Fixed points of an infinite-dimensional operator related to Gibbs measures

We describe fixed points of an infinite-dimensional nonlinear operator related to a hard-core (HC) model with a countable set $$\mathbb N$$ of spin values on a Cayley tree. This operator is defined by a countable set of parameters $$\lambda_i>0$$ , $$a_{ij}\in\{0,1\}$$ , $$i,j\in\mathbb N$$ . We find a sufficient condition on these parameters under which the operator has a unique fixed point. When this condition is not satisfied, we show that the operator may have up to five fixed points. We also prove that every fixed point generates a normalizable boundary law and therefore defines a Gibbs measure for the given HC model.

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