Variational description of Gibbs-non-Gibbs dynamical transitions for the Curie-Weiss model

We perform a detailed study of Gibbs-non-Gibbs transitions for the Curie-Weiss model subject to independent spin-flip dynamics ("infinite-temperature" dynamics). We show that, in this setup, the program outlined in van Enter, Fern\'andez, den Hollander and Redig can be fully completed, namely that Gibbs-non-Gibbs transitions are equivalent to bifurcations in the set of global minima of the large-deviation rate function for the trajectories of the magnetization conditioned on their endpoint. As a consequence, we show that the time-evolved model is non-Gibbs if and only if this set is not a singleton for some value of the final magnetization. A detailed description of the possible scenarios of bifurcation is given, leading to a full characterization of passages from Gibbs to non-Gibbs -and vice versa- with sharp transition times (under the dynamics Gibbsianness can be lost and can be recovered). Our analysis expands the work of Ermolaev and Kulske who considered zero magnetic field and finite-temperature spin-flip dynamics. We consider both zero and non-zero magnetic field but restricted to infinite-temperature spin-flip dynamics. Our results reveal an interesting dependence on the interaction parameters, including the presence of forbidden regions for the optimal trajectories and the possible occurrence of overshoots and undershoots in the optimal trajectories. The numerical plots provided are obtained with the help of MATHEMATICA.Comment: Key words and phrases: Curie-Weiss model, spin-flip dynamics, Gibbs vs. non-Gibbs, dynamical transition, large deviations, action integral, bifurcation of rate functio

Problems with the definition of renormalized Hamiltonians for momentum-space renormalization transformations

For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hamiltonians are ill-defined in certain regions of the phase diagram.Comment: 14 pages, late

Gibbs-non-Gibbs transitions via large deviations: computable examples

We give new and explicitly computable examples of Gibbs-non-Gibbs transitions of mean-field type, using the large deviation approach 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

Low-temperature dynamics of the Curie-Weiss Model: Periodic orbits, multiple histories, and loss of Gibbsianness

We consider the Curie-Weiss model at a given initial temperature in vanishing external field evolving under a Glauber spin-flip dynamics corresponding to a possibly different temperature. 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 bigger than one 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, when the initial temperature is smaller than the temperature of the dynamics, and smaller than 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 there is always symmetry-breaking in the set of bad configurations. These bad configurations 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.Comment: 31 pages, 24 figure

On Random Field Induced Ordering in the Classical XY Model

Consider the classical XY model in a weak random external field pointing along the $Y$ axis with strength $\epsilon$. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension $d \geq 2$ and for all $\epsilon$ sufficiently small, there is a range $L=L(\epsilon)$ so that whenever the inverse temperature $\beta$ is larger than some $\beta(\epsilon)$, there is strong residual ordering along the $X$ direction.Comment: 30 page

Thermodynamic Limit for Spin Glasses. Beyond the Annealed Bound

Using a correlation inequality of Contucci and Lebowitz for spin glasses, we demonstrate existence of the thermodynamic limit for short-ranged spin glasses, under weaker hypotheses than previously available, namely without the assumption of the annealed bound.Comment: 8 page

On the zero-temperature limit of Gibbs states

We exhibit Lipschitz (and hence H\"older) potentials on the full shift $\{0,1\}^{\mathbb{N}}$ such that the associated Gibbs measures fail to converge as the temperature goes to zero. Thus there are "exponentially decaying" interactions on the configuration space $\{0,1\}^{\mathbb Z}$ for which the zero-temperature limit of the associated Gibbs measures does not exist. In higher dimension, namely on the configuration space $\{0,1\}^{\mathbb{Z}^{d}}$, $d\geq3$, we show that this non-convergence behavior can occur for finite-range interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment follow i