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Giorgio Parisi

Publications and source records attributed to Giorgio Parisi.

At least 181 records · Page 10Linked to original sources
arXiv2007

Mosaic length and finite interaction-range effects in a one dimensional random energy model

In this paper we study finite interaction range corrections to the mosaic picture of the glass transition as emerges from the study of the Kac limit of large interaction range for disordered models. To this aim we consider point to set correlation functions, or overlaps, in a one dimensional random energy model as a function of the range of interaction. In the Kac limit, the mosaic length defines a sharp first order transition separating a high overlap phase from a low overlap one. Correspondingly we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic lenght. Nonetheless we find very slow convergence to the Kac limit and we discuss why this could be a problem for measuring the mosaic lenght in realistic models.

cond-mat.dis-nn↗
arXiv2007

Some considerations of finite dimensional spin glasses

In 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 and experimental verifications. I will mainly considered equilibrium properties at zero magnetic field, where the situation is clear and it should be not controversial. I will present the various hypothesis at the basis of the theory and I will discuss their physical status.

cond-mat.dis-nn↗
arXiv2007

On the most compact regular lattice in large dimensions: A statistical mechanical approach

In 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. Using his approach one can see that there are many similarities (and differences) with the problem of computing the entropy of a liquid of perfect spheres. The relation between the two problems is investigated in details. Some conjectures are presented, that need further investigation in order to check their consistency.

cond-mat.stat-mech↗
arXiv2007

Comment on "Ultrametricity in the Edwards-Anderson Model"

In a recent interesting Letter Contucci {\it et al.} have investigated several properties of the three-dimensional (3d) Edwards-Anderson (EA) Ising spin glass. They claim to have found strong numerical evidence for the presence of a complex ultrametric structure similar to the one described by the replica symmetry breaking solution of the mean field model. We illustrate by numerical simulations that the relations used by Contucci {\it et al.} as evidence for an ultrametric structure in the 3d EA model are fulfilled to similar accuracy in the two-dimensional EA model, which is well-described by the droplet picture and has no spin glass phase at finite temperature. We conclude that the data presented in the Contucci {\it et al.} Letter is not sufficient to dismiss the possibility that, e.g., the droplet model might describe the behavior of the 3d EA model.

cond-mat.dis-nn↗
arXiv2007

Large Deviations in the Free-Energy of Mean-Field Spin-Glasses

We compute analytically the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean field model, i.e. we compute the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing the average value of the partition function to the power $n$ as a function of $n$. At zero temperature this absolute prediction displays a remarkable quantitative agreement with the numerical data.

cond-mat.dis-nn↗
arXiv2007

Mean field theory of spin glasses: statics and dynamics

In 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 hypothesis at the basis of the theory and I will discuss their mathematical and physical status.

cond-mat.dis-nn↗
arXiv2006

Finite size corrections to random Boolean networks

Since their introduction, Boolean networks have been traditionally studied in view of their rich dynamical behavior under different update protocols and for their qualitative analogy with cell regulatory networks. More recently, tools borrowed from statistical physics of disordered systems and from computer science have provided a more complete characterization of their equilibrium behavior. However, the largest part of the results have been obtained in the thermodynamic limit, which is often far from being reached when dealing with realistic instances of the problem. The numerical analysis presented here aims at comparing - for a specific family of models - the outcomes given by the heuristic belief propagation algorithm with those given by exhaustive enumeration. In the second part of the paper some analytical considerations on the validity of the annealed approximation are discussed.

cond-mat.dis-nn↗
arXiv2006

On $k$-Core Percolation in Four Dimensions

The $k$-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. For $k=4$ the presence of a discontinuous transition is clearly established but its nature is strictly first order. In particular, the $k$-core density displays no singular behavior before the jump and its correlation length remains finite. For $k=3$ the transition is continuous.

cond-mat.dis-nn↗
arXiv2006

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.

