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Despite the presence of topological disorder, phonons seem to exist also in glasses at very high frequencies (THz) and they remarkably persist into the supercooled liquid. A universal feature of such a systems is the Boson peak, an excess of states over the standard Debye contribution at the vibrational density of states. Exploiting the euclidean random matrix theory of vibrations in amorphous systems we show that this peak is the signature of a phase transition in the space of the stationary points of the energy, from a minima-dominated phase (with phonons) at low energy to a saddle-point dominated phase (without phonons). The theoretical predictions are checked by means of numeric simulations., Comment: to appear in the proceedings of the conference "Slow dynamics in complex sistems", Sendai (Japan) 2003

Glasses are amorphous solids, in the sense that they display elastic behaviour. In crystals, elasticity is associated with phonons, quantized sound-wave excitations. Phonon-like excitations exist also in glasses at very high frequencies (THz), and they remarkably persist into the supercooled liquid. A universal feature of these amorphous systems is the Boson peak: the vibrational density of states $g(\omega)$ has an excess over the Debye (squared frequency) law, seen as a peak in $g(\omega)/\omega^2$. We claim that this peak is the signature of a phase transition in the space of the stationary points of the energy, from a minima-dominated phase (with phonons) at low energy to a saddle-point dominated phase (without phonons). Here, by studying the spectra of inherent structures (local minima of the potential energy), we show that this is the case in a realistic glass model: the Boson peak moves to lower frequencies on approaching the phonon-saddle transition and its height diverges at the critical point. The numerical results agree with Euclidean Random Matrix Theory predictions on the existence of a sharp phase transition between an amorphous elastic phase and a phonon-free one., Comment: 3 postscript figures, to be published on Nature

We show by numerical simulations that the correlation function of the random field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not self-averaging. This is due to the formation of a bound state in the underlying field theory. We argue that this non perturbative phenomenon is not particular to the RFIM in 3-d. It is generic for disordered systems in two dimensions and may also happen in other three dimensional disordered systems.

We study numerically and analytically a simple off-lattice model of scalar harmonic vibrations by means of Euclidean random matrix theory. Since the spectrum of this model shares the most puzzling spectral features with the high-frequency domain of glasses (non-Rayleigh broadening of the Brillouin peak, boson peak and secondary peak), the Euclidean random matrix theory provide a single and fairly simple theoretical framework to their explanation., Comment: 11 pages, 7 postscript figures, Proceedings of Statphys 21

We study, analytically and numerically, an off-lattice model of scalar harmonic vibrations for structural glasses. The model has a Boson-Peak which we argue can be considered as a prototype for materials with a Boson Peak frequency that decreases with lowering temperature. The density evolution of the Boson Peak in silica is qualitatively reproduced. The dispersion relation is linear at the Boson Peak frequency, in agreement with experiments. In our model the Boson Peak is a precursor of a nearby mechanical instability. The Boson Peak is built up by the hybridization of the sound waves with extended, but non propagating (in the sense of a flat dispersion relation), modes., Comment: 4 pages, 3 postscript figures

We study the spectral width as a function of the external momentum for the dynamical structure factor of a disordered harmonic solid, considered as a toy model for supercooled liquids and glasses. Both in the context of single-link coherent potential approximation and of a single-defect approximation, two different regimes are clearly identified: if the density of states at zero energy is zero, the Rayleigh $p^4$ law is recovered for small momentum. On the contrary, if the disorder induces a non vanishing density of states at zero energy, a linear behaviour is obtained. The dynamical structure factor is numerically calculated in lattices as large as $96^3$, and satisfactorily agrees with the analytical computations., Comment: 7 pages plus 4 postscript figures

We compute the thermodynamic properties of the glass phase in a binary mixture of soft spheres. Our approach is a generalization to mixtures of the replica strategy, recently proposed by Mezard and Parisi, providing a first principle statistical mechanics computation of the thermodynamics of glasses. The method starts from the inter-atomic potentials, and translates the problem into the study of a molecular liquid. We compare uor analytical predictions to numerical simulations, focusing onto the values of the thermodynamic transition and the configurational entropy., Comment: 28 pages, 8 figures, revised version, accepted for the publication in the Journal of Chemical Physics

