Abstract:
We consider an infinite system of functional equations for the Potts model with competing interactions of radius $r=2$ and countable spin values $\Phi=\{0,1,\ldots,\}$ on the Cayley tree of order $k=2$. We reduce the problem to the description of the solutions of some infinite system of equations for any $k=2$ and any fixed probability measure $\nu$ with $\nu(i)>0$ on the set of all nonnegative integer numbers. We also give a description of the class of measures $\nu$ on $\Phi$ such that the infinite system of equations has unique solution $\{a^i,\,i=1,2,\ldots\}$, $a\in(0,1)$, with respect to each element of this class.
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