A random matrix approach to lamplighter groups

Let $\Lambda$ be a finitely generated abelian group and $\Gamma=\mathbb Z^{*d}$, we study the Cayley graph of the wreath product $G=\Lambda\wr\Gamma$ 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 $\Gamma=\mathbb Z$, 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.