75 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
Optimization Strategies in Complex Systems
We consider a class of combinatorial optimization problems that emerge in a
variety of domains among which: condensed matter physics, theory of financial
risks, error correcting codes in information transmissions, molecular and
protein conformation, image restoration. We show the performances of two
algorithms, the``greedy'' (quick decrease along the gradient) and
the``reluctant'' (slow decrease close to the level curves) as well as those of
a``stochastic convex interpolation''of the two. Concepts like the average
relaxation time and the wideness of the attraction basin are analyzed and their
system size dependence illustrated.Comment: 8 pages, 3 figure
Correlation Inequalities for Quantum Spin Systems with Quenched Centered Disorder
It is shown that random quantum spin systems with centered disorder satisfy
correlation inequalities previously proved (arXiv:cond-mat/0612371) in the
classical case. Consequences include monotone approach of pressure and ground
state energy to the thermodynamic limit. Signs and bounds on the surface
pressures for different boundary conditions are also derived for finite range
potentials.Comment: 4 page
A mean-field monomer-dimer model with attractive interaction. The exact solution
A mean-field monomer-dimer model which includes an attractive interaction
among both monomers and dimers is introduced and its exact solution rigorously
derived. The Heilmann-Lieb method for the pure hard-core interacting case is
used to compute upper and lower bounds for the pressure. The bounds are shown
to coincide in the thermodynamic limit for a suitable choice of the monomer
density m. The consistency equation characterising m is studied in the phase
space (h, J), where h tunes the monomer potential and J the attractive
potential. The critical point and exponents are computed and show that the
model is in the mean-field ferromagnetic universality class.Comment: 32 pages, 6 figure
Short-range spin glasses and Random Overlap Structures
Properties of Random Overlap Structures (ROSt)'s constructed from the
Edwards-Anderson (EA) Spin Glass model on with periodic boundary
conditions are studied. ROSt's are random matrices whose entries
are the overlaps of spin configurations sampled from the Gibbs measure. Since
the ROSt construction is the same for mean-field models (like the
Sherrington-Kirkpatrick model) as for short-range ones (like the EA model), the
setup is a good common ground to study the effect of dimensionality on the
properties of the Gibbs measure. In this spirit, it is shown, using translation
invariance, that the ROSt of the EA model possesses a local stability that is
stronger than stochastic stability, a property known to hold at almost all
temperatures in many spin glass models with Gaussian couplings. This fact is
used to prove stochastic stability for the EA spin glass at all temperatures
and for a wide range of coupling distributions. On the way, a theorem of Newman
and Stein about the pure state decomposition of the EA model is recovered and
extended.Comment: 27 page
Thermodynamic Limit for Finite Dimensional Classical and Quantum Disordered Systems
We provide a very simple proof for the existence of the thermodynamic limit
for the quenched specific pressure for classical and quantum disordered systems
on a -dimensional lattice, including spin glasses. We develop a method which
relies simply on Jensen's inequality and which works for any disorder
distribution with the only condition (stability) that the quenched specific
pressure is bounded.Comment: 14 pages. Final version, accepted for publication on Rev. Math. Phy
Replica equivalence in the Edwards-Anderson model
After introducing and discussing the "link-overlap" between spin
configurations we show that the Edwards-Anderson model has a
"replica-equivalent" quenched equilibrium state, a property introduced by
Parisi in the description of the mean-field spin-glass phase which generalizes
ultrametricity. Our argument is based on the control of fluctuations through
the property of stochastic stability and works for all the finite-dimensional
spin-glass models.Comment: 12 pages, few remarks added. To appear in Journal of Physics A:
Mathematical and Genera
An Extended Variational Principle for the SK Spin-Glass Model
The recent proof by F. Guerra that the Parisi ansatz provides a lower bound
on the free energy of the SK spin-glass model could have been taken as offering
some support to the validity of the purported solution. In this work we present
a broader variational principle, in which the lower bound, as well as the
actual value, are obtained through an optimization procedure for which
ultrametic/hierarchal structures form only a subset of the variational class.
The validity of Parisi's ansatz for the SK model is still in question. The new
variational principle may be of help in critical review of the issue.Comment: 4 pages, Revtex
Thermodynamics and Universality for Mean Field Quantum Spin Glasses
We study aspects of the thermodynamics of quantum versions of spin glasses.
By means of the Lie-Trotter formula for exponential sums of operators, we adapt
methods used to analyze classical spin glass models to answer analogous
questions about quantum models.Comment: 17 page
Real spin glasses relax slowly in the shade of hierarchical trees
The Parisi solution of the mean-field spin glass has been widely accepted and
celebrated. Its marginal stability in 3d and its complexity however raised the
question of its relevance to real spin glasses. This paper gives a short
overview of the important experimental results which could be understood within
the mean-field solution. The existence of a true phase transition and the
particular behaviour of the susceptibility below the freezing temperature,
predicted by the theory, are clearly confirmed by the experimental results. The
behaviour of the complex order parameter and of the Fluctuation Dissipation
ratio are in good agreement with results of spontaneous noise measurements. The
very particular ultrametric symmetry, the key feature of the theory, provided
us with a simple description of the rejuvenation and memory effects observed in
experiment. Finally, going a step beyond mean-field, the paper shortly
discusses new analyses in terms of correlated domains characterized by their
length scales, as well as new experiments on superspin glasses which compare
well with recent theoretical simulations.Comment: To appear in the proceedings of "Wandering with Curiosity in Complex
Landscapes", a scientific conference in honour of Giorgio Parisi for his 60th
birthday, Roma, September 8-10 2008 (submitted for the special issue of the
Journal of Statistical Physics, 2009
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