233 research outputs found
Complexity of the Sherrington-Kirkpatrick Model in the Annealed Approximation
A careful critical analysis of the complexity, at the annealed level, of the
Sherrington-Kirkpatrick model has been performed. The complexity functional is
proved to be always invariant under the Becchi-Rouet-Stora-Tyutin
supersymmetry, disregarding the formulation used to define it. We consider two
different saddle points of such functional, one satisfying the supersymmetry
[A. Cavagna {\it et al.}, J. Phys. A {\bf 36} (2003) 1175] and the other one
breaking it [A.J. Bray and M.A. Moore, J. Phys. C {\bf 13} (1980) L469]. We
review the previews studies on the subject, linking different perspectives and
pointing out some inadequacies and even inconsistencies in both solutions.Comment: 20 pages, 4 figure
Dynamical critical exponents for the mean-field Potts glass
In this paper we study the critical behaviour of the fully-connected
p-colours Potts model at the dynamical transition. In the framework of Mode
Coupling Theory (MCT), the time autocorrelation function displays a two step
relaxation, with two exponents governing the approach to the plateau and the
exit from it. Exploiting a relation between statics and equilibrium dynamics
which has been recently introduced, we are able to compute the critical slowing
down exponents at the dynamical transition with arbitrary precision and for any
value of the number of colours p. When available, we compare our exact results
with numerical simulations. In addition, we present a detailed study of the
dynamical transition in the large p limit, showing that the system is not
equivalent to a random energy model.Comment: 10 pages, 3 figure
On Spin-Glass Complexity
We study the quenched complexity in spin-glass mean-field models satisfying
the Becchi-Rouet-Stora-Tyutin supersymmetry. The outcome of such study,
consistent with recent numerical results, allows, in principle, to conjecture
the absence of any supersymmetric contribution to the complexity in the
Sherrington-Kirkpatrick model. The same analysis can be applied to any model
with a Full Replica Symmetry Breaking phase, e.g. the Ising -spin model
below the Gardner temperature. The existence of different solutions, breaking
the supersymmetry, is also discussed.Comment: 4 pages, 2 figures; Text changed in some parts, typos corrected,
Refs. [17],[21] and [22] added, two Refs. remove
On the critical slowing down exponents of mode coupling theory
A method is provided to compute the parameter exponent yielding the
dynamic exponents of critical slowing down in mode coupling theory. It is
independent from the dynamic approach and based on the formulation of an
effective static field theory. Expressions of in terms of third order
coefficients of the action expansion or, equivalently, in term of six point
cumulants are provided. Applications are reported to a number of mean-field
models: with hard and soft variables and both fully-connected and dilute
interactions. Comparisons with existing results for Potts glass model, ROM,
hard and soft-spin Sherrington-Kirkpatrick and p-spin models are presented.Comment: 4 pages, 1 figur
Large Deviations of the Free-Energy in Diluted Mean-Field Spin-Glass
Sample-to-sample free energy fluctuations in spin-glasses display a markedly
different behaviour in finite-dimensional and fully-connected models, namely
Gaussian vs. non-Gaussian. Spin-glass models defined on various types of random
graphs are in an intermediate situation between these two classes of models and
we investigate whether the nature of their free-energy fluctuations is Gaussian
or not. It has been argued that Gaussian behaviour is present whenever the
interactions are locally non-homogeneous, i.e. in most cases with the notable
exception of models with fixed connectivity and random couplings . We confirm these expectation by means of various analytical
results. In particular we unveil the connection between the spatial
fluctuations of the populations of populations of fields defined at different
sites of the lattice and the Gaussian nature of the free-energy fluctuations.
On the contrary on locally homogeneous lattices the populations do not
fluctuate over the sites and as a consequence the small-deviations of the free
energy are non-Gaussian and scales as in the Sherrington-Kirkpatrick model
Analysis of the infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model
In this work we analyse the Parisi's infinity-replica symmetry breaking
solution of the Sherrington - Kirkpatrick model without external field using
high order perturbative expansions. The predictions are compared with those
obtained from the numerical solution of the infinity-replica symmetry breaking
equations which are solved using a new pseudo-spectral code which allows for
very accurate results. With this methods we are able to get more insight into
the analytical properties of the solutions. We are also able to determine
numerically the end-point x_{max} of the plateau of q(x) and find that lim_{T
--> 0} x_{max}(T) > 0.5.Comment: 15 pages, 11 figures, RevTeX 4.
Bond chaos in the Sherrington-Kirkpatrick model
We calculate the probability distribution of the overlap between a spin glass
and a copy of itself in which the bonds are randomly perturbed in varying
degrees. The overlap distribution is shown to go to a delta distribution in the
thermodynamic limit for arbitrarily small perturbations (bond chaos) and we
obtain the scaling behaviour of the distribution with system size N in the high
and low temperature phases and exactly at the critical temperature. The results
are relevant for the free energy fluctuations in the Sherrington-Kirkpatrick
model.Comment: 12 pages, no figure
Chaos in temperature in the Sherrington-Kirkpatrick model
We prove the existence of chaos in temperature in the
Sherringhton-Kirkpatrick model. The effect is exceedingly small, namely of the
ninth order in perturbation theory. The equations describing two systems at
different temperatures constrained to have a fixed overlap are studied
analytically and numerically, yielding information about the behaviour of the
overlap distribution function in finite-size systems.Comment: REVTEX, 6 pages, 2 figure
Quenched Computation of the Complexity of the Sherrington-Kirkpatrick Model
The quenched computation of the complexity in the
Sherrington-Kirkpatrick model is presented. A modified Full Replica
Symmetry Breaking Ansatz is introduced in order to study the complexity
dependence on the free energy. Such an Ansatz corresponds to require
Becchi-Rouet-Stora-Tyutin supersymmetry. The complexity computed this way is
the Legendre transform of the free energy averaged over the quenched disorder.
The stability analysis shows that this complexity is inconsistent at any free
energy level but the equilibirum one. The further problem of building a
physically well defined solution not invariant under supersymmetry and
predicting an extensive number of metastable states is also discussed.Comment: 19 pages, 13 figures. Some formulas added corrected, changes in
discussion and conclusion, one figure adde
Magnetic field chaos in the SK Model
We study the Sherrington--Kirkpatrick model, both above and below the De
Almeida Thouless line, by using a modified version of the Parallel Tempering
algorithm in which the system is allowed to move between different values of
the magnetic field h. The behavior of the probability distribution of the
overlap between two replicas at different values of the magnetic field h_0 and
h_1 gives clear evidence for the presence of magnetic field chaos already for
moderate system sizes, in contrast to the case of temperature chaos, which is
not visible on system sizes that can currently be thermalized.Comment: Latex, 16 pages including 20 postscript figure
- …