A random matrix approach to lamplighter groups

Abstract

Let $\hat{I}›$ be a finitely generated abelian group and $\hat{I}“={\mathbb{Z}}^{\ast d}$, we study the Cayley graph of the wreath product $G=\hat{I}›\wr \hat{I}“$ with natural set of generators and their inverse $S$. First, we establish a random matrix model $X_N={\sum }_s{X}_N^{(s)}$ where the sum is indexed by the set $S$. As the size $N$ of the matrices goes to infinity, the traffic distribution of the $X_N^{(s)}$'s converges to that of the image of these generators in the reduced $C^{\ast }$-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 $\hat{I}“=\mathbb{Z}$, 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.

32 pages, 2 figures