关于随机最小有向生成树的总长
On the total length of the random minimal directed spanning tree
摘要：
在Bhatt和Roy的最小有向生成树（MDST）中，对于单位正方形中的点的随机偏序集，所有的边必须遵守“坐标方向”偏序，并且从每个顶点到极小元必须有一条有向路.本文研究了具有幂权边的图的总长度的渐近性质。极限分布由远离边界的正常分量和由边界效应引入的贡献之和给出，其可以由不动点方程表征，并且使人想起某些算法的概率分析中产生的极限。随着幂权重指数的增加，分布经历了从正常贡献占主导地位的边界效应占主导地位的相变。在权是简单欧几里得长度的临界情况下，这两种效应都对极限定律有显著贡献。我们还给出了图的总权的一个大数定律。
英文摘要：
In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square,all edges must respect the ``coordinatewise'' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.
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