随机树权图
Random tree-weighted graphs
摘要：
对于每个$n \ge 1$，令$\mathrm{d}^n=(d^{n}(i),1 \le i \le n)$是具有偶数和$\sum_{i=1}^n d^n(i) \ge 2n$的正整数序列。设$(G_n,T_n,Γ_n)$均匀分布在具有度序列$\mathrm{d}^n$的简单图$G_n$的集合上，赋予生成树$T_n$，并以$G_n$的有向边$Γ_n$为根，该边不是$T_n$的边.在$G_n$度的有限方差假设下，我们证明了，重新缩放后，$T_n$在分布上收敛到布朗连续随机树为$n \to \infty$。我们的主要工具是新版本的Pitman的添加剂聚结（https：//doi.org/10.1006/jcta.1998.2919），它可以用来构建具有固定度序列的随机树，以及具有固定度序列的随机树加权图。作为输入的证明，我们还推导出一个泊松近似定理的循环和多个边缘的数量在叠加的固定图和随机图与给定的度序列采样根据配置模型，我们发现这是独立的兴趣。
英文摘要：
For each $n \ge 1$, let $\mathrm{d}^n=(d^{n}(i),1 \le i \le n)$ be a sequence of positive integers with even sum $\sum_{i=1}^n d^n(i) \ge 2n$. Let $(G_n,T_n,Γ_n)$ be uniformly distributed over the set of simple graphs $G_n$ with degree sequence $\mathrm{d}^n$, endowed with a spanning tree $T_n$ and rooted along an oriented edge $Γ_n$ of $G_n$ which is not an edge of $T_n$. Under a finite variance assumption on degrees in $G_n$, we show that, after rescaling, $T_n$ converges in distribution to the Brownian continuum random tree as $n \to \infty$. Our main tool is a new version of Pitman's additive coalescent (https://doi.org/10.1006/jcta.1998.2919), which can be used to build both random trees with a fixed degree sequence, and random tree-weighted graphs with a fixed degree sequence. As an input to the proof, we also derive a Poisson approximation theorem for the number of loops and multiple edges in the superposition of a fixed graph and a random graph with a given degree sequence sampled according to the configuration model; we find this to be of independent interest.
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