A general renormalization procedure on the one-dimensional lattice and decay of correlations

Abstract

We present a general form of Renormalization operator R acting on potentials V : {0,1} N → R. We exhibit the analytical expression of the fixed point potential V for such operator R. This potential can be expressed in a naturally way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by A. Baraviera, R. Leplaideur and A. Lopes where the fixed point potential V was of Hofbauer type.For the potentials of Hofbauer type (a well known case of phase transition) the decay is like n −γ , γ > 0 Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential V of our general renormalization procedure. In some cases we get polynomial decay like n −γ , γ > 0, and in others a decay faster than n e − √ n , when n → ∞. The potentials g we consider here are elements of the so called family of Walters' potentials on {0,1} N which generalizes a family of potentials considered initially by F. Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known.In a final section we also show that given any choice d n → 0 of real numbers varying with n ∈ N there exist a potential g on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

The involution kernel and the dual potential for functions in the Walters family

“…In this case the equilibrium states are unique (no phase transition) and the pressure function is differentiable. The family described by Definition 1.3 contains a subfamily of potentials called of the Hofbauer potentials (see [19], [21], [30], [15], [23] and [8]) which are not of Hölder class. For this subclass, in some cases, there exists more than one equilibrium state (phase transition happens) and the pressure function may not be differentiable.…”

“…, where ζ(γ) is the Riemman zeta function. Questions related to renormalization for this class of potentials appear in [3] and [23]; the Hofbauer potential is a fixed point for the renormalization operator.…”

“…The variety of possibilities of the functions on the Walters' family of potentials is so rich that given a certain sequence c n , n ∈ N, describing a possible decay of correlations (under some mild assumptions), one can find a potential in the family such the equilibrium probability for this potential has this decay of correlation (see [23]).…”