We investigate the phase transition in a strongly disordered short-range three-spin interaction model characterized by the absence of time reversal symmetry in the Hamiltonian. In the mean-field limit the model is well described by the Adam-Gibbs-DiMarzio scenario for the glass transition; however in the short-range case this picture turns out to be modified. The model presents a finite temperature continuous phase transition characterized by a divergent spin-glass susceptibility and a negative specific heat exponent. We expect the nature of the transition in this 3-spin model to be the same as the transition in the Edwards-Anderson model in a magnetic field, with the advantage that the strong crossover effects present in the latter case are absent., Comment: 19 pages, Latex, 16 ps figures

We propose a first principle computation of the equilibrium thermodynamics of simple fragile glasses starting from the two body interatomic potential. A replica formulation translates this problem into that of a gas of interacting molecules, each molecule being built of m atoms, and having a gyration radius (related to the cage size) which vanishes at zero temperature. We use a small cage expansion, valid at low temperatures, which allows to compute the cage size, the specific heat (which follows the Dulong and Petit law), and the configurational entropy., Comment: Latex, 40 pages, 9 figures, corrected misprints, improved presentation

We propose a first principle computation of the thermodynamics of simple fragile glasses starting from the two body interatomic potential. A replica formulation translates this problem into that of a gas of interacting molecules, each molecule being built of $m$ atoms, and having a gyration radius (related to the cage size) which vanishes at zero temperature. We use a small cage expansion, valid at low temperatures, which allows to compute the cage size, the specific heat (which follows the Dulong and Petit law), and the configurational entropy., Comment: Latex, 13 pages, 4 figures

In off-equilibrium dynamics we define a dynamical correlation length which is proportional to the size of the region in which the atoms move in a correlated way. General arguments indicate that this dynamical correlation length diverges at large times in the glassy phase. Numerical simulations for binary mixtures point toward the correctness of this prediction., Comment: 4 pages with 4 eps figures

In this work we calculate the dynamical fluctuations at O(1/N) in the low temperature phase of the $p=2$ spherical spin glass model. We study the large-times asymptotic regimes and we find, in a short time-differences regime, a fluctuation dissipation relation for the four-point correlation functions. This relation can be extended to the out of equilibrium regimes introducing a function $X_{t}$ which, for large time $t$, we find scales as $t^{-1/2}$ as in the case of the two-point functions., Comment: Latex, 8 pages

We use a constrained Monte Carlo technique to analyze ultrametric features of a 4 dimensional Edwards-Anderson spin glass with quenched couplings J=\pm 1. We find that in the large volume limit an ultrametric structure emerges quite clearly in the overlap of typical equilibrium configurations., Comment: 8 one column pages, latex, 4 figures with epsfig.sty

We discuss Gaussian fluctuations in a spin glass model with one replica symmetry breaking. We show how non-perturbative fluctuations of the break-point parameter can be included in the longitudinal propagator within linear response theory., Comment: 7 pages, latex source file

In this paper we study the on and off-equilibrium properties of the four dimensional Gaussian spin glass. In the static case we determine with more precision that in previous simulations both the critical temperature as well as the critical exponents. In the off-equilibrium case we settle the general form of the autocorrelation function, and show that is possible to obtain dynamically, for the first time, a value for the order parameter., Comment: 16 pages and 13 figures, uses epsfig.sty and rotate.sty. Some minor grammatical changes. Also available at http://chimera.roma1.infn.it/index_papers_complex.html

We rederive the finite size scaling formula for the apparent critical temperature by using Mean Field Theory for the Ising Model above the upper critical dimension. We have also performed numerical simulations in five dimensions and our numerical data are in a good agreement with the Mean Field theoretical predictions, in particular, with the finite size exponent of the connected susceptibility and with the value of the Binder cumulant., Comment: 9 pages and 3 figures, available at http://chimera.roma1.infn.it/index_papers_complex.html

We show that the predictions of spontaneously broken replica symmetry are in perfect agreement with the recent rigorous results obtained by Newman and Stein., Comment: 2 pages, aps style

We have carried out numerical simulations of the three-dimensional Ising spin glass model with first neighbour Gaussian couplings using three replicas for each sample of couplings. We have paid special attention to the measure of two types of Binder cumulant that can be constructed from the three possible overlaps between the replicas. We obtain new information about the possible phase transition and perform an initial analysis of the ultrametricity issue., Comment: 14 pages and 7 figures, available at http://chimera.roma1.infn.it/index_papers_complex.html

We study the analytical solution of the Monte Carlo dynamics in the spherical Sherrington-Kirkpatrick model using the technique of the generating function. Explicit solutions for one-time observables (like the energy) and two-time observables (like the correlation and response function) are obtained. We show that the crucial quantity which governs the dynamics is the acceptance rate. At zero temperature, an adiabatic approximation reveals that the relaxational behavior of the model corresponds to that of a single harmonic oscillator with an effective renormalized mass., Comment: Uuencoded file including: REVTEX (33 pages) and 7 figures (PostScript).

