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On the number of metastable states in spin glassesJun 12 1995In 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 ... More

Numerical estimate of the Kardar Parisi Zhang universality class in (2 + 1) dimensionsNov 25 2016We study the Restricted Solid on Solid model for surface growth in spatial dimension $d=2$ by means of a multi-surface coding technique that allows to produce a large number of samples of samples in the stationary regime in a reasonable computational ... More

A numerical study of the overlap probability distribution and its sample-to-sample fluctuations in a mean-field modelAug 03 2011In this paper we study the fluctuations of the probability distributions of the overlap in mean field spin glasses in the presence of a magnetic field on the De Almeida-Thouless line. We find that there is a large tail in the left part of the distribution ... More

The crossover region between long-range and short-range interactions for the critical exponentsJul 12 2014It is well know that systems with an interaction decaying as a power of the distance may have critical exponents that are different from those of short-range systems. The boundary between long-range and short-range is known, however the behavior in the ... More

Spatial correlation functions and dynamical exponents in very large samples of 4D spin glassesDec 27 2013The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite size effects they show. By exploiting at the best the capabilities of standard ... More

Glassy Critical Points and Random Field Ising ModelMar 21 2012Dec 29 2012We consider the critical properties of points of continuous glass transition as one can find in liquids in presence of constraints or in liquids in porous media. Through a one loop analysis we show that the critical Replica Field Theory describing these ... More

Diluted Mean-Field Spin-Glass Models at CriticalityJan 08 2014Feb 10 2015We present a method derived by cavity arguments to compute the spin-glass and higher-order susceptibilities in diluted mean-field spin-glass models. The divergence of the spin-glass susceptibility is associated to the existence of a non-zero solution ... More

Large Deviations of Correlation Functions in Random MagnetsAug 09 2013Jul 01 2014We present a large deviations theory of the spin-spin correlation functions in the Random Field Ising Model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical ... More

The backtracking survey propagation algorithm for solving random K-SAT problemsAug 20 2015Oct 06 2016Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features ... More

On the probabilistic formulation of the replica approach to spin glassesJan 09 1998In this paper we review the predictions of the replica approach on the probability distribution of the overlaps among replicas and on the sample to sample fluctuations of this probability. We stress the role of replica equivalence in obtaining relations ... More

A divergent correlation length in off-equilibrium glassesJan 06 1998In 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 ... More

Short time aging in binary glassesJan 04 1997We present some simple computer simulations that indicate that at short time aging is realized in a simple model of binary glasses. It is interesting to note that modest computer simulations are enough to evidenziate this effect. We also find indications ... More

A Conjecture on random bipartite matchingJan 17 1998In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution.

Slow dynamics of glassy systemsMay 30 1997In these lectures I will present an introduction to the modern way of studying the properties of glassy systems. I will start from soluble models of increasing complications, the Random Energy Model, the $p$-spins interacting model and I will show how ... More

$D$-dimensional Arrays of Josephson Junctions, Spin Glasses and $q$-deformed Harmonic OscillatorsOct 25 1994We 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; ... More

The mean field theory of spin glasses: the heuristic replica approach and recent rigorous resultsDec 06 2009The 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 ... More

Some considerations of finite dimensional spin glassesNov 02 2007In talk I will review the theoretical results that have been obtained for spin glasses, paying a particular attention to finite dimensional spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical ... More

Complexity and intelligenceApr 06 2005In this paper I will discuss the properties of the Algorithmic Complexity, presenting the most relevant properties. The related concept of logical depth is also introduced. These properties will be used to study the problem of learning from example, paying ... More

On the origine of the Boson peakJan 16 2003We show that the phonon-saddle transition in the ensemble of generalized inherent structures (minima and saddles) happens at the same point as the dynamical phase transition in glasses, that has been studied in the framework of the mode coupling approximation. ... More

Glasses, replicas and all thatJan 10 2003In these lectures I will review the approach to glasses based on the replica formalism. Many of the physical ideas are very similar to those of older approaches. The replica approach has the advantage of describing in an unified setting both the behaviour ... More

The physical Meaning of Replica Symmetry BreakingMay 18 2002In this talk I will presente the physical meaning of replica symmetry breaking stressing the physical concepts. After introducing the theoretical framework and the experimental evidence for replica symmetry breaking, I will describe some of the basic ... More

The marginally stable Bethe lattice spin glass revisitedSep 17 2016Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only ... More

