AI 中文总结
本文通过随机矩阵方法研究灯手群的Cayley图谱性质，推导了谱测度收敛性及中心极限定理。
AI 中文摘要
设Λ是一个有限生成的阿贝尔群，Γ=ℤ*^d，我们研究wreath积群G=Λ≀Γ的Cayley图，其自然生成集及其逆元S。首先，我们建立了一个随机矩阵模型X_N=Σ_s X^{(s)}_N，其中求和索引由集合S确定。当矩阵大小N趋于无穷时，X^{(s)}_N的交通分布收敛于这些生成元在G的简化C*代数中的图像。特别是，X_N的谱测度收敛于G的Cayley图的谱测度。此外，在Γ=ℤ的情况下，我们为该随机矩阵模型的线性统计量建立了中心极限定理。然后，我们推导了渐近R变换的公式，并以其极限一阶分布来表示极限的二阶分布。
英文摘要
Let $Λ$ be a finitely generated abelian group and $Γ=\mathbb Z^{*d}$, we study the Cayley graph of the wreath product $G=Λ\wrΓ$ with natural set of generators and their inverse $S$. First, we establish a random matrix model $X_N=\sum_s X^{(s)}_N$ where the sum is indexed by the set $S$. As the size $N$ of the matrices goes to infinity, the traffic distribution of the $X^{(s)}_N$'s converges to that of the image of these generators in the reduced $C^*$-algebra of $G$. In particular, the spectral measure of $X_N$ converges toward that of the Cayley graph of $G$ with generators $S$. Moreover, in the case $Γ=\mathbb Z$, we establish a central limit theorem for linear statistics of this random matrix model. Then, we exhibit a formula for the asymptotic $R$-transform and derive the second-order distribution of the limit of $X_N$ in terms of its limiting first-order distribution.
Comments32 pages, 2 figures
