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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.

MSC:
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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