5,544 research outputs found
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
General properties of overlap probability distributions in disordered spin systems. Toward Parisi ultrametricity
For a very general class of probability distributions in disordered Ising
spin systems, in the thermodynamical limit, we prove the following property for
overlaps among real replicas. Consider the overlaps among s replicas. Add one
replica s+1. Then, the overlap q(a,s+1) between one of the first s replicas,
let us say a, and the added s+1 is either independent of the former ones, or it
is identical to one of the overlaps q(a,b), with b running among the first s
replicas, excluding a. Each of these cases has equal probability 1/s.Comment: LaTeX2e, 11 pages. Submitted to Journal of Physics A: Mathematical
and General. Also available at
http://rerumnatura.zool.su.se/stefano/ms/ghigu.p
Interpolating the Sherrington-Kirkpatrick replica trick
The interpolation techniques have become, in the past decades, a powerful
approach to lighten several properties of spin glasses within a simple
mathematical framework. Intrinsically, for their construction, these schemes
were naturally implemented into the cavity field technique, or its variants as
the stochastic stability or the random overlap structures. However the first
and most famous approach to mean field statistical mechanics with quenched
disorder is the replica trick. Among the models where these methods have been
used (namely, dealing with frustration and complexity), probably the best known
is the Sherrington-Kirkpatrick spin glass: In this paper we are pleased to
apply the interpolation scheme to the replica trick framework and test it
directly to the cited paradigmatic model: interestingly this allows to obtain
easily the replica-symmetric control and, synergically with the broken replica
bounds, a description of the full RSB scenario, both coupled with several minor
theorems. Furthermore, by treating the amount of replicas as an
interpolating parameter (far from its original interpretation) this can be
though of as a quenching temperature close to the one introduce in
off-equilibrium approaches and, within this viewpoint, the proof of the
attended commutativity of the zero replica and the infinite volume limits can
be obtained.Comment: This article is dedicated to David Sherrington on the occasion of his
seventieth birthda
Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique
During the last years, through the combined effort of the insight, coming
from physical intuition and computer simulation, and the exploitation of
rigorous mathematical methods, the main features of the mean field
Sherrington-Kirkpatrick spin glass model have been firmly established. In
particular, it has been possible to prove the existence and uniqueness of the
infinite volume limit for the free energy, and its Parisi expression, in terms
of a variational principle, involving a functional order parameter. Even the
expected property of ultrametricity, for the infinite volume states, seems to
be near to a complete proof. The main structural feature of this model, and
related models, is the deep phenomenon of spontaneous replica symmetry breaking
(RSB), discovered by Parisi many years ago. By expanding on our previous work,
the aim of this paper is to investigate a general frame, where replica symmetry
breaking is embedded in a kind of mechanical scheme of the Hamilton-Jacobi
type. Here, the analog of the "time" variable is a parameter characterizing the
strength of the interaction, while the "space" variables rule out
quantitatively the broken replica symmetry pattern. Starting from the simple
cases, where annealing is assumed, or replica symmetry, we build up a
progression of dynamical systems, with an increasing number of space variables,
which allow to weaken the effect of the potential in the Hamilton-Jacobi
equation, as the level of symmetry braking is increased. This new machinery
allows to work out mechanically the general K-step RSB solutions, in a
different interpretation with respect to the replica trick, and lightens easily
their properties as existence or uniqueness.Comment: 24 pages, no figure
The mean field theory of spin glasses: the heuristic replica approach and recent rigorous results
The mathematically correct computation of the spin glasses free energy in the
infinite range limit crowns 25 years of mathematic efforts in solving this
model. The exact solution of the model was found many years ago by using a
heuristic approach; the results coming from the heuristic approach were crucial
in deriving the mathematical results. The mathematical tools used in the
rigorous approach are quite different from those of the heuristic approach. In
this note we will review the heuristic approach to spin glasses in the light of
the rigorous results; we will also discuss some conjectures that may be useful
to derive the solution of the model in an alternative way.Comment: 12 pages, 1 figure; lecture at the Flato Colloquia Day, Thursday 27
November, 200
A Statistical Interpretation of Space and Classical-Quantum duality
By defining a prepotential function for the stationary Schr\"odinger equation
we derive an inversion formula for the space variable as a function of the
wave-function . The resulting equation is a Legendre transform that
relates , the prepotential , and the probability density. We
invert the Schr\"odinger equation to a third-order differential equation for
and observe that the inversion procedure implies a -
duality. This phenomenon is related to a modular symmetry due to the
superposition of the solutions of the Schr\"odinger equation. We propose that
in quantum mechanics the space coordinate can be interpreted as a macroscopic
variable of a statistical system with playing the role of a scaling
parameter. We show that the scaling property of the space coordinate with
respect to is determined by the
``beta-function''. We propose that the quantization of the inversion formula is
a natural way to quantize geometry. The formalism is extended to higher
dimensions and to the Klein-Gordon equation.Comment: 11 pages. Standard Latex. Final version to appear in Physical Review
Letters. Revised and extended version. The formalism is extended to higher
dimensions and to the Klein-Gordon equation. A possible connection with
string theory is considered. The duality is emphasized by a minor
change in the title. The new title is: Duality of and and a
statistical interpretation of space in quantum mechanic
How glassy are neural networks?