We propose a method to study quantitatively the glass transition in a system of interacting particles. In spite of the absence of any quenched disorder, we introduce a replicated version of the hypernetted chain equations. The solution of these equations, for hard or soft spheres, signals a transition to the glass phase. However the predicted value of the energy and specific heat in the glass phase are wrong, calling for an improvement of this method., Comment: 9 pages, four postcript figures attached

By studying the structure of infrared divergences in a toy propagator in the replica approach to the Ising spin glass below $T_c$, we suggest a possible cancellation mechanism which could decrease the degree of singularity in the loop expansion., Comment: 13 pages, Latex , revised version

We define a probabilistic scheme to compute the distributions of periods, transients and weigths of attraction basins in Kauffman networks. These quantities are obtained in the framework of the annealed approximation, first introduced by Derrida and Pomeau. Numerical results are in good agreement with the computed values of the exponents of average periods, but show also some interesting features which can not be explained whithin the annealed approximation., Comment: latex, 36 pages, figures added in uufiles format,error in epsffile name

By numerical simulations of the $3d$ Ising spin glass we find evidence that spontaneous replica symmetry breaking theory and not the droplet model describes with good accuracy the equilibrium behavior of the system., Comment: PHYSREV format, 2 .ps figures added with figure command in uufiles format

In this note we present an exact solution of the Monte Carlo dynamics of the spherical Sherrington-Kirkpatrick spin-glass model. We obtain the dynamical equations for a generalized set of moments which can be exactly closed. Only in a certain particular limit the dynamical equation of the energy coincides with that of the Langevin dynamics., Comment: 11 Pages + 2 Figures, Revtex

We show that computing the coefficients of the Taylor expansion of the solution of the off-equilibrium dynamical equations characterizing models with quenched disorder is a very effective way to understand the long time asymptotic behavior. We study the $p=3$ spherical spin glass model, and we compute the asymptotic energy (in the critical region and down to $T=0$) and the coefficients of the time decay of the energy., Comment: 9 pages, LaTeX, 3 uuencoded figures

We present a model of spheres moving in a high-dimensional compact space. We relate it to a mixed matrix model with a $O(N)$ invariant model plus a $P(N)$ invariant perturbation. We then study the low pressure regime by performing a diagrammatic expansion of this matrix model. Finally, we show the results from numerical simulations and we present evidence for a glassy regime at high pressures., Comment: 19 pages, 5 figures included, Z-compressed postscript

In this letter, we show that the formulae of Bray and Moore for the average logarithm of the number of metastable states in spin glasses can be obtained by calculating the partition function with $m$ coupled replicas with the symmetry among these explicitly broken according to a generalization of the `two-group' ansatz. This equivalence allows us to find solutions of the BM equations where the lower `band-edge' free energy equals the standard static free energy. We present these results for the Sherrington-Kirkpatrick model, but we expect them to apply to all mean-field spin glasses., Comment: 6 pages, LaTeX, no figures. Postscript directly available http://chimera.roma1.infn.it/index_papers_complex.html

In this paper we develop a method introduced by one of us to study metastable states in spin glasses. We consider a `potential function' defined as the free energy of a system at a given temperature $T$ constrained to have a fixed overlap with a reference configuration of equilibrium at temperature $T'$. We apply the method to the spherical p-spin glass and to some generalization of this model in the range of temperatures between the dynamic and the static transition. The analysis suggests a correspondence among local minima of the potential and metastable states. This correspondence is confirmed studying the relaxation dynamics at temperature $T$ of a system starting from an initial configuration equilibrated at a different temperature $T'$., Comment: 21 pages, latex, 4 uuencoded figures.

