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In the study of disordered models like spin glasses the key object of interest is the rugged energy hypersurface defined in configuration space. The statistical mechanics calculation of the Gibbs–Boltzmann partition function gives the information necessary to understand the equilibrium behavior of the system as a function of the temperature but is not enough if we are interested in the more general aspects of the hypersurface: it does not give us, for instance, the different degrees of ruggedness at different scales. In the context of the replica symmetry breaking (RSB) approach we discuss here a rather simple extension that can provide a much more detailed picture. The attractiveness of the method relies on the fact that it is conceptually transparent and the additional calculations are rather straightforward. We think that this approach reveals an ultrametric organisation with many levels in models like p-spin glasses when we include saddle points. In this first paper we present detailed calculations for the spherical p-spin glass model where we discover that the corresponding decreasing Parisi function q(x) codes this hidden ultrametric organisation.

Abdus Salam was a great man in more than one dimension. The conception and building of the ICTP system required much more than the intelligence of a great scientist. I will stress those other facets that made him such a unique personality: the optimism that coloured his views about men and women, his love for his people and his commitment to the less favoured peoples of the world and a crucial ingredient, his deep and complex sense of ethical values that pushed him towards engagement in the political reality. Endowed with a formidable power of persuasion and a healthy lack of respect for norms or rules that are not based on justice he made a big difference on many of us and will remain as an icon for future generations. I will also describe the final touches that Salam’s collaborators and successors had to add to keep his heritage flourishing.

We study the replica free energy surface for a spin glass model near the glassy temperature. In this model the simplicity of the equilibrium solution hides non trivial metastable saddle points. By means of the stability analysis performed for one and two real replicas constrained, an interpretation for some of them is achieved., Comment: 10 pages and 3 figures upon request, Univerista` di Roma I preprint 94/1008

In the context of the p-spin spherical model we show how saddle points of the n - 0 replica free energy surface can be interpreted as metastable states or barriers.

We apply the technique introduced in a previous paper, to study the violation of ultrametricity in a Spin Glass model. We consider 3 real replicas of a Spin Glass near the glassy temperature, constrained in such a way that their overlaps violate ultrametricity. We compute the free energy of such a system and find that the additional free energy required by the constraint is extensive (proportional to N). This result is likely to be independent of the model chosen, and to be a general feature in Mean-Field Spin Glass systems.

In this note we introduce a method to calculate the finite volume corrections to the mean field results for the free energy when replica symmetry is broken at one-step. We find that the naive results are modified by the presence of additional corrections: these corrections can be interpreted as arising from fluctuations in the size of the blocks in the replica approach. The computation suggests a new approach for deriving the replica broken results in a rigorous way., 10 pages

In order to study the limits of validity of ultrametricity we imagine a system composed of two SK models weakly coupled among themselves. For this case one assumes, as usual, that each one of them separately obeys ultrametricity while it is an open question whether the relevant configurations of the total system intended as a whole can be so classified. For zero coupling the andwer is trivially negative. For arbitrarily small interaction energy, provided it is extensive, an analysis of the associated generalized random energy models suggests that ultrametricity is recovered.

We display a synaptic matrix that can efficiently store, in attractor neural networks (AMN) and perceptrons, patterns organized in uncorrelated classes. We find a storage capacity limit increasing with m, the overlap of a pattern with its class ancestor, and diverging as m→1. The probability distributions of the local stability parameters is studied, leading to a complete analysis of the performance of a perceptron with this synaptic matrix, and to a qualitative understanding of the behavior of the corresponding AMN. The analysis of the retrieval attractor of the ANN is completed via statistical mechanics. The motivation for the construction of this matrix was to make possible a study of a model for prosopagnosia, i.e. the shift from individual to class recall, under lesion, i.e. a rand On met en evidence une matrice synaptique qui stocke efficacement les patterns organises en categories non correlees dans les reseaux neuronaux a attracteurs et les perceptrons. La capacite de stockage limite augmente avec le recouvrement m d'un pattern avec sa categorie ancestrale, et diverge lorsque m tend vers 1. La distribution de probabilite des parametres de stabilite locaux est etudiee, et conduit a une analyse complete des performances d'un perceptron en fonction de sa matrice synaptique, ainsi qu'a une comprehension qualitative du comportement du reseau neuronal correspondant. L'analyse de l'attracteur du reseau est completee a l'aide de la mecanique statistique. La motivation d'une telle construction est de rendre possible l'etude d'un modele de prosopagnosie: le passage du rappel individuel a celui de categories lors de lesions, c'est-a-dire d'une deterioration aleatoire des efficacites synaptiques. Les proprietes de rappel du modele en fonction de la matrice synaptique proposee sont etudiees en detail. Enfin nous comparons notre matrice synaptique a une matrice generique dont tous les parametres de stabilite sont positifs

The historical path leading to the so-called Virasoro algebra is recalled, and the associated physical context is briefly discussed.

