论文标题
关于增加树木的特征值的分布
On the distribution of eigenvalues of increasing trees
论文作者
论文摘要
我们证明，在$ n $ vertices上的随机递归树中固定特征值$α$的多样性分别以平均值和方差等于$μ__αn $和$σ^2_αN$。还表明，$μ_α$和$σ^2_α$对于每个完全真实的代数整数都是正的。这些证明是基于霍姆格伦和詹森引起的添加树功能的一般结果。对于特征值$ 0 $，可以通过生成功能明确确定常数$μ_0$和$σ^2_0 $。也获得了针对拉普拉斯特征值和二元增加树木的类似结果。
We prove that the multiplicity of a fixed eigenvalue $α$ in a random recursive tree on $n$ vertices satisfies a central limit theorem with mean and variance asymptotically equal to $μ_α n$ and $σ^2_α n$ respectively. It is also shown that $μ_α$ and $σ^2_α$ are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue $0$, the constants $μ_0$ and $σ^2_0$ can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.
