最小生成树中的固定森林与三次体积增长
Fixed forests in the minimum spanning tree and cubic volume growth
AI总结：
本文针对带均匀边权完全图的最小生成树，研究固定森林出现的概率渐近性质，给出相关函数的递归描述与显式结果，还得到最小生成树中球的期望大小渐近值及指数尾界。
AI中文摘要：
设$M_n$为带独立同分布均匀边权的完全图$K_n$的最小生成树。对于具有连通分支$T_1, \boldsymbol{\text{...}}, T_d$的固定森林$F$，本文证明存在有限树上的函数$\boldsymbol{\text{Ψ}}$，使得$n^{|E(F)|} \boldsymbol{\text{P}}_n(F \boldsymbol{\text{⊆}} M_n) \to \boldsymbol{\text{∏}}_{i=1}^d \boldsymbol{\text{Ψ}}(T_i)$；给出$\boldsymbol{\text{Ψ}}$的递归描述并对多个小树显式计算，对星型图$S_k$和路径$P_k$分别证明$\boldsymbol{\text{Ψ}}(S_k) \boldsymbol{\text{∼}} \boldsymbol{\text{ζ}}(2)^k$、$\boldsymbol{\text{Ψ}}(P_k) \boldsymbol{\text{∼}} k^2/12$，还证明半径为$r$的球的期望大小渐近于$r^3/36$，并给出指数尾界。
英文摘要：
Let $M_n$ be the minimum spanning tree of the complete graph $K_n$ with i.i.d.\ uniform edge weights. For a fixed forest $F$ with connected components $T_1, \ldots, T_d$, we show that there exists a function $Ψ$ on finite trees such that $$ n^{|E(F)|} \mathbb{P}_n(F \subseteq M_n) \longrightarrow \prod_{i=1}^d Ψ(T_i). $$ We give a recursive description of $Ψ$ and calculate it explicitly for several small trees. For the star $S_k$ and the path $P_k$, we prove that $Ψ(S_k) \sim ζ(2)^k$ and $Ψ(P_k) \sim k^2/12$, respectively. We also show that the expected size of a ball of radius $r$ is asymptotic to $r^3/36$, and give exponential tail bounds.
