NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2016, 7 (5), P. 893–899
Translation-invariant Gibbs measures for a model with logarithmic potential on a Cayley tree
Yu. Kh. Eshkabilov – National University of Uzbekistan, Tashkent, Uzbekistan; firstname.lastname@example.org
Sh. P. Bobonazarov – Tashkent Institute of Irrigation and Melioration, Tashkent, Uzbekistan; email@example.com
R. I. Teshaboev – Termez State University, Termez, Uzbekistan; firstname.lastname@example.org
In this paper, we consider a model with logarithmical potential and with the set [0, 1] of spin values, on a Cayley tree Γk of the order k. In the case k = 2; 3, we shall prove that, there is a unique translation-invariant splitting Gibbs measure for this model. For the case k = 4, we show that there are three translation-invariant Gibbs measures for this model.
Keywords: Cayley tree, configuration, translation-invariant Gibbs measure, fixed point, nonlinear operator.