139 research outputs found
Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models
It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional probabilities in finite volume of the following mean field models: (a) joint measures of random field Ising, (b) joint measures of dilute Ising, (c) decimation of ferromagnetic Ising. The conditional probabilities are functions of the empirical mean of the conditionings; so we look at the large volume behavior of these functions to discover non-trivial limiting objects. For (a) we find (1) discontinuous dependence for almost any realization and (2) dependence of the conditional probabilities on the phase. In contrast to that we see continuous behavior for (b) and (c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.
Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures
We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional disorder field, and prove concentration w.r.t. the disorder field. Both fields are assumed to be in the uniqueness regime, allowing in particular for non-independent disorder fields. The modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the Dobrushin contraction coefficient, and are explicitly computable.
On the basis of these inequalities, we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in Euclidean space, restricted to a finite volume. Here we also allow for thermal dislocations of the scatterers around their equilibrium positions. Extending recent results for independent scatterers, we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean. The bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set.
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
Uniqueness of gradient Gibbs measures with disorder
We consider - in 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 though 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 , 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.
Van Enter and Kuelske proved in 2008 that adding a disorder term as in model
(A) prohibits the existence of such gradient Gibbs measures for general
interaction potentials in . In Cotar and Kuelske (2012) we proved the
existence of shift-covariant random gradient Gibbs measures for model (A) when
, the disorder is i.i.d and has mean zero, and for model (B) when
and the disorder has stationary distribution.
In the present paper, we prove existence and uniqueness of shift-covariant
random gradient Gibbs measures with a given expected tilt and with
the corresponding annealed measure being ergodic: for model (A) when
and the disordered random fields are i.i.d. and symmetrically-distributed, and
for model (B) when and for any stationary disorder dependence
structure. We also compute for both models for any gradient Gibbs measure
constructed as in Cotar and Kuelske (2012), when the disorder is i.i.d. and its
distribution satisfies a Poincar\'e inequality assumption, the optimal decay of
covariances with respect to the averaged-over-the-disorder gradient Gibbs
measure.Comment: 39 pages. arXiv admin note: text overlap with arXiv:1012.437
Gibbsian representation for point processes via hyperedge potentials
We consider marked point processes on the d-dimensional euclidean space,
defined in terms of a quasilocal specification based on marked Poisson point
processes. We investigate the possibility of constructing absolutely-summable
Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al.
These potentials are a natural generalization of physical multi-body potentials
which are useful in models of stochastic geometry. We prove that such
representations can be achieved, under appropriate locality conditions of the
specification. As an illustration we also provide such potential
representations for the Widom-Rowlinson model under independent spin-flip
time-evolution. Our paper draws a link between the abstract theory of point
processes in infinite volume, the study of measures under transformations, and
statistical mechanics of systems of point particles.Comment: 21 pages, 2 figure, 1 tabl
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
Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry
We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete analysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature β^−1 ≥ 1, we prove that the time-evolved measure is always Gibbsian. For ⅔ ≤ β^−1 < 1, the time-evolved measure loses its Gibbsian character at a sharp transition time. For β^−1 < ⅔, we observe the new phenomenon of symmetry-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.
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
Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures
The concept of metastate measures on the states of a random spin system was
introduced to be able to treat the large-volume asymptotics for complex
quenched random systems, like spin glasses, which may exhibit chaotic volume
dependence in the strong-coupling regime. We consider the general issue of the
extremal decomposition for Gibbsian specifications which depend measurably on a
parameter that may describe a whole random environment in the infinite volume.
Given a random Gibbs measure, as a measurable map from the environment space,
we prove measurability of its decomposition measure on pure states at fixed
environment, with respect to the environment. As a general corollary we obtain
that, for any metastate, there is an associated decomposition metastate, which
is supported on the extremes for almost all environments, and which has the
same barycenter.Comment: 12 page
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
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