Uniqueness of Gibbs states in lattice models with one ground state

A Classical WR Model with $$q$$ q Particle Types

“…Our approach is based on papers [11,25]. (We also use constructions from [13,14].) In our opinion, constructions performed here yield simplified versions of those used in [11,25].…”

“…13) which can be verified for z large enough (with respect to b) using the same enumerating arguments as in the proof of Lemma 3.1, together with estimates (3.10) and(3.11).For a polymer S = [[ S ]] formed by small contours S (see Appendix) we define the base B( S ) = ∪ S ∈ S B( S ) and the statistical weight w( S ) as in (6.5). Then, due to shift-invariance of the statistical weights w( ), the representation (6.4) can be rewritten as log ( i; S) = υ( ) f (i; S) + r ( i; S).…”

Regularity properties and pathologies of position-space renormalization-group transformations

“…To prove the uniqueness of the Gibbs measure for the system with all image spins \+", we provide two arguments. First argument, proving uniqueness only at low temperature: Pirogov-Sinai theory 328,260,322] implies that the phase diagram at low enough temperature is a small deformation of that at zero temperature, but in this case there is only one ground state (namely, all spins \+"). Second argument, proving uniqueness at all temperatures: The internal-spin system is an Ising model on a periodic lattice, with nearest-neighbor coupling J > 0 and a periodic magnetic eld h x = h n.i.…”