A random matrix approach to lamplighter groups

The lamplighter group $G$ is defined as the wreath product

$\mathbb{Z}/2\mathbb{Z}\wr \mathbb{Z}$ and can be seen as the group of

lamp configurations on the set of integers, where a lamp is either on or

off, together with a specified position of the so-called lamplighter. It

was the first exemple of the Cayley graph of a group having a pure point

spectrum for a specific choice of generators.

We exhibit a random matrix model whose spectral measure converges to

that of the Cayley graph of $G$ for a more natural choice of generators.

This motivates the use of random matrix tools in order to obtain

information on the spectral measure of $G$ with this set of generators.