Untitled

Abstract

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite range lattice gases, when suitable strong mixing conditions are satisfied. Using block decimation procedure, cluster expansion (like in [HK]) and detailed comparison between statistical ensembles, we are able to prove Gibbsianess and convergence to a trivial (i.e. Gaussian and product) fixed point. Our results apply to 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.

Perturbative Analysis of Disordered Ising Models Close to Criticality

“…In [6] we shall prove a similar result in a more general context and, by using the combinatorial approach in [4], deduce an exponential decay of correlations from the convergence of the graded cluster expansion.…”

Metastability of Synchronous and Asynchronous Dynamics

“…In [72], they posed for this model the question of Gibbsianess [73,74], and it was proved that the total variation distance between the invariant measure of the PCA defined in (36) and the Ising Gibbs measure goes to zero as the volume and the control parameter q goes to infinity. Notice that these approximation results do not apply directly to the model considered in the present review, which, indeed, is recovered for q = h = 0 and Jij = J ij .…”

Relative Entropy, Gaussian Concentration and Uniqueness of Equilibrium States

“…As an example of application of Corollary (1), we mention the iteration of renormalization group transformations in the high-temperature regime [ 14 ], where convergence of the renormalized potentials can be established, and as a consequence, we obtain convergence of the relative entropy density. Then, Corollary (1) implies that the renormalized measures converge in the metric d as least as fast as the potentials.…”

“…In the context of stochastic dynamics, i.e., where is a time-evolved measure (at time n), it is usually not simple to obtain the convergence . In the high-temperature setting (high-temperature dynamics, high-temperature initial measure) this can be obtained with similar means as in [ 14 ].…”