Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (where $\bar{\eta}(\tilde{m})=1 $ iff $\tilde{m}=\tilde{0}$ and $\tilde{e_i}$ are the standard basis of the lattice $\mathbb{Z}^d$).

For a simple random walk on $G_d$ (measure on $G_d$ is uniformly distributed on $\Gamma$), it is known that for $d \geq 3$ the Furstenberg-Poisson (FP) boundary is non-trivial. The FP boundary is identified by the space of eventual lamp configurations i.e $\bar{H_d}=\prod_{\mathbb{Z}^d}\mathbb{Z}_2$. We get a group action of $G_d$ on $\bar{H_d}$ which makes the factor map equivariant, where the group action of $G_d$ on $G_d^\mathbb{N}$ is by left multiplication. This group action is quasi-invariant.

I am trying to get a sharp estimate for the Radon-Nikodym derivative of this quasi-invariant action of $G_d$ on $\bar{H_d}$. Can anyone please provide me with any references (apart from the book ''Probability on Trees and Networks '' by Lyons, Peres) that may help me to get some ideas regarding the aforementioned problem.