论文标题
在给定度序列的树木的轮廓上
On the profile of trees with a given degree sequence
论文作者
论文摘要
学位序列是一个序列$ {\ bf s} =（n_i，i \ geq 0）$的非阴性整数满足$ 1+\ sum_i in_i in_i = \ sum_i n_i n_i n_i n_i <\ infty $。我们对均匀分布$ \ mathbb {p} _ {\ bf s} $上的均匀分布感兴趣，其学位序列等于$ {\ bf s} $，为树木的大小转化为无限。这提供了更一般的公式和由于Aldous引起的猜想的概率证明（1991）。我们的配方包含并扩展了Drmota和Gittenberger（1997）和Kersting（2011）先前获得的该方向的结果。需要技术结果来确保具有法律的树木$ \ Mathbb {p} _ {\ bf s} $在第一代中具有足够的个人，并且通过新颖的路径变换和可交换增量过程的波动理论来处理。结果，我们获得了Aldous，Miermont和Pitman（2004）引入的不均匀连续性随机树的有界标准。
A degree sequence is a sequence ${\bf s}=(N_i,i\geq 0)$ of non-negative integers satisfying $1+\sum_i iN_i=\sum_i N_i<\infty$. We are interested in the uniform distribution $\mathbb{P}_{\bf s}$ on rooted plane trees whose degree sequence equals ${\bf s}$, giving conditions for the convergence of the profile (sequence of generation sizes) as the size of the tree goes to infinity. This provides a more general formulation and a probabilistic proof of a conjecture due to Aldous (1991). Our formulation contains and extends results in this direction obtained previously by Drmota and Gittenberger (1997) and Kersting (2011). A technical result is needed to ensure that trees with law $\mathbb{P}_{\bf s}$ have enough individuals in the first generations, and this is handled through novel path transformations and fluctuation theory of exchangeable increment processes. As a consequence, we obtain a boundedness criterion for the inhomogeneous continuum random tree introduced by Aldous, Miermont and Pitman (2004).
