Abstract:
Cayley forests and products of Cayley trees of order k⩾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.
\Bibitem{GanRoz03}
\by N.~N.~Ganikhodzhaev, U.~A.~Rozikov
\paper Group representation of the Cayley forest and some of its applications
\jour Izv. Math.
\yr 2003
\vol 67
\issue 1
\pages 17--27
\mathnet{http://mi.mathnet.ru/eng/im416}
\crossref{https://doi.org/10.1070/IM2003v067n01ABEH000416}
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\zmath{https://zbmath.org/?q=an:1065.82005}
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https://www.mathnet.ru/eng/im416
https://doi.org/10.1070/IM2003v067n01ABEH000416
https://www.mathnet.ru/eng/im/v67/i1/p21
This publication is cited in the following 6 articles:
Rustamjon Khakimov, Muhtorjon Makhammadaliev, Kamola Umirzakova, “Alternative Gibbs measure for fertile three-state Hard-Core models on a Cayley tree”, Phase Transitions, 2024, 1
R. M. Khakimov, M. T. Makhammadaliev, F. H. Haydarov, “New class of Gibbs measures for two-state hard-core model on a Cayley tree”, Infin. Dimens. Anal. Quantum. Probab. Relat. Top., 26:04 (2023)
U. A. Rozikov, M. M. Rahmatullaev, R. M. Khakimov, “Periodic Gibbs measures for the Potts model in translation-invariant and periodic external fields on the Cayley tree”, Theoret. and Math. Phys., 210:1 (2022), 135–153
M. M. Rahmatullaev, “On weakly periodic Gibbs measures of the Potts model with a special external field on a Cayley tree”, Zhurn. matem. fiz., anal., geom., 12:4 (2016), 302–314
Rakhmatullaev M.M., “On Weakly Periodic Gibbs Measures for the Potts Model with External Field on the Cayley Tree”, Ukr. Math. J., 68:4 (2016), 598–611
É. P. Normatov, U. A. Rozikov, “A description of harmonic functions via properties of the group representation of the Cayley tree”, Math. Notes, 79:3 (2006), 399–407
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