Random walks on dense subgroups of locally compact groups

Let Γ be a countable discrete group, H a lcsc totally disconnected group and ρ:Γ→H a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup L<H and bi-L-invariant probability measure θ on H, a Furstenberg discretization τ of θ such that the Poisson boundary of (H,θ) is a τ-boundary. Among other things, this technique allows us to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not L^p-irreducible for any p≥1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group Γ and two spread-out probability measures τ₁ and τ₂ on Γ such that the boundary entropy spectrum of (Γ,τ₁) is an interval, while the boundary entropy spectrum of (Γ,τ₂) is a Cantor set.