大型平面三角剖分和四边形剖分中船体周长统计的若干结果
Some results on the statistics of hull perimeters in large planar triangulations and quadrangulations
摘要：
在一个平面地图中，距离d处的船体周长是将这两个顶点分开并与第一个顶点相距d（d<k）的闭曲线的长度。研究了随机平面三角剖分和四边形剖分中船体周长随k和d的变化规律。在无限大k的极限下，得到了距离d处船体周长的概率密度以及距离d1和d2处船体周长的联合概率密度的显式表达式。我们还考虑了测量船体周长的距离d对应于k的有限分数的情况。我们得到的各种法律是相同的三角剖分和四边形剖分，直到一个全球重新标度。我们的方法是基于递归关系最近推出的作者确定的生成函数的所谓的切片，即件的地图与适当的距离约束。它确实表明，这些递归关系的基础上的地图分解是密切相关的船体周长的概念，并提供了一个简单的方法来完全控制它们。
英文摘要：
The hull perimeter at distance d in a planar map with two marked vertices at distance k from each other is the length of the closed curve separating these two vertices and lying at distance d from the first one (d<k). We study the statistics of hull perimeters in large random planar triangulations and quadrangulations as a function of both k and d. Explicit expressions for the probability density of the hull perimeter at distance d, as well as for the joint probability density of hull perimeters at distances d1 and d2, are obtained in the limit of infinitely large k. We also consider the situation where the distance d at which the hull perimeter is measured corresponds to a finite fraction of k. The various laws that we obtain are identical for triangulations and for quadrangulations, up to a global rescaling. Our approach is based on recursion relations recently introduced by the author which determine the generating functions of so-called slices, i.e. pieces of maps with appropriate distance constraints. It is indeed shown that the map decompositions underlying these recursion relations are intimately linked to the notion of hull perimeters and provide a simple way to fully control them.
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