# Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups

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.

• What exactly do you want to do? What do you mean by a "sharp estimate"? – R W Jan 6 '18 at 12:22
• Yes. The notion of sharp estimate here is different (which I am yet to understand) since Radon-Nikodym derivative here is a function from the space of $\mathbb{Z}_2$ valued functions on $\mathbb{Z}^d$ to real numbers. But first of all is there any estimate which is known? Are there any references for these estimates? – Bharath Jan 9 '18 at 6:31
• I still don't understand what kind of estimates you have in mind - can you give an example for other groups? – R W Jan 9 '18 at 7:53

## 1 Answer

As R W explained, it depends on what you call "sharp estimates".

Maybe you will be interested in a description of the Martin boundary. For $d=1$, this is done by Brofferio and Woess in the paper Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs (Ann. I. H. Poincaré, 2005).

There are some precise estimates of the Green kernel and of the Martin kernel in there.

• Thanks for the reference. I was looking for computations of asymptotic bounds for the Green's function. If there are any more references like this, please let me know. – Bharath Feb 19 '18 at 11:58