Limiting and gradient Gibbs measures on lattice system

The talk consists of the following results:
$\bullet$ to construct a new specification with a family of non-probability kernels on lattice systems; to describe a set of gradient Gibbs measures with periodic boundary laws for the SOS model with a countable set of spin values and construct infinitely many gradient Gibbs measures for the HC model with a countable set of spin values on lattice systems;
$\bullet$ to find conditions of existence, uniqueness, and non-uniqueness of Gibbs measures on the Cayley tree by using connections between the limiting Gibbs measures for continuous spin models and the fixed points of nonlinear integral operators;
$\bullet$ to compute the free energies of translation-invariant and periodic boundary conditions for the Ising model with an external field and obtain a sufficiency condition on the invariance property of subgroups of the group representation of Cayley trees.