7,538 research outputs found
The replica symmetric behavior of the analogical neural network
In this paper we continue our investigation of the analogical neural network,
paying interest to its replica symmetric behavior in the absence of external
fields of any type. Bridging the neural network to a bipartite spin-glass, we
introduce and apply a new interpolation scheme to its free energy that
naturally extends the interpolation via cavity fields or stochastic
perturbations to these models. As a result we obtain the free energy of the
system as a sum rule, which, at least at the replica symmetric level, can be
solved exactly. As a next step we study its related self-consistent equations
for the order parameters and their rescaled fluctuations, found to diverge on
the same critical line of the standard Amit-Gutfreund-Sompolinsky theory.Comment: 17 page
Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model
In a region above the Almeida-Thouless line, where we are able to control the
thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica
symmetry, we show that the fluctuations of the overlaps and of the free energy
are Gaussian, on the scale N^{-1/2}, for N large. The method we employ is based
on the idea, we recently developed, of introducing quadratic coupling between
two replicas. The proof makes use of the cavity equations and of concentration
of measure inequalities for the free energy.Comment: 18 page
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
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
Mean field dilute ferromagnet I. High temperature and zero temperature behavior
We study the mean field dilute model of a ferromagnet. We find and prove an
expression for the free energy density at high temperature, and at temperature
zero. We find the critical line of the model, separating the phase with zero
magnetization from the phase with symmetry breaking. We also compute exactly
the entropy at temperature zero, which is strictly positive. The physical
behavior at temperature zero is very interesting and related to infinite
dimensional percolation, and suggests possible behaviors at generic low
temperatures. Lastly, we provide a complete solution for the annealed model.
Our results hold both for the Poisson and the Bernoulli versions of the model.Comment: 38 page
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
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
Structural Properties of the Disordered Spherical and other Mean Field Spin Models
We extend the approach of Aizenman, Sims and Starr for the SK-type models to
their spherical versions. Such an extension has already been performed for
diluted spin glasses. The factorization property of the optimal structures
found by Guerra for the SK model, which holds for diluted models as well, is
verified also in the case of spherical systems, with the due modifications.
Hence we show that there are some common structural features in various mean
field spin models. These similarities seem to be quite paradigmatic, and we
summarize the various techniques typically used to prove the structural
analogies and to tackle the computation of the free energy per spin in the
thermodynamic limit.Comment: 24 page
Local Spin Glass Order in 1D
We study the behavior of one dimensional Kac spin glasses as function of the
interaction range. We verify by Montecarlo numerical simulations the crossover
from local mean field behavior to global paramagnetism. We investigate the
behavior of correlations and find that in the low temperature phase
correlations grow at a faster rate then the interaction range. We completely
characterize the growth of correlations in the vicinity of the mean-field
critical region
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