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...xamples of non-Gibbsian measures, to what extent the failure of the Gibbsian property has to be taken serious, and what suitable generalizations of Gibbsianness should be (see e.g. [F],[E],[DS],[BKL],=-=[MRM]-=-, references therin, and the basic paper [EFS]). While this discussion still does not seem to be finished, the answers seem to depend on the specific situation. Our point in this context is less a gen...

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...construction of [Koz]). For general information about scenarios of the failure of the Gibbsian property for lattice measures and possible generalizations 2of Gibbsianness see e.g. [F],[E],[DS],[BKL],=-=[MRM]-=-, [MRSM], references therein, and the basic paper [EFS]. In the first mathematical paper [EMSS] which studied a joint measure of a quenched random system it was shown that the joint measure resulting ...

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...e start, R. L. Dobrushin was interested and involved in this development; indeed, Dobrushin and Shlosman [2], [3] proposed a programme of Gibbsian restoration, based on the idea that the pathological bad configurations of a transformed Gibbs measure (i.e., the essential points of discontinuity of some of its finite-set, e.g., single-site, conditional probabilities) are exceptional in the measure-theoretic sense (i.e., they form a set of measure zero). This has led to two extended notions of Gibbs measures: weakly Gibbsian measures and almost Gibbsian measures (see Maes, Redig and Van Moffaert [24]). Later, several refined notions were proposed, such as intuitively weakly Gibbs (van Enter and Verbitskiy [10]) and right-continuous conditional probabilities. In van Enter, Fernandez, den Hollander and Redig [4], the behavior of a Gibbs measure µ subject to a high-temperature Glauber spin-flip dynamics was considered. A guiding example is the case where we start from the low-temperature plus-phase of the Ising model, and we run a high-temperature dynamics, modeling the fast heating up of a cold system. The question of Gibbsianness of the measure µt at time t > 0 can then be interpreted as ...

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...ditional probabilities and Gibbs measures via the DLR equations. Since Dobrushin and Shlosman [4], many articles have shown the “weak Gibbs” property of renormalized low-temperature phases (see e.g., =-=[3, 17, 19, 21]-=-) and of joint measures of disordered spin systems [13, 14]. Parallel to this, Fernández and Pfister [6] developed ideas about generalized regularity properties of the conditional probabilities. They ...

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...t follows from (2.1) that φ(ω) is a tail function (under translations), in fact measurable with respect to the σ-algebra at +∞ (or −∞). Thus, if µ is a Gibbs state for some potential, or weakly Gibbs =-=[16]-=-, then its cyclic components should also be Gibbs states, in the appropriate sense, for that potential. 2.b Results Our first result concerns systems of particles in the continuum. The two cases of pr...

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...known to be non-Gibbsian below the critical temperature, see [35], [10]. Following suggestions of Dobrushin it was recently shown that this restriction is in fact weakly Gibbsian, see [5], [6], [34], =-=[30]-=-. We state and discuss ∗ Onderzoeksleider FWO, Flanders. Email: christian.maes@fys.kuleuven.ac.be † Postdoctoraal onderzoeker FWO, Flanders. Email: frank.redig@fys.kuleuven.ac.be ‡ Aspirant FWO, Fland...

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... transformations of interest, and (ii) extension of the thermodynamic formalism to these broader class of measures. The first issue motivated the notions of weak Gibbsianness and almost quasilocality =-=[1, 2, 11, 12, 13, 14, 15]-=-. A measure is called almost quasilocal if it admits a version of its conditional probabilities which is continuous on a set of measure one. A measure is called weakly Gibbs if there exists a potentia...