Strong coupling of the multi-partial wave meson isotriplet, B. Sakita generalization of dual models, J.-L. Gervais and B. Sakita functional approach to strong-coupling theory in static models I - charge-scalar models, G.C. Branco et al gauge degrees of freedom, external changes, quark confinement criterion in A0=0 canonical formalism, J.-L. Gervais and B. Sakita SO(2N) grand unification in an SU(N) basis, R.N. Mohapatra and B. Sakita chiral symmetry and chiral anomaly in incommensurate charge-density-wave system, Z.-B. Su and B. Sakita local chiral symmetry and charge-density waves in one-dimensional conductors, B. Sakita and K. Shizuya. (Part contents)

We compare neural network behaviour with human behaviour in a quasiregular environment. We find both interesting similarities and essential differences. The most striking difference is that a generic neural network does not use the simplest rule as a default rule. Our analysis applies to large classes of networks.

The hierarchical organization of the pure states of a S.K. spin glass (ultrametricity) is analysed in terms of self-averaging distributions of local magnetizations. We show that every pure state α defines an ultrametric distance Dα(i,j) among the N sites. Given two states α, β with overlap q there is a minimum distande d m such that for two sites i, j with Dα(i,j)≥d m the two distances Dα and Dβ coincide. It follows that the sites can be partitioned in disjoint cells inside which the total magnetization is the same for all the states with mutual overlap q. For this same family of states we then define an «ancestor» that has, inside each cell, constant local magnetization equal to the average magnetization of the descendants. The ancestors satisfy mean field like equations. The functional dependence of the local magnetization in terms of the local field is given by the solution of the diffusion equation in x space which is given a purely static interpretation Analyse de l'organisation hierarchique des etats purs d'un verre de spin de Sherrington-Kirkpatrick en termes de la distribution des aimantations locales: chaque etat pur α definit une distance ultrametrique Dα(i,j) entre les N sites et, etant donnes deux etats α, β dont le recouvrement est q, il y a une distance minimale d m telle que, pour chaque paire de site i, j verifiant Dα(i,j)≥d m , les deux distances Dα et Dβ coincident

In the original formulation of Hopfield's memory model, the learning rule setting the interaction strengths is best suited for orthogonal words. From the point of view of categorization, this feature is not convenient unless we reinterpret these words as primordial categories. But then one has to complete the model so as to be able to store a full hierarchical tree of categories embodying subcategories and so on. We use recent results on the spin glass mean field theories to show that this completion can be done in a natural way with a minimal modification of Hebb's rule for learning. Categorization emerges naturally from an encoding stage structured in layers.

We show that explicit replica symmetry breaking can be implemented by averaging over a small random magnetic field the partition function to an appropriate power. These results give a new insight on the physical meaning of replica symmetry breaking and a new way to compute the spin glass order parameter q(x) On montre qu'une brisure explicite de la symetrie des repliques peut etre realisee en moyennant sur un petit champ magnetique aleatoire une puissance convenable de la fonction de partition. Ces resultats jettent un nouvel eclairage sur la signification physique de la brisure de symetrie des repliques et donnent une nouvelle maniere de calculer le parametre d'ordre des verres de spin q(x)

A probability distribution has been proposed recently by one of us as an order parameter for spin glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology Etude d'une distribution de probabilite, proposee recemment, comme parametre d'onde pour les verres de spin: la probabilite depend de la realisation particuliere des couplages, meme a la limite thermodynamique. Etude de la distribution de cette probabilite. Mise en evidence de la topologie ultrametrique de l'espace des etats

We introduce a new method, which does not use replicas, from which we recover all the results of the replica symmetry-breaking solution of the Sherrington-Kirkpatrick model.

