论文标题
Erlang加权树，一个新的分支过程
The Erlang Weighted Tree, A New Branching Process
论文作者
论文摘要
在本文中，我们研究了一棵新的离散树和所得的分支过程，我们称之为\ textbf {e} rlang \ textbf {w}八ed \ textbf {t} ree（\ textbf {ewt}）。 EWT显示为在〜\ cite {la2015}中提出的随机图模型的局部弱极限。与众所周知的随机图模型的局部弱极限相反，EWT具有相互依赖的结构。特别是，其顶点编码了一个多种类型的多类分支过程。 我们得出了EWT的主要特性，例如灭绝，生长速率等的概率。我们表明灭绝的概率是操作员的最小固定点。然后，我们采用一个点过程的观点并分析增长率运算符。我们得出了Krein-Rutman特征值$β_0$和增长算子的相应特征函数，并证明灭绝的概率在且仅当$β_0\ leq 1 $时。
In this paper, we study a new discrete tree and the resulting branching process, which we call the \textbf{E}rlang \textbf{W}eighted \textbf{T}ree(\textbf{EWT}). The EWT appears as the local weak limit of a random graph model proposed in~\cite{La2015}. In contrast to the local weak limit of well-known random graph models, the EWT has an interdependent structure. In particular, its vertices encode a multi-type branching process with uncountably many types. We derive the main properties of the EWT, such as the probability of extinction, growth rate, etc. We show that the probability of extinction is the smallest fixed point of an operator. We then take a point process perspective and analyze the growth rate operator. We derive the Krein--Rutman eigenvalue $β_0$ and the corresponding eigenfunctions of the growth operator, and show that the probability of extinction equals one if and only if $β_0 \leq 1$.
