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Almost Gibbsianness and Parsimonious Description of the Decimated 2d-Ising Model

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Abstract

In this paper, we complete and provide details for the existing characterizations of the decimation of the Ising model on \(\mathbb{Z}^{2}\) in the generalized Gibbs context. We first recall a few features of the Dobrushin program of restoration of Gibbsianness and present the construction of global specifications consistent with the extremal decimated measures. We use them to prove that these renormalized measures are almost Gibbsian at any temperature and to analyse in detail its convex set of DLR measures. We also recall the weakly Gibbsian description and complete it using a potential that admits a quenched correlation decay, i.e. a well-defined configuration-dependent length beyond which this potential decays exponentially. We use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences.

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Notes

  1. When Ω is equipped with the product topology, see [11, 17].

  2. We briefly write μ[f] for the expectation \(\mathbb{E}_{\mu}[f]\) of a measurable function f under a measure μ.

  3. Introduced by Dobrushin [6], Lanford and Ruelle [26] in the late 60’s.

  4. Formally introduced by Föllmer [14] and Preston [40] in the mid 70’s.

  5. Taken as definition of consistency by Kozlov [24].

  6. For standard Borel spaces, see [17] or the constructions of Goldstein [18], Preston [41] and Sokal [48].

  7. There exist indeed specifications γ for which \(\mathcal{G}(\gamma)=\emptyset\), see [17, 28].

  8. One mostly reserves the terminology “phase” to the t.i. extremal elements of \(\mathcal{G}(\gamma)\).

  9. More precisely, the set \(\mathcal{G}(\gamma)\) is even a non-empty compact convex subset of \(\mathcal{M}_{1}^{+}\) [17].

  10. Tail-measurability is required to insure that the partition function is well-defined.

  11. Such a potential is also called a lattice-gas potential, see e.g. the terminology of [23].

  12. They prove that there exists disjoint sets Ω +,Ω −⊂Ω such that ν +(Ω +)=ν −(Ω −)=1 and a translation invariant interaction Φ with Ω Φ =Ω +∪Ω − such that ν + and ν − are weakly Gibbs for the pair (Φ,Ω Φ ). In their framework ν + and ν − share the same interaction but concentrate on different configurations.

  13. In general, and in particular in our set-up, this set is always a Choquet simplex, i.s. a convex set where each element is determined by a unique convex combination of the extremal elements, see e.g. [17, 28]. Moreover, these extreme points are the DLR measures that are trivial on the tail σ-algebra \(\mathcal{F}_{\infty}\).

  14. Consider e.g. a path that essentially avoids all contours of the configuration [33].

  15. It is known that at β c the correlations decay sub-exponentially according to a power law, while typical exponentially decaying scenarii for the behavior of the systems in certain regimes are studied in [45].

  16. The former is automatically of non-zero \(\nu^{+}_{\beta}\)-measure in our topological settings and thus modifying conditional probabilities on a negligible set cannot make them continuous, see also [28, 53].

  17. One sometimes speaks about a constrained or hidden phase transitions.

  18. This set has been originally introduced in [3] to get correlation estimates for the +-phase.

  19. The conditioning acts as an external field h 2i =+1 on the even sites, known to lead to uniqueness with the +-phase as an equilibrium state.

  20. A similar construction can be done for any γ monotonicity-preserving and right- or left-continuous.

  21. One says that the configuration is frozen in ω on S c.

  22. See e.g. [12], p. 1304 or proceed like in pp. 1294–1295.

  23. The freezing acts as an all positive/negative magnetic field, for which uniqueness holds from Theorem 1.

  24. A more complete variational principle has been established in [25], but we do not need it here.

  25. A complete interpretation and characterization in the original Gibbsian context can be also found in [23].

  26. Consistent with a t.i. UAC potential. It does not imply its existence for t.i. quasilocal measure, because the potential built by Kozlov from a t.i. quasilocal specification is not necessarily translation invariant [25].

  27. Introduced by Pfister [37] to state general large deviation principles.

  28. See e.g. [53], p. 970 for a one-line proof.

  29. To be true, this re-writing requires absolute convergence, checked next section.

  30. See e.g. Proposition 4.24 in [11]. Here, we have even more because the specification is right-continuous.

  31. This is not the case if one removes the condition k≤i.

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  1. Laboratoire de Mathématiques et d’Analyse Appliquées (LAMA—UMR CNRS 8050), Université Paris-Est Créteil (UPEC), 91 avenue du général de Gaulle, 94010, Créteil, France

    Arnaud Le Ny

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A. Le Ny is on leave from Université de Paris-Sud, Laboratoire de Mathématiques d’Orsay (LMO UMR CNRS 8628).

Work supported by the ANR project ANR-JCJC-0139-RANDYMECA.

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Le Ny, A. Almost Gibbsianness and Parsimonious Description of the Decimated 2d-Ising Model. J Stat Phys 152, 305–335 (2013). https://doi.org/10.1007/s10955-013-0773-1

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