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 2--D 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 2--D case, complete analyticity from which, as it has been 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 DSU condition is verified for a large enough volume $V$ for all constrained models.Comment: 39 pages, teX file, 4 Postscript figures, 1 TeX figur

A combinatorial proof of tree decay of semi-invariants

We consider finite range Gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi--invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions. This hypothesis holds for completely analytical Gibbs fields; in this context the tree decay of semi--invariants has been proven via analyticity arguments. However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griffiths' phase when analyticity arguments fail

Renormalization Group in the uniqueness region: weak Gibbsianity and convergence

We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale-adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

The ScS precursors for the study of the lowermost mantle

The exploration of the lowermost-mantle structures by means of body waveform modeling allows the small-scale detection of heterogeneity and anomalous layers. In some regions the D00 layer presents a discontinuity at its top that seems to be a local feature. This anomalous reflector may be recognized by the detection of a small core-reflected phases precursor. These studies may present different order of problems. The main difficulties, are connected to the identification of the precursor and its association to the D00 region. Misunderstandings often result because of phases produced by heterogeneity and anisotropy along and in the vicinity of the ray paths, in the crust and mantle structures. These complexities are increased when large dataset and recording arrays, which may facilitate the waveform analysis, are not available. In this paper we discuss the body waveform modeling of lower-mantle phases for the study of the D00 with particular focus on the case of sparse data with only few events and stations available

Competitive nucleation in reversible Probabilistic Cellular Automata

The problem of competitive nucleation in the framework of Probabilistic Cellular Automata is studied from the dynamical point of view. The dependence of the metastability scenario on the self--interaction is discussed. An intermediate metastable phase, made of two flip--flopping chessboard configurations, shows up depending on the ratio between the magnetic field and the self--interaction. A behavior similar to the one of the stochastic Blume--Capel model with Glauber dynamics is found