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11
Low-temperature dynamics of the CurieWeiss model: periodic orbits, multiple histories, and loss of Gibbsianness
- J. Stat. Phys
, 2010
"... We consider the Curie-Weiss model at initial temperature 0 < β−1 ≤ ∞ in van-ishing external field evolving under a Glauber spin-flip dynamics with temperature 0 < β′−1 ≤ ∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of disco ..."
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Cited by 15 (4 self)
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We consider the Curie-Weiss model at initial temperature 0 < β−1 ≤ ∞ in van-ishing external field evolving under a Glauber spin-flip dynamics with temperature 0 < β′−1 ≤ ∞. We study the limiting conditional probabilities and their continuity properties and discuss their set of points of discontinuity (bad points). We provide a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending earlier work for the case of independent spin-flip dynamics. For initial temperature β−1> 1 we prove that the time-evolved measure stays Gibbs forever, for any (possibly low) temperature of the dynamics. In the regime of heating to low-temperatures from even lower temperatures, 0 < β−1 < min{β′−1, 1} we prove that the time-evolved measure is Gibbs initially and becomes non-Gibbs after a sharp transition time. We find this regime is further divided into a region where only symmetric bad configurations exist, and a region where this symmetry is broken. In the regime of further cooling from low-temperatures, β′−1 < β−1 < 1 there is always symmetry-breaking in the set of bad configurations. These bad configura-tions are created by a new mechanism which is related to the occurrence of periodic orbits for the vector field which describes the dynamics of Euler-Lagrange equations for the path large deviation functional for the order parameter. To our knowledge this is the first example of the rigorous study of non-Gibbsian phenomena related to cooling, albeit in a mean-field setup.
A largedeviation view on dynamical gibbs-non-gibbs transitions
- Moscow Mathematical Journal
"... Abstract. We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, we show that the tr ..."
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Cited by 15 (5 self)
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Abstract. We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, 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. 2000 Math. Subj. Class. Primary: 60F10, 60G60, 60K35; Secondary: 82B26, 82C22.
Variational description of Gibbs-non-Gibbs dynamical transitions for spin-flip systems with a Kac-type interaction
, 2013
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Gibbs-non-Gibbs properties for evolving Ising models on trees
, 2010
"... In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from ..."
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In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad. For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again. AMS 2000 subject classification:.
Renormalization group transformations near the critical point: Some rigorous results
- 2011 J. Math. Phys
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A LARGE-DEVIATION VIEW ON DYNAMICAL GIBBS-NON-GIBBS TRANSITIONS
"... Abstract. We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, we show that the tr ..."
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Abstract. We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spinflip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, 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. 2000 Math. Subj. Class. Primary: 60F10, 60G60, 60K35; Secondary: 82B26, 82C22.
Gibbs-non-Gibbs transitions via large deviations: computable examples
, 2012
"... We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation ap-proach introduced in [4]. These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birt ..."
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We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation ap-proach introduced in [4]. These examples include Brownian motion with small variance and related diffusion processes, such as the Ornstein-Uhlenbeck process, as well as birth and death processes. We show for a large class of initial measures and diffusive dynamics both short-time conservation of Gibbsianness and dynamical Gibbs-non-Gibbs transitions. 1