cond-mat.dis-nn↗
arXiv2006

Computing the number of metastable states in infinite-range models

In 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 large number of stationary points of a random function. Quite surprisingly supersymmetry plays a crucial role in this problem. A careful analysis of the physical implication of supersymmetry and of supersymmetry breaking will be presented: the most spectacular one is that in the Sherrington-Kirkpatrick model for spin glasses most of the stationary points are saddles, as predicted long time ago.

cond-mat.dis-nn↗
arXiv2005

Loop expansion around the Bethe-Peierls approximation for lattice models

We 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 saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetisation and correlation function.

cond-mat.stat-mech↗
arXiv2005

Euclidean random matrices: solved and open problems

In 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 finds a phase transition between a {\sl phonon} phase and a {\sl saddle} phase. If we apply these considerations to the study of the Hessian of the Hamiltonian of the particles of a fluid, we find that this phonon-saddle transition corresponds to the dynamical phase transition in glasses, that has been studied in the framework of the mode coupling approximation. The Boson peak observed in glasses at low temperature is a remanent of this transition. We finally present some recent results obtained with a new approach where one deeply uses some hidden supersymmetric properties of the problem.

cond-mat.stat-mech↗
arXiv2005

Complexity and intelligence

In 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 a special attention to machine learning. We introduce the propensity of a machine to learn a rule and we use it define the intelligence of a machine.

cond-mat.stat-mech↗
arXiv2005

On local equilibrium equations for clustering states

In 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 where the solutions has cost zero (e.g. K-satisfiability). On a random graph the solutions of the local equilibrium equations are associated to clusters of configurations (clustering states). On a random graph the local equilibrium equations have solutions almost everywhere in the uncolored phase; in this case we have to introduce the concept quasi-solution of the local equilibrium equations.

cs.CC↗
arXiv2004

Zero-Temperature Limit of the SUSY-breaking Complexity in Diluted Spin-Glass Models

We study the SUSY-breaking complexity of the Bethe Lattice Spin-Glass in the zero temperature limit. We consider both the Gaussian and the bimodal distribution of the coupling constants. For $J_{ij}=\pm 1$ the SUSY breaking theory yields fields distributions that concentrate on integer values at low temperatures, at variance with the unbroken SUSY theory. This concentration takes place both in the quenched as well as in the simpler annealed formulation.

cond-mat.dis-nn↗
arXiv2004

Cavity Method for Supersymmetry Breaking Spin Glasses

The spontaneous supersymmetry-breaking that takes place in certain spin-glass models signals a particular fragility in the structure of metastable states of such systems. This fragility is due to the presence of at least one marginal mode in the Hessian of the free energy, that makes the states highly susceptible under external perturbations. The cavity method is a technique that recursively describes the property of a system with N+1 spins in terms of those of a system with N spins. To do so, the cavity method assumes a certain degree of stability when adding a new spin to the system, i.e. it assumes that for a generic choice of the parameters there is an one-to-one correspondence between the metastable states of the system with N spins and the metastable states of the system with N+1 spins. In systems where the supersymmetry is broken such a correspondence does not exist, and an alternative formulation of the cavity method must be devised. We introduce a generalized cavity approach that takes care of this problem and we apply it to the computation of the probability distribution of the local magnetizations in the Sherrington-Kirkpatrick model. Our findings agree with the correct supersymmetry-breaking result.

cond-mat.stat-mech↗
arXiv2003

Scale Invariance and Self-averaging in disordered systems

In a previous paper we found that in the random field Ising model at zero temperature in three dimensions the correlation length is not self-averaging near the critical point and that the violation of self-averaging is maximal. This is due to the formation of bound states in the underlying field theory. We present a similar study for the case of disordered Potts and Ising ferromagnets in two dimensions near the critical temperature. In the random Potts model the correlation length is not self-averaging near the critical temperature but the violation of self-averaging is weaker than in the random field case. In the random Ising model we find still weaker violations of self-averaging and we cannot rule out the possibility of the restoration of self-averaging in the infinite volume limit.

cond-mat.stat-mech↗