Almost Gibbsian versus weakly Gibbsian measures

Maes, Christian; Redig, Frank; Moffaert, A. Van; Leuven, KU

doi:10.1016/s0304-4149(98)00083-0

Cited by 44 publications

(39 citation statements)

References 9 publications

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“…This implies as usual (see e.g. [50]) weak Gibbsianness with an a.s. convergent potential as the telescoping one given in [53]. The latter possesses extra asymptotic properties such as a uniform polynomial decay that should be weaker here.…”

Section: Extensions Related Issues and Commentsmentioning

confidence: 60%

“…Estimating the measure of the discontinuity points leads one to the question of "almost Gibbsian" [50], "intuitively weakly Gibbsian" [14] and "weakly Gibbsian" properties [50]. The analysis of [21] and [49] extends, due to monotonicity and right-continuity properties, to prove almost Gibbsianness of the transformed measures both with and without a field.…”

Section: Extensions Related Issues and Commentsmentioning

confidence: 99%

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Decimation of the Dyson–Ising ferromagnet

Enter

2017

Stochastic Processes and their Applications

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We study the decimation to a sublattice of half the sites of the one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair potentials of the form 1 |i−j| α , deep in the phase transition region (1 < α ≤ 2 and low temperature). We prove non-Gibbsianness of the decimated measures at low enough temperatures by exhibiting a point of essential discontinuity for the (finite-volume) conditional probabilities of decimated Gibbs measures. This result complements previous work proving conservation of Gibbsianness for fastly decaying potentials (α > 2) and provides an example of a "standard" non-Gibbsian result in one dimension, in the vein of similar results in higher dimensions for short-range models. We also discuss how these measures could fit within a generalized (almost vs. weak) Gibbsian framework. Moreover we comment on the possibility of similar results for some other transformations.

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Section: Extensions Related Issues and Commentsmentioning

confidence: 60%

Section: Extensions Related Issues and Commentsmentioning

confidence: 99%

Decimation of the Dyson–Ising ferromagnet

Enter

2017

Stochastic Processes and their Applications

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“…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 [21]). Later, several refined notions were proposed, such as intuitively weakly Gibbs (Van Enter and Verbitskiy [9]) and right-continuous conditional probabilities.…”

Section: 1 Dynamical Gibbs-non-gibbs Transitionsmentioning

confidence: 99%

“…In that case D * α = ν x with ν x the translation-invariant product measure with ν x , H A = x |A| . The latter follows from the identity 21) and the rate function becomes…”

Section: Optimal Trajectoriesmentioning

confidence: 99%

A Large-Deviation View on Dynamical Gibbs-Non-Gibbs Transitions

Enter

Fernández

Hollander

et al. 2010

MMJ

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We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gärtner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical measures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition.MSC2010: Primary 60F10, 60G60, 60K35; Secondary 82B26, 82C22.

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“…The bad configurations are the discontinuity points of the conditional probability distribution of Y 0 , as made precise by the following proposition (see [5], Proposition 6, and [3], Theorem 1.2). Proposition 1.2.…”

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confidence: 94%

A crossover for the bad configurations of random walk in random scenery

Blachère¹,

Hollander²,

Steif³

2011

Ann. Probab.

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In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-p), respectively, with p in [1/2,1] and epsilon in [0,1), and where the scenery assigns the color black or white to the sites of Z with probability 1/2 each. We show that, remarkably, the set of bad configurations exhibits a crossover: for epsilon=0 and p in (1/2,4/5) all configurations are bad, while for (p,epsilon) in an open neighborhood of (1,0) all configurations are good. In addition, we show that for epsilon=0 and p=1/2 both bad and good configurations exist. We conjecture that for all epsilon in [0,1) the crossover value is unique and equals 4/5. Finally, we suggest an approach to handle the seemingly more difficult case where epsilon>0 and p in [1/2,4/5), which will be pursued in future work.Comment: Published in at http://dx.doi.org/10.1214/11-AOP664 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Almost Gibbsian versus weakly Gibbsian measures

Cited by 44 publications

References 9 publications

Decimation of the Dyson–Ising ferromagnet

Decimation of the Dyson–Ising ferromagnet

A Large-Deviation View on Dynamical Gibbs-Non-Gibbs Transitions

A crossover for the bad configurations of random walk in random scenery

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