论文标题
根生和无根的距离简单生成的树
The distance profile of rooted and unrooted simply generated trees
论文作者
论文摘要
众所周知，在适当的归一化之后，具有有限后代方差的临界条件的加尔顿 - 沃特森树的高度轮廓与标准布朗时代的当地时间收敛。在这项工作中，我们研究距离轮廓，定义为顶点对之间所有距离的轮廓。我们表明，在正确重新重新缩放距离之后，距离轮廓会收敛到连续的随机函数，该函数可以描述为布朗连续脉随机树中随机点之间的距离密度。 我们证明了此限制功能A.S. Hölder是否连续$α<1 $连续，并且是A.E.可区分。我们注意到，它在$ 0 $的情况下不能以$ 0 $为单位，而是作为悬而未决的问题，无论是Lipschitz，以及在半行$（0，\ infty）$上是否持续可区分。 距离轮廓自然也定义为无根的树木，与针对植根树设计的高度轮廓相反。这是在我们的证明中使用的，我们证明了随机无根生成的树的距离轮廓的相应收敛结果。作为当前工作的次要目的，我们还将未根生成的树木的概念正式化，并包括一些简单的结果，将它们与植根的简单生成的树木相关联，这可能具有独立的兴趣。
It is well-known that the height profile of a critical conditioned Galton-Watson tree with finite offspring variance converges, after a suitable normalization, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $α<1$, and that it is a.e. differentiable. We note that it cannot be differentiable at $0$, but leave as open questions whether it is Lipschitz, and whether is continuously differentiable on the half-line $(0,\infty)$. The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.