In this paper we continue our investigation on the high storage regime of a
neural network with Gaussian patterns. Through an exact mapping between its
partition function and one of a bipartite spin glass (whose parties consist of
Ising and Gaussian spins respectively), we give a complete control of the whole
annealed region. The strategy explored is based on an interpolation between the
bipartite system and two independent spin glasses built respectively by
dichotomic and Gaussian spins: Critical line, behavior of the principal
thermodynamic observables and their fluctuations as well as overlap
fluctuations are obtained and discussed. Then, we move further, extending such
an equivalence beyond the critical line, to explore the broken ergodicity phase
under the assumption of replica symmetry and we show that the quenched free
energy of this (analogical) Hopfield model can be described as a linear
combination of the two quenched spin-glass free energies even in the replica
symmetric framework
A note on the Guerra and Talagrand theorems for Mean Field Spin Glasses: the simple case of spherical models
The aim of this paper is to discuss the main ideas of the Talagrand proof of
the Parisi Ansatz for the free-energy of Mean Field Spin Glasses with a
physicist's approach. We consider the case of the spherical -spin model,
which has the following advantages: 1) the Parisi Ansatz takes the simple ``one
step replica symmetry breaking form'', 2) the replica free-energy as a function
of the order parameters is simple enough to allow for numerical maximization
with arbitrary precision. We present the essential ideas of the proof, we
stress its connections with the theory of effective potentials for glassy
systems, and we reduce the technically more difficult part of the Talagrand's
analysis to an explicit evaluation of the solution of a variational problem.Comment: 20 pages, 5 figures. Added references and minor language correction
Crowd-based cognitive perception of the physical world: Towards the internet of senses
This paper introduces a possible architecture and discusses the research directions for the realization of the Cognitive Perceptual Internet (CPI), which is enabled by the convergence of wired and wireless communications, traditional sensor networks, mobile crowd-sensing, and machine learning techniques. The CPI concept stems from the fact that mobile devices, such as smartphones and wearables, are becoming an outstanding mean for zero-effort world-sensing and digitalization thanks to their pervasive diffusion and the increasing number of embedded sensors. Data collected by such devices provide unprecedented insights into the physical world that can be inferred through cognitive processes, thus originating a digital sixth sense. In this paper, we describe how the Internet can behave like a sensing brain, thus evolving into the Internet of Senses, with network-based cognitive perception and action capabilities built upon mobile crowd-sensing mechanisms. The new concept of hyper-map is envisioned as an efficient geo-referenced repository of knowledge about the physical world. Such knowledge is acquired and augmented through heterogeneous sensors, multi-user cooperation and distributed learning mechanisms. Furthermore, we indicate the possibility to accommodate proactive sensors, in addition to common reactive sensors such as cameras, antennas, thermometers and inertial measurement units, by exploiting massive antenna arrays at millimeter-waves to enhance mobile terminals perception capabilities as well as the range of new applications. Finally, we distillate some insights about the challenges arising in the realization of the CPI, corroborated by preliminary results, and we depict a futuristic scenario where the proposed Internet of Senses becomes true
Genetic effects of static magnetic fields. Body size increase and lethal mutations induced in populations of Drosophila melanogaster after chronic exposure
- …