We use the mapping between Burgers' equation and the problem of a directed polymer in a random medium in order to study the fully developped turbulence in the $N$ dimensional forced Burgers' equation. The stirring force corresponds to a quenched (spatio temporal) random potential for the polymer. The properties of the inertial regime are deduced from a study of the directed polymer on length scales smaller than the correlation length of the potential. From this study we propose an Ansatz for the velocity field in the large Reynolds number limit of the forced Burgers' equation in $N$ dimensions. This Ansatz allows us to compute exactly the full probability distribution of the velocity difference $u(r)$ between points separated by a distance $r$ much smaller than the correlation length of the forcing. We find that the moments $$ scale as $r^{\zeta(q)}$ with $\zeta(q) \equiv 1$ for all $q \geq 1$. This strong `intermittency' is related to the large scale singularities of the velocity field, which is concentrated on a $N-1$ dimensional froth-like structure., Comment: 35 pages latex, 4 ps figures in separate uufiles package.

In this letter we show strong numerical evidence that the four dimensional Diluted Ising Model for a large dilution is not described by the Mean Field exponents. These results suggest the existence of a new fixed point with non-gaussian exponents., Comment: 9 pages. compressed ps-file (uufiles)

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e.\ the fully-frustrated model on an infinite dimensional hypercube and the so-called sine-model. We use the mean-field (or {\sc tap}) equations which we derive by resuming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the {\sc tap} equations. We find that the configurational entropy (or complexity) is extensive in the range $T_{\mbox{\tiny RSB}}

We study the statistical mechanics of a $D$-dimensional array of Josephson junctions in presence of a magnetic field on a lattice of side $2$. In the high temperature region the thermodynamical properties can be computed in the limit $D \to \infty$. A conjectural form of the thermodynamic properties in the low temperature phase is obtained by assuming that they are the same of an appropriate spin glass system, based on quenched disordered couplings. Numerical simulations show that this conjecture is very accurate in one regime of the magnetic field, while it is probably slightly inaccurate in a second regime., Comment: latex 35 pages, including figures added in uufiles format

We study the statistical mechanics of a model describing the coevolution of species interacting in a random way. We find that at high competition replica symmetry is broken. We solve the model in the approximation of one step replica symmetry breaking and we compare our findings with accurate numerical simulations., Comment: 12 pages, TeX, 5 postscript figures are avalaible upon request, submitted to Journal of Physics A: Mathematical and General

In this note we study the zero temperature dynamics of the Sherrington Kirkppatric model and we investigate the statistical properties of the configurations that are obtained in the large time limit. We find that the replica symmetry is broken (in a weak sense). We also present some general considerations on the synchronic approach to the off-equilibrium dynamics, which have motivated the present study., Comment: 13 pages, 6 figures

In this paper, after a telegraphic introduction to modern immunology, we present a simple model for the idiotypic network among antibodies and we study its relevance for the maintenance of immunological memory. We also consider the problem of computing the memory capacity of such a model., Comment: 7 pages and no figures

We consider the mean field theory of the Random Field Ising Model obtained by weighing the many solutions of the mean field equations with Boltzmann-like factors. These solutions are found numerically in three dimensions and we observe critical behavior arising from the weighted sum. The resulting exponents are calculated., Comment: 15 pages of tex using harvmac. 8 postscript figures (fig3.ps is large) in a separate .uu file

We will review some of the theoretical progresses that have been recently done in the study of slow dynamics of glassy systems: the general techniques used for studying the dynamics in the mean field approximation and the emergence of a pure dynamical transition in some of these systems. We show how the results obtained for a random Hamiltonian may be also applied to a given Hamiltonian. These two results open the way to a better understanding of the glassy transition in real systems.

In this lecture I will present some models of neural networks that have been developed in the recent years. The aim is to construct neural networks which work as associative memories. Different attractors of the network will be identified as different internal representations of different objects. At the end of the lecture I will present a comparison among the theoretical results and some of the experiments done on real mammal brains.

We will review some of the theoretical progresses that have been in the study of complex systems in physics and of their applications to biology.

In this talk I will review the approach to spin glasses based on the spontaneously broken replica symmetry. I will concentrate my attention mostly on more general ideas, skipping technical details and stressing the characteristic predictions of this approach. After the introduction of the replica method, the predicted structure of states is investigated in details, paying a particular attention to the local overlaps and to the structure of the clusters. I will finally study the behaviour of the system near the lower critical dimension and I will show that the technique of coupling real replicas is able to give relevant information.

