175 research outputs found
Stochastic Stability: a Review and Some Perspectives
A review of the stochastic stability property for the Gaussian spin glass
models is presented and some perspectives discussed.Comment: 12 pages, typos corrected, references added. To appear in Journal of
Statistical Physics, Special Issue for the 100th Statistical Mechanics
Meetin
Inequalities for the Local Energy of Random Ising Models
We derive a rigorous lower bound on the average local energy for the Ising
model with quenched randomness. The result is that the lower bound is given by
the average local energy calculated in the absence of all interactions other
than the one under consideration. The only condition for this statement to hold
is that the distribution function of the random interaction under consideration
is symmetric. All other interactions can be arbitrarily distributed including
non-random cases. A non-trivial fact is that any introduction of other
interactions to the isolated case always leads to an increase of the average
local energy, which is opposite to ferromagnetic systems where the Griffiths
inequality holds. Another inequality is proved for asymmetrically distributed
interactions. The probability for the thermal average of the local energy to be
lower than that for the isolated case takes a maximum value on the Nishimori
line as a function of the temperature. In this sense the system is most stable
on the Nishimori line.Comment: 10 pages. Submitted to J. Phys. Soc. Jp
On the Stability of the Quenched State in Mean Field Spin Glass Models
While the Gibbs states of spin glass models have been noted to have an
erratic dependence on temperature, one may expect the mean over the disorder to
produce a continuously varying ``quenched state''. The assumption of such
continuity in temperature implies that in the infinite volume limit the state
is stable under a class of deformations of the Gibbs measure. The condition is
satisfied by the Parisi Ansatz, along with an even broader stationarity
property. The stability conditions have equivalent expressions as marginal
additivity of the quenched free energy. Implications of the continuity
assumption include constraints on the overlap distribution, which are expressed
as the vanishing of the expectation value for an infinite collection of
multi-overlap polynomials. The polynomials can be computed with the aid of a
"real"-replica calculation in which the number of replicas is taken to zero.Comment: 17 pages, LaTex, Revised June 5, 199
Interaction Flip Identities for non Centered Spin Glasses
We consider spin glass models with non-centered interactions and investigate
the effect, on the random free energies, of flipping the interaction in a
subregion of the entire volume. A fluctuation bound obtained by martingale
methods produces, with the help of integration by parts technique, a family of
polynomial identities involving overlaps and magnetizations
Comment on ``Both site and link overlap distributions are non trivial in 3-dimensional Ising spin glasses'', cond-mat/0608535v2
We comment on recent numerical experiments by G.Hed and E.Domany
[cond-mat/0608535v2] on the quenched equilibrium state of the Edwards-Anderson
spin glass model. The rigorous proof of overlap identities related to replica
equivalence shows that the observed violations of those identities on finite
size systems must vanish in the thermodynamic limit. See also the successive
version cond-mat/0608535v
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
Spin-Glass Stochastic Stability: a Rigorous Proof
We prove the property of stochastic stability previously introduced as a
consequence of the (unproved) continuity hypothesis in the temperature of the
spin-glass quenched state. We show that stochastic stability holds in
beta-average for both the Sherrington-Kirkpatrick model in terms of the square
of the overlap function and for the Edwards-Anderson model in terms of the bond
overlap. We show that the volume rate at which the property is reached in the
thermodynamic limit is V^{-1}. As a byproduct we show that the stochastic
stability identities coincide with those obtained with a different method by
Ghirlanda and Guerra when applyed to the thermal fluctuations only.Comment: 12 pages, revised versio
Thermodynamic Limit for Mean-Field Spin Models
If the Boltzmann-Gibbs state of a mean-field -particle system
with Hamiltonian verifies the condition for every decomposition , then its free
energy density increases with . We prove such a condition for a wide class
of spin models which includes the Curie-Weiss model, its p-spin generalizations
(for both even and odd p), its random field version and also the finite pattern
Hopfield model. For all these cases the existence of the thermodynamic limit by
subadditivity and boundedness follows.Comment: 15 pages, few improvements. To appear in MPE
Griffiths Inequalities for Ising Spin Glasses on the Nishimori Line
The Griffiths inequalities for Ising spin glasses are proved on the Nishimori
line with various bond randomness which includes Gaussian and bond
randomness. The proof for Ising systems with Gaussian bond randomness has
already been carried out by Morita et al, which uses not only the gauge theory
but also the properties of the Gaussian distribution, so that it cannot be
directly applied to the systems with other bond randomness. The present proof
essentially uses only the gauge theory, so that it does not depend on the
detail properties of the probability distribution of random interactions. Thus,
the results obtained from the inequalities for Ising systems with Gaussian bond
randomness do also hold for those with various bond randomness, especially with
bond randomness.Comment: 13pages. Submitted to J. Phys. Soc. Jp
Ultrametricity in the Edwards-Anderson Model
We test the property of ultrametricity for the spin glass three-dimensional
Edwards-Anderson model in zero magnetic field with numerical simulations up to
spins. We find an excellent agreement with the prediction of the mean
field theory. Since ultrametricity is not compatible with a trivial structure
of the overlap distribution our result contradicts the droplet theory.Comment: typos correcte
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