大四边形中测地线的统计
Statistics of geodesics in large quadrangulations
摘要：
我们研究了随机平面四边形中测地线的统计性质，即最小长度的路径。我们扩展舍费尔的标记树双射的情况下，四边形与标记测地线，导致的概念“脊柱树”，服从直接枚举。我们得到的生成函数的四边形标记测地线的固定长度，以及与一组“合流测地线”，即一组不相交的最小路径连接两个给定的点。在面积为n的四边形剖分的极限中，我们特别发现两个距离i>>1的固定点之间的测地线平均数为3*2^i。我们发现，对于通用端点，两个合流测地线保持彼此接近，并有广泛的接触。这个属性在一些“例外”端点上失败，这些端点可以通过真正不同的测地线连接。给出了有限长度i和标度极限i ~ n^（1/4）下的结果。特别地，我们给出了例外点的标度分布。
英文摘要：
We study the statistical properties of geodesics, i.e. paths of minimal length, in large random planar quadrangulations. We extend Schaeffer's well-labeled tree bijection to the case of quadrangulations with a marked geodesic, leading to the notion of "spine trees", amenable to a direct enumeration. We obtain the generating functions for quadrangulations with a marked geodesic of fixed length, as well as with a set of "confluent geodesics", i.e. a collection of non-intersecting minimal paths connecting two given points. In the limit of quadrangulations with a large area n, we find in particular an average number 3*2^i of geodesics between two fixed points at distance i>>1 from each other. We show that, for generic endpoints, two confluent geodesics remain close to each other and have an extensive number of contacts. This property fails for a few "exceptional" endpoints which can be linked by truly distinct geodesics. Results are presented both in the case of finite length i and in the scaling limit i ~ n^(1/4). In particular, we give the scaling distribution of the exceptional points.
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