Local fluctuation dissipation relationAug 04 2002In this letter I show that the recently proposed local version of the fluctuation dissipation relations follows from the general principle of stochastic stability in a way that is very similar to the usual proof of the fluctuation dissipation theorem ... More

Two spaces looking for a geometerJul 13 2002Jul 20 2002In this talk I will introduces two spaces: the first space is the usual n-dimensional vector space with the unusual feature that n is non-integer, the second space is composed by the linear matrices acting on the previous space (physicists are particularly ... More

Spin Glass Theory: numerical and experimental results in three-dimensional systemsOct 04 2007Here I will review the theoretical results that have been obtained for spin glasses. I will concentrate my attention on the predictions of the mean field approach in three dimensional systems and on its numerical and experimental verifications.

Some remarks on the survey decimation algorithm for K-satisfiabilityJan 16 2003In this note we study the convergence of the survey decimation algorithm. An analytic formula for the reduction of the complexity during the decimation is derived. The limit of the converge of the algorithm are estimated in the random case: interesting ... More

Euclidean random matrices, the glass transition and the Boson peakJan 16 2003In this paper I will describe some results that have been recently obtained in the study of random Euclidean matrices, i.e. matrices that are functions of random points in Euclidean space. In the case of translation invariant matrices one generically ... More

The Physics of the Glass TransitionJan 24 2000In this talk, after a short phenomenological introduction on glasses, I will describe some recent progresses that have been done in glasses using the replica method in the definition and in the evaluation of the configurational entropy (or complexity). ... More

On the replica method for glassy systemsMay 31 1998Jun 02 1998In this talk we review our theoretical understanding of spin glasses paying a particular attention to the basic physical ideas. We introduce the replica method and we describe its probabilistic consequences (we stress the recently discovered importance ... More

An introduction to the immune networkJan 12 1995In 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 ... More

Gauge Theories, Spin Glasses and Real GlassesNov 28 1994In 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) ... More

Entanglement critical length at the many-body localization transitionOct 28 2016Nov 18 2016We study the details of the distribution of the entanglement spectrum (eigenvalues of the reduced density matrix) of a disordered spin chain exhibiting a many-body localization (MBL) transition. In the thermalizing region we identify the evolution under ... More

Hopping and the Stokes-Einstein relation breakdown in simple glass formersJul 21 2014Sep 15 2014One of the most actively debated issues in the study of the glass transition is whether a mean-field description is a reasonable starting point for understanding experimental glass formers. Although the mean-field theory of the glass transition -- like ... More

Field Theory of Fluctuations in GlassesSep 22 2011We develop a field-theoretical description of dynamical heterogeneities and fluctuations in supercooled liquids close to the (avoided) MCT singularity. Using quasi-equilibrium arguments we eliminate time from the description and we completely characterize ... More

Non-equilibrium fluctuation dissipation relations in binary glassesApr 05 1997We show that at short time aging is realized in a simple model of binary glasses. We find that below the transition point the off-equilibrium correlation functions and response functions are compatible with the relations that were originally derived Cugliandolo ... More

The overlap in glassy systemsOct 20 2013In this paper I will consider many of the various definitions of the overlap and of its probability distribution that have been introduced in the literature starting from the original papers of Edwards and Anderson; I will present also some of the most ... More

Constraint Optimization and Statistical MechanicsDec 05 2003In these lectures I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to simplify the presentation ... More

Stochastic StabilityJul 22 2000In this talk I will introduce the principle of stochastic stability and discussing its consequences both at equilibrium and off-equilibrium.

Soft modes in jammed hard spheres (I): Mean field theory of the isostatic transitionJan 17 2014In this paper we consider different models for soft modes in jammed hard spheres. We show how one can construct mean field models that can be solved analytically. The analytic solution of these models displays an excess of low energy soft modes that become ... More

Statistical properties of Random Matrices and the replica methodJan 07 1997I present here some results on the statistical behaviour of large random matrices in an ensemble where the probability distribution is not a function of the eigenvalues only. The perturbative expansion can be cast in a closed form and the limits of validity ... More

Computing the number of metastable states in infinite-range modelsFeb 15 2006In these notes I will review the results that have been obtained in these last years on the computation of the number of metastable states in infinite-range models of disordered systems. This is a particular case of the problem of computing the exponentially ... More

Euclidean random matrices: solved and open problemsNov 30 2005In this paper I will describe some results that have been recently obtained in the study of random Euclidean matrices, i.e. matrices that are functions of random points in Euclidean space. In the case of {\sl translation invariant} matrices one generically ... More

On the survey-propagation equations for the random K-satisfiability problemDec 07 2002In this note we study the existence of a solution to the survey-propagation equations for the random K-satisfiability problem for a given instance. We conjecture that when the number of variables goes to infinity, the solution of these equations for a ... More

Complex Systems: a Physicist's ViewpointMay 14 2002I present my viewpoint on complexity, stressing general arguments and using a rather simple language.

