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1.
Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures 总被引:1,自引:1,他引:0
A.C.D. van Enter R. Fernández F. den Hollander F. Redig 《Communications in Mathematical Physics》2002,226(1):101-130
We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a
reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study
the Gibbsian character of the measure νS(t) at time t and show the following:
(1) For all ν and μ, νS(t) is Gibbs for small t.
(2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.
(3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.
(4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t.
The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.
Received: 26 April 2001 / Accepted: 10 October 2001 相似文献
2.
J. -R. Chazottes P. Collet C. Külske F. Redig 《Probability Theory and Related Fields》2007,137(1-2):201-225
We present a new and simple approach to concentration inequalities in the context of dependent random processes and random
fields. Our method is based on coupling and does not use information inequalities. In case one has a uniform control on the
coupling, one obtains exponential concentration inequalities. If such a uniform control is no more possible, then one obtains
polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, both at
high and low temperatures and in particular to the low-temperature Ising model which is a concrete example of non-uniformity
of the coupling.
相似文献
3.
We consider a specific continuous-spin Gibbs distribution μt=0 for a double-well potential that allows for ferromagnetic ordering. We study the time-evolution of this initial measure under
independent diffusions.
For `high temperature' initial measures we prove that the time-evoved measure μt is Gibbsian for all t. For `low temperature' initial measures we prove that μt stays Gibbsian for small enough times t, but loses its Gibbsian character for large enough t. In contrast to the analogous situation for discrete-spin Gibbs measures, there is no recovery of the Gibbs property for
large t in the presence of a non-vanishing external magnetic field. All of our results hold for any dimension d≥2. This example suggests more generally that time-evolved continuous-spin models tend to be non-Gibbsian more easily than
their discrete-spin counterparts.
Research carried out at EURANDOM and supported by Deutsche Forschungsgemeinschaft 相似文献
4.
Let {X
t:0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew
–1(X
t), where {w(x):xd} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering function
F(q,t)=E
0
(qd) is completely monotonic int (E
0 denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factor
S(q, w)=2
0
cos(t)F(q, t) exists for 0 and is strictly positive. Ind=1, 2 it diverges as 1/||1/2, resp. –ln(||), in the limit 0; ind3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that ||D |q|2, whereD is the diffusion constant. 相似文献
5.
F. Redig 《Journal of statistical physics》1994,74(3-4):815-827
We consider a symmetric translation-invariant random walk on thed-dimensional lattice ? d . The walker moves in an environment of moving traps. When the walker hits a trap, he is killed. The configuration of traps in the course of time is a reversible Markov process satisfying a level-2 large-deviation principle. Under some restrictions on the entropy function, we prove an exponential upper bound for the survival probability, i.e., $$\mathop {lim sup}\limits_{t \to \infty } \frac{1}{t}\log \mathbb{P}(T \geqslant t)< 0$$ whereT is the survival time of the walker. As an example, our results apply to a random walk in an environment of traps that perform a simple symmetric exclusion process. 相似文献
6.
Cristian Giardinà Jorge Kurchan Frank Redig Kiamars Vafayi 《Journal of statistical physics》2009,135(1):25-55
In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions
and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then
the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which
the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large
family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing. 相似文献
7.
Non-Fellerian particle systems are characterized by nonlocal interactions, somewhat analogous to non-Gibbsian distributions.
They exhibit new phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional
non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and
in another case, so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate,
there is a sharp transition from a non-trivial invariant measure to the trivial invariant measure of the sandpile process.
For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state. 相似文献
8.
We reconsider the discrete dual of the immediate exchange model and define a more general class of models where mass is split, exchanged and merged. We relate the splitting process to the symmetric inclusion process via thermalization and from that obtain symmetries and self-duality for it and its generalization. We show that analogous properties hold for models where the splitting is related to the symmetric exclusion process or to independent random walkers. 相似文献
9.
We continue our study of the exponential law for occurrences and returns of patterns in the context of Gibbsian random fields.
For the low-temperature plus-phase of the Ising model, we prove exponential laws with error bounds for occurrence, return,
waiting and matching times. Moreover we obtain a Poisson law for the number of occurrences of large cylindrical events and
a Gumbel law for the maximal overlap between two independent copies. As a by-product, we derive precise fluctuation results
for the logarithm of waiting and return times. The main technical tool we use, in order to control mixing, is disagreement
percolation 相似文献
10.
Journal of Statistical Physics - We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these... 相似文献