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, $p^n(x,y)\sim C_{x,y}{R}^{-n}{n}^{-3/2}$.

This paper studies the on- and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

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...