Replica and GlassesJul 04 1999In these two lectures I review our theoretical understanding of spin glasses paying a particular attention to the basic physical ideas. We introduce the replica method and we describe its probabilistic consequences (we stress the recently discovered importance ... More

The Random Fractional Matching ProblemFeb 08 2018We consider two formulations of the random-link fractional matching problem, a relaxed version of the more standard random-link (integer) matching problem. In one formulation, we allow each node to be linked to itself in the optimal matching configuration. ... More

Quasi equilibrium construction for the long time limit of glassy dynamicsSep 23 2015In this paper we review a recent proposal to understand the long time limit of glassy dynamics in terms of an appropriate Markov Chain. [1]. The advantages of the resulting construction are many. The first one is that it gives a quasi equilibrium description ... More

A backtracking survey propagation algorithm for K-satisfiabilityAug 25 2003In this paper we present a backtracking version of the survey propagation algorithm. We show that the introduction of the simplest form of backtracking greatly improves the ability of the original survey propagation algorithm in solving difficult random ... More

Physics of glassy systemsOct 23 1999In this talk I present some of the recent theoretical results that have been obtained on glassy systems like spin glasses or structural glasses. The physical principles at the basis of the theory are explained in a simple language (without using replicas) ... More

New ideas in glass transitionsDec 07 1997In this talk I will review some of the recent applications of the replica theory to glasses. I will firstly describe the basic assumptions and I will show that they can be considered as a precise reformulations of old ideas. The relation of this approach ... More

Numerical indications for the existence of a thermodynamic transition in binary glassesJan 15 1997In this note we present numerical simulations of binary mixtures and we find indications for a thermodynamic transition to a glassy phase. We find that below the transition point the off equilibrium correlation functions and response functions seems to ... More

Recent rigorous results support the predictions of spontaneously broken replica symmetry for realistic spin glassesMar 14 1996We show that the predictions of spontaneously broken replica symmetry are in perfect agreement with the recent rigorous results obtained by Newman and Stein.

On the replica scenario for the glass transitionNov 12 2009In this letter we study a lattice gas system that undergoes a glassy transition. When we approach the glass transition we find both a divergence of a point to set correlation length and the vanishing of the thermodynamic potential. These findings are ... More

Mean field theory of spin glasses: statics and dynamicsJun 01 2007In these lectures I will review some theoretical results that have been obtained for spin glasses. I will concentrate my attention on the formulation of the mean field approach and on its numerical and experimental verifications. I will present the various ... More

Planck's Legacy to Statistical MechanicsJan 18 2001In this talk I will describe the deep influence Planck had on the development of statistical mechanics. At this end I will first outline the theoretical situation of statistical mechanics before Planck. I will then describe his main contributions to this ... More

On the approach to equilibrium of an Hamiltonian chain of anharmonic oscillatorsApr 25 1997In this note we study the approach to equilibrium of a chain of anharmonic oscillators. We find indications that a sufficiently large system always relaxes to the usual equilibrium distribution. There is no sign of an ergodicity threshold. The time however ... More

Attractor Neural NetworksDec 06 1994In 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 ... More

Complexity in BiologyDec 05 1994We will review some of the theoretical progresses that have been in the study of complex systems in physics and of their applications to biology.

Ensemble renormalization group for disordered systemsNov 29 2011Apr 28 2013We propose and study a renormalization group transformation that can be used also for models with strong quenched disorder, like spin glasses. The method is based on a mapping between disorder distributions, chosen such as to keep some physical properties ... More

On local equilibrium equations for clustering statesDec 18 2002Mar 13 2005In this note we show that local equilibrium equations (the generalization of the TAP equations or of the belief propagation equations) do have solutions in the colorable phase of the coloring problem. The same results extend to other optimization problems ... More

Local overlaps, heterogeneities and the local fluctuation dissipation relationsNov 26 2002In this paper I introduce the probability distribution of the local overlap in spin glasses. The properties of the local overlaps are studied in details. These quantities are related to the recently proposed local version of the fluctuation dissipation ... More

