Gibbs states and Gibbsian specifications on the space ℝ^ℕ

Abstract

We are interested in the study of Gibbs and equilbrium probabilities on the lattice R N . Consider the unilateral full-shift defined on the noncompact set R N and an α-Hölder continuous potential A from R N into R. From a suitable class of a priori probability measures ν (over the Borelian sets of R) we define the Ruelle operator associated to A (using an adequate extension of this operator to the compact set R N = (S 1 ) N ) and we show the existence of eigenfunctions, conformal probability measures and equilibrium states associated to A. The above, can be seen as a generalization of the results obtained in the compact case for the XY-model. We also introduce an extension of the definition of entropy and show the existence of A-maximizing measures. Moreover, we prove the existence of an involution kernel for A. Finally, we build a Gibbsian specification for the Borelian sets on the set R N and we show that this family of probability measures satisfies a FKGinequality.

Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

“…For the shift map in the non-compact setting, similar results were proved showing the existence of ground states (the existence of accumulation points at zero temperature). Results in this direction were obtained for countable Markov shifts satisfying the so-called BIP property, topologically transitive countable Markov shifts, and full shifts defined on the lattice R N (see for instance [FV18,JMU05,Kem11,LMMS15,LV20a,SV20]).…”

Existence of Gibbs states and maximizing measures on a general one-dimensional lattice system with markovian structure

“…In [Sar99] and [MU01] these results were generalized for the non-compact setting of countable Markov shifts. Another interesting model, known as XY model, was studied in [BCL+11] and [LMST09], from which was derived some interesting generalizations for compact, bounded and even non-bounded metric spaces (see for instance [CSS19,LMMS15,LMV19,LV19]).…”

“…The uniqueness of the ground state was proved in [Kem11] in the setting of countable Markov shifts satisfying the BIP property, however, that problem is still open for the topologically transitive case. In the setting of XY models, problems of selection and non-selection at zero temperature were studied in [BCL+11] and [LMST09] for the classical approach on the interval [0, 1], and these results were generalized to compact metric spaces in [LMMS15] and to a non-compact bounded metric spaces in [CSS19] and [LV19].…”

On involution kernels and large deviations principles on $ \beta $-shifts

“…Under those assumptions, we use the notation A (y) := A (y, x) and we call the potential A : Σ β → R as the transpose potential of A. Some references with lots of details and examples related to the involution kernel in classical settings of the XY model, the full-shift on finite alphabets and XY models with Markovian structure appear in [1], [2], [20] and [26].…”

“…The following lemma gives some important tools to prove the main theorem of this paper. More specifically, it provides a characterization of the value γ defined in (20) in terms of limits depending on the strictly increasing sequence (t n ) n≥1 satisfying (17). The statement of the result is the following.…”