A probability distribution has been proposed recently by one of us as an order parameter for spin-glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology.

The free energies of the pure states in the spin glass phase are studied in the mean field theory. They are shown to be independent random variables with an exponential distribution. Physical implications concerning the fluctuations from sample to sample are worked out. The physical nature of the mean field theory is fully characterized.

An input-output map in which the patterns are divided into classes is considered for the perceptron. The statistical mechanical analysis with a finite number of classes turns out to give the same results as the case of only one class of patterns; the limit of capacity and the relevant order parameters are calculated in a mean field approach. The analysis is then extended to the Derrida Gardner canonical ensemble in which the perceptron can be studied beyond the limit of capacity. We complete the analysis with numerical simulations with the perceptron learning rule. The revelance of those results to the possible emergence of spontaneous categorization is finally discussed Nous considerons une application entree-sortie pour un perceptron dans laquelle les formes sont divisees en classe. L'analyse par la mecanique statistique dans le cas d'un nombre fini de classe donne les memes resultats que pour une seule classe; nous calculons la limite de capacite et les parametres d'ordre pertinents en champ moyen. Nous generalisons ensuite l'analyse a l'ensemble canonique de Derrida-Gardner dans lequel le perceptron peut etre etudie au-dela de sa limite de capacite. Nous completons l'analyse en etudiant numeriquement la regle d'apprentissage du perceptron. Nous discutons finalement la relevance de ces resutlats a l'emergence possible d'une categorisation spontanee

Ultrametricity is a simple topological concept, but its appearance in the language of physicists is recent. This review provides all the elementary background (from mathematics, taxonomy, and statistical physics) and surveys the main fields of development (spin glasses, optimization theory). Static and dynamic aspects are covered. From present knowledge, one can already draw some tentative conclusions on the common causes for the occurrence of ultrametric structures in nature. Some perspectives on unresolved problems in physics and biology are also presented.

The height of barriers separating metastable states is studied for the infinite range Ising spin-glass model. A barrier configuration corresponds to a saddle point {m i , i=1,..., N, -1≤m i ≤1} of an energy surface that smoothly interpolates the energy in the hypercube. For low energy barriers, which are relevant for the dynamics, the number of independent descending directions from the saddle point is finite when N→∞. It is found that these descending directions are contained in the linear subspace of the hypercube [-1,1] N generated by directions for which Σ j J ij m j =0. Numerical estimates were performed by two distinct algorithms which lead to consistent results. Assuming barrier heights growing with system size N as Nα we find α=0.34±0.08 Nous etudions les hauteurs de barriere separant des etats metastables pour le modele de verre de spin avec symetrie d'Ising et portee infinie. Une configuration de barriere correspond a un col de {m i , i=1,..., N, -1≤m i ≤1} de la surface d'energie qui interpole de facon reguliere l'energie dans l'hypercube. Pour des barrieres d'energie faibles, qui sont importantes pour la dynamique, le nombre de directions descendantes au col est fini dans la limite N→∞. Nous trouvons que ces directions sont contenues dans un sous-espace lineaire de l'hypercube [-1,1] N engendre par les directions pour lesquelles Σ j J ij m j =0. Nous avons fait des simulations numeriques par deux algorithmes differents. Les resultats sont coherents. Si nous supposons que les hauteurs de barriere croissent avec la taille N du systeme comme Nα, nous trouvons α=0,34±0,08

We use zero temperature mean field equations to study numerically several properties of the Sherrington-Kirkpatrick model. In particular we observe the fluctuations from sample to sample of the order parameter q(x) and see partial evidence of the ultrametricity structure among pure states A partir des equations de champ moyen a temperature nulle on etudie numeriquement les proprietes du modele de Sherrington-Kirkpatrick. En particulier on observe les fluctuations du parametre d'ordre q(x) quand on change de couplages J ij et on voit partiellement la structure ultrametrique parmi les etats purs.

We re-examine Nielsen's interpretation of the Veneziano amplitudes from the point of view of the functional formulation. It is shown that the sum of a large number of "fishnet" Feynman graphs of very high order can be approximated by generalized Veneziano amplitudes.