Off-equilibrium fluctuation dissipation relation in binary mixturesMar 25 1997Apr 08 1997In this note we present numerical simulations of binary mixtures. We study the diffusion of particles and the response to an external driving force. We find evidence for the validity of the Cugliandolo Kurchan off-equilibrium fluctuation dissipation relation. ... More

Slow Dynamics in GlassesDec 07 1994We 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 ... More

On the most compact regular lattice in large dimensions: A statistical mechanical approachOct 03 2007In this paper I will approach the computation of the maximum density of regular lattices in large dimensions using a statistical mechanics approach. The starting point will be some theorems of Roger, which are virtually unknown in the community of physicists. ... More

On the probabilistic approach to the random satisfiability problemAug 05 2003In this note I will review some of the recent results that have been obtained in the probabilistic approach to the random satisfiability problem. At the present moment the results are only heuristic. In the case of the random 3-satisfiability problem ... More

A pedagogical introduction to the replica method for fragile glassesMay 23 1999May 24 1999In this note I present a simplified version of the recent computation (Mezard and Parisi 1998, 1999) of the properties of glasses in the low temperature phase in the framework of the replica theory, using an extension of the tools used in liquid theory. ... More

Testing replica predictions in experimentsJan 02 1998We review the predictions of the replica approach both for the statics and for the off-equilibrium dynamics. We stress the importance of the Cugliandolo-Kurchan off-equilibrium fluctuation-dissipation relation in providing a bridge between the statics ... More

On the replica approach to glassesJan 10 1997Here we review the approach to glassy systems based on the replica method and we introduce the main ingredients of replica symmetry breaking. We explain why the replica method has been successful in spin glass and why it should be successful for real ... More

On the mean field approach to glassy systemsJan 07 1997In these lectures I will study some properties that are shared by many glassy systems. I will shown how some of these properties can be understood in the framework of the mean field approach based on the replica method and I will discuss which are the ... More

On the Statistical Properties of the Large Time Zero Temperature Dynamics of the SK ModelJan 12 1995In 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 ... More

On the solution of a `solvable' model of an ideal glass of hard spheres displaying a jamming transitionNov 23 2010Feb 21 2011We discuss the analytical solution through the cavity method of a mean field model that displays at the same time an ideal glass transition and a set of jamming points. We establish the equations describing this system, and we discuss some approximate ... More

Replica Field Theory of the Dynamical Transition in Glassy SystemsMay 26 2011The critical behaviour of the dynamical transition of glassy system is controlled by a Replica Symmetric action with n=1 replicas. The most divergent diagrams in the loop expansion correspond at all orders to the solutions of a stochastic equation leading ... More

Relations between Short Range and Long Range Ising modelsJan 27 2014Jan 30 2014We perform a numerical study of the long range (LR) ferromagnetic Ising model with power law decaying interactions ($J \propto r^{-d-\sigma}$) both on a one-dimensional chain ($d=1$) and on a square lattice ($d=2$). We use advanced cluster algorithms ... More

Glasses and replicasOct 15 2009We review the approach to glasses based on the replica formalism. The replica approach presented here is a first principle's approach which aims at deriving the main glass properties from the microscopic Hamiltonian. In contrast to the old use of replicas ... More

The cavity method at zero temperatureJul 04 2002Oct 10 2002In this note we explain the use of the cavity method directly at zero temperature, in the case of the spin glass on a Bethe lattice. The computation is done explicitly in the formalism equivalent to 'one step replica symmetry breaking'; we compute the ... More

P-adic numbers and replica symmetry breakingJun 07 1999The p-adic formulation of replica symmetry breaking is presented. In this approach ultrametricity is a natural consequence of the basic properties of the p-adic numbers. Many properties can be simply derived in this approach and p-adic Fourier transform ... More

A Study of Activated Processes in Soft Sphere GlassJan 08 1997On the basis of long simulations of a binary mixture of soft spheres just below the glass transition, we make an exploratory study of the activated processes that contribute to the dynamics. We concentrate on statistical measures of the size of the activated ... More

On Toy AgingAug 03 1993We consider the dynamics of a simple one dimensional model and we discuss the phenomenon of aging (i.e., the strong dependence of the dynamical correlation functions over the waiting time). Our model is the so-called random random walk, the toy model ... More

A renormalization group computation of the critical exponents of hierarchical spin glassesJun 29 2010Nov 05 2010The infrared behaviour of a non-mean field spin-glass system is analysed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the epsilon-expansion technique with two independent methods. Both ... More

