Mathematical aspects of mean field spin glass theory

A comprehensive review will be given about the rich mathematical structure of mean field spin glass theory, mostly developed, until now, in the frame of the methods of theoretical physics, based on deep physical intuition and hints coming from numerical simulation. Central to our treatment is a very simple and yet powerful interpolation method, allowing to compare different probabilistic schemes, by using convexity and positivity arguments. In this way we can prove the existence of the thermodynamic limit for the free energy density of the system, a long standing open problem. Moreover, in the frame of a generalized variational principle, we can show the emergency of the Derrida-Ruelle random probability cascades, leading to the form of free energy given by the celebrated Parisi \textit {Ansatz}. All these results seem to be in full agreement with the mechanism of spontaneous replica symmetry breaking as developed by Giorgio Parisi.Comment: proceedings of the "4th European Congress of Mathematics", Stockholm, 2004. 17 page

The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature

We study a spin system on a large box with both Ising interaction and Sherrington-Kirpatrick couplings, in the presence of an external field. Our results are: (i) existence of the pressure in the limit of an infinite box. When both Ising and Sherrington-Kirpatrick temperatures are high enough, we prove that: (ii) the value of the pressure is given by a suitable replica symmetric solution, and (iii) the fluctuations of the pressure are of order of the inverse of the square of the volume with a normal distribution in the limit. In this regime, the pressure can be expressed in terms of random field Ising models

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

New expression for the K-shell ionization

A new expression for the total K-shell ionization cross section by electron impact based on the relativistic extension of the binary encounter Bethe (RBEB) model, valid from ionization threshold up to relativistic energies, is proposed. The new MRBEB expression is used to calculate the K-shell ionization cross sections by electron impact for the selenium atom. Comparison with all, to our knowledge, available experimental data shows good agreement

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

Surface terms on the Nishimori line of the Gaussian Edwards-Anderson model

For the Edwards-Anderson model we find an integral representation for some surface terms on the Nishimori line. Among the results are expressions for the surface pressure for free and periodic boundary conditions and the adjacency pressure, i.e., the difference between the pressure of a box and the sum of the pressures of adjacent sub-boxes in which the box can been decomposed. We show that all those terms indeed behave proportionally to the surface size and prove the existence in the thermodynamic limit of the adjacency pressure.Comment: Final version with minor corrections. To appear in Journal of Statistical Physic

Responding to Morally Flawed Historical Philosophers and Philosophies

Many historically-influential philosophers had profoundly wrong moral views or behaved very badly. Aristotle thought women were “deformed men” and that some people were slaves “by nature.” Descartes had disturbing views about non-human animals. Hume and Kant were racists. Hegel disparaged Africans. Nietzsche despised sick people. Mill condoned colonialism. Fanon was homophobic. Frege was anti-Semitic; Heidegger was a Nazi. Schopenhauer was sexist. Rousseau abandoned his children. Wittgenstein beat his young students. Unfortunately, these examples are just a start. These philosophers are famous for their intellectual accomplishments, yet they display serious moral or intellectual flaws in their beliefs or actions. At least, some of their views were false, ultimately unjustified and, perhaps, harmful. How should we respond to brilliant-but-flawed philosophers from the past? Here we explore the issues, asking questions and offering few answers. Any insights gained here might be applicable to contemporary imperfect philosophers, scholars in other fields, and people in general

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