A general formulation of duality theory is presented that includes nonplanar Feynman-like diagrams. All diagrams, planar as well as nonplanar, are so classified that the diagrams in a given class are mutually connected by duality. A prescription is given for constructing an integral representation of the scattering amplitude for each class. Some fundamental properties of the duality relations are discussed.

We investigate the possible subsidiary conditions compatible with the equations of motion of dual-resonance models for the unrealistic choice of $\ensuremath{\alpha}(0)=1$. In the language of four-dimensional harmonic oscillators, we find one subsidiary condition for each mode of oscillation. All time components (in the c.m. system) can therefore be eliminated. We discuss the possibility of relaxing the condition $\ensuremath{\alpha}(0)=1$.

We propose a perturbative approach in which the Veneziano representation plays the role of a Born term. We interpret Veneziano's formula as describing only the contribution of one-particle intermediate states. We then add to it the contribution of many-particle intermediate states by means of Feynman-like diagrams. The rules for writing the integrals corresponding to any planar diagram are given. Crossing symmetry, duality, and Reggeization are explicitly taken into account. We find the asymptotic behavior of each Feynman-like diagram. We sum them and prove that the whole amplitude has Regge behavior. The new trajectory, however, is no longer linear, and it incorporates correctly the elastic unitarity constraint. We argue that this approach will ultimately provide a framework in which generalized unitarity (in Cutkosky's sense) can be imposed.

We generalize Veneziano's formula for the five-point function by writing an integral formula on a two-dimensional space and imposing on it the requirements implied by crossing symmetry, analyticity, and Regge behavior.

A representation for the scattering amplitude that contains Regge behavior, crossing symmetry, and analyticity is discussed. It is shown that it provides a different ghost-eliminating mechanism (the Mandelstam-Wang one) from that given by Veneziano's proposal. Furthermore, it does not restrict the external masses, but reduces to Veneziano's formula when $\ensuremath{\alpha}(s)+\ensuremath{\alpha}(t)+\ensuremath{\alpha}(u)$ equals a particular integer that depends on the reaction. Several examples are discussed. The behavior of the Regge residue ${\ensuremath{\beta}}_{\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}}(t)$ at ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(t)=0$ is proposed as a test that distinguishes the two representations.

The hierarchical organization of the pure states of a S.K. spin glass (ultrametricity) is analysed in terms of self-averaging distributions of local magnetizations. We show that every pure state α defines an ultrametric distance Dα(i,j) among the N sites. Given two states α, β with overlap q there is a minimum distande d m such that for two sites i, j with Dα(i,j)≥d m the two distances Dα and Dβ coincide. It follows that the sites can be partitioned in disjoint cells inside which the total magnetization is the same for all the states with mutual overlap q. For this same family of states we then define an «ancestor» that has, inside each cell, constant local magnetization equal to the average magnetization of the descendants. The ancestors satisfy mean field like equations. The functional dependence of the local magnetization in terms of the local field is given by the solution of the diffusion equation in x space which is given a purely static interpretation Analyse de l'organisation hierarchique des etats purs d'un verre de spin de Sherrington-Kirkpatrick en termes de la distribution des aimantations locales: chaque etat pur α definit une distance ultrametrique Dα(i,j) entre les N sites et, etant donnes deux etats α, β dont le recouvrement est q, il y a une distance minimale d m telle que, pour chaque paire de site i, j verifiant Dα(i,j)≥d m , les deux distances Dα et Dβ coincident

A probability distribution has been proposed recently by one of us as an order parameter for spin glasses. We show that this probability depends on the particular realization of the couplings even in the thermodynamic limit, and we study its distribution. We also show that the space of states has an ultrametric topology Etude d'une distribution de probabilite, proposee recemment, comme parametre d'onde pour les verres de spin: la probabilite depend de la realisation particuliere des couplages, meme a la limite thermodynamique. Etude de la distribution de cette probabilite. Mise en evidence de la topologie ultrametrique de l'espace des etats

In the original formulation of Hopfield's memory model, the learning rule setting the interaction strengths is best suited for orthogonal words. From the point of view of categorization, this feature is not convenient unless we reinterpret these words as primordial categories. But then one has to complete the model so as to be able to store a full hierarchical tree of categories embodying subcategories and so on. We use recent results on the spin glass mean field theories to show that this completion can be done in a natural way with a minimal modification of Hebb's rule for learning. Categorization emerges naturally from an encoding stage structured in layers.