Replica symmetry breaking in and around six dimensionsNov 14 2011Two, replica symmetry breaking specific, quantities of the Ising spin glass --- the breakpoint x1 of the order parameter function and the Almeida-Thouless line --- are calculated in six dimensions (the upper critical dimension of the replicated field ... More

Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networksMar 18 1998Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at ... More

Local Spin Glass Order in 1DApr 19 2006Jun 20 2006We 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 ... More

Loop expansion around the Bethe-Peierls approximation for lattice modelsDec 21 2005We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe-Peierls approximation appears naturally as the saddle point approximation. The corrections to the ... More

On nonlinear susceptibility in supercooled liquidsMay 04 2000In this paper, we discuss theoretically the behavior of the four point nonlinear susceptibility and its associated correlation length for supercooled liquids close to the Mode Coupling instability temperature $T_c$. We work in the theoretical framework ... More

Toy model for the mean-field theory of hard-sphere liquidsMar 13 2000We investigate a toy model of liquid, based on simplified HNC equations in very large spatial dimension D. The model does not exhibit a phase transition, but several regimes of the behavior when D\to\infty can be observed in different intervals of the ... More

The simplest model of jammingJan 14 2015We study a well known machine learning model -the perceptron- as a simple model of jamming of hard objects. We exhibit two regimes: 1) a convex optimisation regime where jamming is hypostatic and non-critical. 2) a non convex optimisation regime where ... More

Universality Classes of Critical Points in Constrained GlassesJul 18 2013Oct 28 2013We analyze critical points that can be induced in glassy systems by the presence of constraints. These critical points are predicted by the Mean Field Thermodynamic approach and they are precursors of the standard glass transition in absence of constraints. ... More

Effective potential in glassy systems: theory and simulationsNov 21 1997We study the phase diagram of glassy systems in presence of an attractive coupling among real replicas. We find competition among a localized and a delocalized phase, that are separated by a coexistence line as in ordinary first order phase transitions. ... More

A tentative Replica Study of the Glass TransitionFeb 01 1996We 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, ... More

Simulated Tempering: A New Monte Carlo SchemeMay 22 1992We propose a new global optimization method ({\em Simulated Tempering}) for simulating effectively a system with a rough free energy landscape (i.e. many coexisting states) at finite non-zero temperature. This method is related to simulated annealing, ... More

A first principle computation of the thermodynamics of glassesDec 10 1998Mar 22 1999We 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 ... More

On the 1/D expansion for directed polymersDec 17 1997We present a variational approach for directed polymers in $D$ transversal dimensions which is used to compute the corrections to the mean field theory predictions with broken replica symmetry. The trial function is taken to be a symmetrized version of ... More

Phase diagram of glassy systems in an external fieldJan 07 1997We study the mean-field phase diagram of glassy systems in a field pointing in the direction of a metastable state. We find competition among a ``magnetized'' and a ``disordered'' phase, that are separated by a coexistence line as in ordinary first order ... More

Cross correlations of the American baby namesOct 10 2014May 14 2015The quantitative description of cultural evolution is a challenging task. The most difficult part of the problem is probably to find the appropriate measurable quantities that can make more quantitative such evasive concepts as, for example, dynamics ... More

Finite size corrections to disordered systems on Erdös-Rényi random graphsMay 14 2013Oct 01 2013We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase ... More

On the Effects of a Bulk Perturbation on the Ground State of 3D Ising Spin GlassesJul 29 2000We compute and analyze couples of ground states of 3D spin glasses before and after applying a volume perturbation which adds to the Hamiltonian a repulsion from the true ground state. The physical picture based on Replica Symmetry Breaking is in excellent ... More

Thermodynamics of glasses: a first principle computationJul 31 1998We 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 ... More

Critical properties of a three dimensional p-spin modelMay 07 1998In this paper we study the critical properties of a finite dimensional generalization of the p-spin model. We find evidence that in dimension three, contrary to its mean field limit, the glass transition is associated to a diverging susceptibility (and ... More

On $k$-Core Percolation in Four DimensionsSep 29 2006The $k$-core percolation on the Bethe lattice has been proposed as a simple model of the jamming transition because of its hybrid first-order/second-order nature. We investigate numerically $k$-core percolation on the four-dimensional regular lattice. ... More

Mean-Field Equations for Spin Models with Orthogonal Interaction MatricesMar 01 1995Mar 27 1995We 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 ... More