In this talk I will show that usual spin glasses are a peculiar kind of Abelian gauge theory. I will shortly review the techniques used to study them. At the end I will consider more general models (e.g. spin glasses based on non Abelian gauge group) and I will discuss the relevance of these models to real glasses. Finally I will derive from first principles a generalised Vogel-Fulcher law for the divergence of the characteristic time near the glass transition.

We discuss the behavior of the fully frustrated hypercubic cell in the infinite dimensional mean-field limit. In the Ising case the system undergoes a glass transition, well described by the random orthogonal model. Under the glass temperature aging effects show clearly. In the $XY$ case there is no sign of a phase transition, and the system is always a paramagnet., Comment: Figures added in uufiles format, and epsf included

We study the statistical mechanics of a $D$-dimensional array of Josephson junctions in presence of a magnetic field. In the high temperature region the thermodynamical properties can be computed in the limit $D \to \infty$, where the problem is simplified; this limit is taken in the framework of the mean field approximation. Close to the transition point the system behaves very similar to a particular form of spin glasses, i.e. to gauge glasses. We have noticed that in this limit the evaluation of the coefficients of the high temperature expansion may be mapped onto the computation of some matrix elements for the $q$-deformed harmonic oscillator.

We study the static as well as the glassy or dynamical transition in the mean-field $p$-state Potts glass. By numerical solution of the saddle point equations we investigate the static and the dynamical transition for all values of $p$ in the non-perturbative regime $p>4$. The static and dynamical Edwards-Anderson parameter increase with $p$ logarithmically. This makes the glassy transition temperature lie very close to the static one. We compare the main predictions of the theory with the numerical simulations., Comment: 24 pages + 12 figures, LateX, figures uuencoded at the end of file

We show that the dynamical order parameters can be reexpressed in terms of the distribution of the staggered auto-correlation and response functions. We calculate these distributions for the out of equilibrium dynamics of the Sherrington-Kirpatrick model at long times. The results suggest that the landscape this model visits at different long times in an out of equilibrium relaxation process is, in a sense, self-similar. Furthermore, there is a similarity between the landscape seen out of equilibrium at long times and the equilibrium landscape. The calculation is greatly simplified by making use of the superspace notation in the dynamical approach. This notation also highlights the rather mysterious formal connection between the dynamical and replica approaches., Comment: 25 pages, Univ. di Roma I preprint #1049 (we replaced the file by the RevTex file, figures available upon request)

We present a family of solvable models of interacting particles in high dimensionalities without quenched disorder. We show that the models have a glassy regime with aging effects. The interaction is controlled by a parameter $p$. For $p=2$ we obtain matrix models and for $p>2$ `tensor' models. We concentrate on the cases $p=2$ which we study analytically and numerically., Comment: 10 pages + 2 figures, Univ.Roma I, 1038/94, ROM2F/94/27

We introduce and study a model which admits a complex landscape without containing quenched disorder. Continuing our previous investigation we introduce a disordered model which allows us to reconstruct all the main features of the original phase diagram, including a low $T$ spin glass phase and a complex dynamical behavior., Comment: 35 pages with uu figures, Roma 1027

We analyse the Langevin dynamics of the random walk, the scalar field, the X-Y model and the spinoidal decomposition. We study the deviations from the equilibrium dynamics theorems (FDT and homogeneity), the asymptotic behaviour of the systems and the aging phenomena. We compare the results with the dynamical behaviour of (random) spin-glass mean-field models., Comment: 26 pages, Universit\`a di Roma I preprint N. 1028

We study systems without quenched disorder with a complex landscape, and we use replica symmetry theory to describe them. We discuss the Golay-Bernasconi-Derrida approximation of the low autocorrelation model, and we reconstruct it by using replica calculations. Then we consider the full model, its low $T$ properties (with the help of number theory) and a Hartree-Fock resummation of the high-temperature series. We show that replica theory allows to solve the model in the high $T$ phase. Our solution is based on one-link integral techniques, and is based on substituting a Fourier transform with a generic unitary transformation. We discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder., Comment: 42 pages, uufile with eps figures added in figures, ROM2F/94/15

In this paper we try to estimate the lower critical dimension for replica symmetry breaking in spin glasses through the calculation of the additional free-energy required to create a domain wall between two different phases. This mechanism alone would say that replica symmetry would be restored at the lower critical dimension $D_c =2.5$., Comment: 14 pages, LaTeX, NORDITA preprint 94/21