Phase transitions for models with a continuum set of spin values on a Bethe lattice

Yu. Kh. Eshkabilov^a, G. I. Botirov^b, F. H. Haydarov<sup>c

a</sup> Karshi State University, Kashkhadaryo, Uzbekistan
^b Institute of Mathematics, Tashkent, Uzbekistan
^c National University of Uzbekistan, Tashkent, Uzbekistan

Abstract: We consider a model with nearest-neighbor interactions and the set $[0,1]$ of spin values on a Bethe lattice {(}Cayley tree{\rm)} of arbitrary order. This model depends on a continuous parameter $\theta$ and is a generalization of known models. For all values of $\theta$, we give a complete description of the set of translation-invariant Gibbs measures of this model.

Keywords: Cayley tree, spin value, Gibbs measure, Hammerstein's equation, fixed point.

Received: 23.02.2020
Revised: 10.04.2020

DOI: 10.4213/tmf9893

 English version: Theoretical and Mathematical Physics, 2020, 205:1, 1372–1380

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