Abstract
Cayley forests and products of Cayley trees of order are represented as subgroups in the free product of cyclic groups () of order 2. The automorphism groups of these objects are determined. We give a complete description of the sets of translation-invariant and periodic Gibbs measures for the Ising model on a Cayley forest. We construct a new class of limiting Gibbs measures for the inhomogeneous Ising model on a Cayley tree. We find sufficient conditions for random walks in periodic random media on the forest to be never returning provided that the jumps of the walking particle are bounded.
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