The cases 1 < α < ∞ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors, while the geodesics in the case α = ∞ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conjectures for these two situations. New results for 1 < α < ∞ (and any d) include inequalities on the fluctuation exponents for the metric (X ≤ 1/2) and for the geodesies (ξ ≤ 3/4) in strong enough versions to yield conclusions not yet obtained for lattice FPP: almost surely, every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic (from every q). For d = 2 and 2 < α < ∞ further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces.