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Random walks on the affine group of local fields and of homogeneous trees. (English) Zbl 0809.60010
Summary: The affine group of a local field acts on the tree \({\mathbb{T}}({\mathfrak F})\) (the Bruhat-Tits building of \(\text{GL} (2,{\mathfrak F})\)) with a fixed point in the space of ends \(\partial{\mathbb{T}}(F)\). More generally, we define the affine group \(\text{Aff}({\mathfrak F})\) of any homogeneous tree \({\mathbb{T}}\) as the group of all automorphisms of \({\mathbb{T}}\) with a common fixed point in \(\partial{\mathbb{T}}\), and establish main asymptotic properties of random products in \(\text{Aff}({\mathfrak F})\): (1) law of large numbers and central limit theorem; (2) convergence to \(\partial{\mathbb{T}}\) and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary with \(\partial{\mathbb{T}}\), which gives a description of the space of bounded \(\mu\)-harmonic functions. Our methods strongly rely on geometric properties of homogeneous trees as discrete counterparts of rank one symmetric spaces.

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
05C05 Trees
43A85 Harmonic analysis on homogeneous spaces
22E35 Analysis on \(p\)-adic Lie groups
31C20 Discrete potential theory
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