It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$. It is also shown that Ancona’s inequalities extend to $R$, and therefore that the Martin boundary for $R$-potentials coincides with the natural geometric boundary $S^1$, and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, $p^n(x,y)\sim C_{x,y}{R}^{-n}{n}^{-3/2}$.

We establish the lower bound $p_{2t}(e,e)\succsim exp\left(-t^{1/3}\right)$, for the large times asymptotic behaviours of the probabilities $p_{2t}(e,e)$ of return to the origin at even times $2t$, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer $r$, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to $r$.)

Let $G={*}_{j=1}^{q+1}{G}_{n_j+1}$ be the product of $q+1$ finite groups each having order $n_j+1$ and let $\mu$ be the probability measure which takes the value $p_j/n_j$ on each element of $G_{n_j+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_j$. We also compute the continuous spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$,...

The affine group of a local field acts on the tree $𝕋\left(𝔉\right)$ (the Bruhat-Tits building of $\mathrm{GL}(2,𝔉)$) with a fixed point in the space of ends $\partial 𝕋\left(F\right)$. More generally, we define the affine group $Aff\left(𝔉\right)$ of any homogeneous tree $𝕋$ as the group of all automorphisms of $𝕋$ with a common fixed point in $\partial 𝕋$, and establish main asymptotic properties of random products in $Aff\left(𝔉\right)$: (1) law of large numbers and central limit theorem; (2) convergence to $\partial 𝕋$ and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...

Let N be a simply connected nilpotent Lie group and let $S=N⋊{\left(ℝ⁺\right)}^d$ be a semidirect product, ${\left(ℝ⁺\right)}^d$ acting on N by diagonal automorphisms. Let (Qₙ,Mₙ) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xₙ = MₙXn-1 + Qₙ. We prove that for an appropriate homogeneous norm on N there is χ₀ such that $li{m}_{t\to \infty }{t}^{\chi ₀}\nu x:\left|x\right|>t=C>0$. In particular, this applies to classical Poisson kernels on symmetric spaces,...