59 research outputs found
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
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
Metastates in mean-field models with random external fields generated by Markov chains
We extend the construction by Kuelske and Iacobelli of metastates in
finite-state mean-field models in independent disorder to situations where the
local disorder terms are are a sample of an external ergodic Markov chain in
equilibrium. We show that for non-degenerate Markov chains, the structure of
the theorems is analogous to the case of i.i.d. variables when the limiting
weights in the metastate are expressed with the aid of a CLT for the occupation
time measure of the chain. As a new phenomenon we also show in a Potts example
that, for a degenerate non-reversible chain this CLT approximation is not
enough and the metastate can have less symmetry than the symmetry of the
interaction and a Gaussian approximation of disorder fluctuations would
suggest.Comment: 20 pages, 2 figure
Nonexistence of random gradient Gibbs measures in continuous interface models in
We consider statistical mechanics models of continuous spins in a disordered
environment. These models have a natural interpretation as effective interface
models. It is well known that without disorder there are no interface Gibbs
measures in infinite volume in dimension , while there are ``gradient
Gibbs measures'' describing an infinite-volume distribution for the increments
of the field, as was shown by Funaki and Spohn. In the present paper we show
that adding a disorder term prohibits the existence of such gradient Gibbs
measures for general interaction potentials in . This nonexistence result
generalizes the simple case of Gaussian fields where it follows from an
explicit computation. In where random gradient Gibbs measures are
expected to exist, our method provides a lower bound of the order of the
inverse of the distance on the decay of correlations of Gibbs expectations
w.r.t. the distribution of the random environment.Comment: Published in at http://dx.doi.org/10.1214/07-AAP446 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction
We consider a class of infinite-dimensional diffusions where the interaction
between the components is both spatial and temporal. We start the system from a
Gibbs measure with finite-range uniformly bounded interaction. Under suitable
conditions on the drift, we prove that there exists such that the
distribution at time is a Gibbs measure with absolutely summable
interaction. The main tool is a cluster expansion of both the initial
interaction and certain time-reversed Girsanov factors coming from the
dynamics
Discrete approximations to vector spin models
We strengthen a result of two of us on the existence of effective
interactions for discretised continuous-spin models. We also point out that
such an interaction cannot exist at very low temperatures. Moreover, we compare
two ways of discretising continuous-spin models, and show that, except for very
low temperatures, they behave similarly in two dimensions. We also discuss some
possibilities in higher dimensions.Comment: 12 page
Uniqueness of gradient Gibbs measures with disorder
We consider—in a uniformly strictly convex potential regime—two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters through the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite volume in dimension (Formula presented.), while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn (Commun Math Phys 185:1–36, 1997). Van Enter and Külske proved in (Ann Appl Probab 18(1):109–119, 2008) that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in (Formula presented.). In Cotar and Külske (Ann Appl Probab 22(5):1650–1692, 2012) we proved the existence of shift-covariant random gradient Gibbs measures for model (A) when (Formula presented.), the disorder is i.i.d and has mean zero, and for model (B) when (Formula presented.) and the disorder has a stationary distribution. In the present paper, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt(Formula presented.) and with the corresponding annealed measure being ergodic: for model (A) when (Formula presented.) and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when (Formula presented.) and for any stationary disorder-dependence structure. We also compute for both models for any gradient Gibbs measure constructed as in Cotar and Külske (Ann Appl Probab 22(5):1650–1692, 2012), when the disorder is i.i.d. and its distribution satisfies a Poincaré inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure
Multilayer parking with screening on a random tree
In this paper we present a multilayer particle deposition model on a random
tree. We derive the time dependent densities of the first and second layer
analytically and show that in all trees the limiting density of the first layer
exceeds the density in the second layer. We also provide a procedure to
calculate higher layer densities and prove that random trees have a higher
limiting density in the first layer than regular trees. Finally, we compare
densities between the first and second layer and between regular and random
trees.Comment: 15 pages, 2 figure
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