Abstract:
Cayley forests and products of Cayley trees of order $k\geqslant 1$ are represented as subgroups in the free product of $m$ cyclic groups ($m>k$) 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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