Fractal Failure of Quasilocality for a Majority Rule Transformation on a Tree

We provide a transformation of the Ising model on a Cayley tree leading to non-Gibbsianness at any temperature, i.e. even within the uniqueness regime. We also introduce a new type of pathologies of renormalized Gibbs measures, called the fractal failure of quasilocality, and exhibit a concrete example.     

Renormalization group at criticality and complete analyticity of constrained models: A numerical study

We study the majority rule transformation applied to the Gibbs measure for the 2D Ising model at the critical point. The aim is to show that the renormalized Hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman uniqueness (DSU) finite-size condition for the constrained models corresponding to different configurations of the image system. It is known that DSU implies, in our 2D case, complete analyticity from which, as recently shown by Haller and Kennedy. Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite-volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed the DSU condition is verified for a large enough volumeV for all constrained models.     

Some remarks on pathologies of renormalization-group transformations for the Ising model

The results recently obtained by van Enter, Fernandez, and Sokal on non-Gibbsianness of the measurev =T _{b ,h} arising from the application of a single decimation transformationT _b , with spacingb, to the Gibbs measure _,h , of the Ising model, for suitably chosen large inverse temperature and nonzero external fieldh, are critically analyzed. In particular, we show that if, keeping fixed the same values of, h, andb, one iterates a sufficiently large number of timesn the transformationT _b , one obtains a new measurev = (T _b )_n,h which is Gibbsian and moreover very weakly coupled.     

Instability of renormalization-group pathologies under decimation

We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.