arXiv subjects

Giorgio Parisi

Publications and source records attributed to Giorgio Parisi.

At least 199 records · Page 11Linked to original sources

arXiv2003

Numerical study of metastable states in Ising spin glasses

We study numerically the structure of metastable states in the Sherrington-Kirkpatrick spin glass. We find that all non-paramagnetic stationary points of the free energy are organized into pairs, consisting in a minimum and a saddle of order one, which coalesce in the thermodynamic limit. Within the annealed approximation, the entropic contribution of these states, that is the complexity, is compatible with the supersymmetry-breaking calculation performed in [A.J. Bray and M.A. Moore, J. Phys. C, 13 L469 (1980)]. This result indicates that the supersymmetry is spontaneously broken in the Sherrington-Kirkpatrick model.

cond-mat.stat-mech↗

arXiv2003

Constraint Optimization and Statistical Mechanics

In 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 I will describe in details only the problems related to the coloring of a random graph.

cs.CC↗

arXiv2003

A backtracking survey propagation algorithm for K-satisfiability

In 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 problems near the sat-unsat transition.

cond-mat.dis-nn↗

arXiv2003

Instability of one-step replica-symmetry-broken phase in satisfiability problems

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of [A.Montanari and F. Ricci-Tersenghi, Eur.Phys.J. B 33, 339 (2003)] to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clauses density alpha or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for alpha< alpha_m, with alpha_m\approx 4.153. On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB.

cond-mat.dis-nn↗

arXiv2003

On the probabilistic approach to the random satisfiability problem

In 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 a phase transition from the satisfiable to the unsatisfiable phase is found at $α=4.267$. There are other values of $α$ that separates different regimes and they will be described in details. In this context the properties of the survey decimation algorithm will also be discussed.

cs.CC↗

arXiv2003

The role of the Becchi-Rouet-Stora-Tyutin supersymmetry in the calculation of the complexity for the Sherrington-Kirkpatrick model

The Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry is a powerful tool for the calculation of the complexity of metastable states in glassy systems, and it is particularly useful to uncover the relationships between complexity and standard thermodynamics. In this work we compute the Thouless-Anderson-Palmer complexity of the Sherrington-Kirkpatrick model at the quenched level, by using the BRST supersymmetry. We show that the complexity calculated at K steps of replica symmetry breaking is strictly related to the static free energy at K+1 steps of replica symmetry breaking. The supersymmetry therefore provides a prescription to obtain the complexity of the TAP states from the standard static free energy, even in models which are solved by more than one step of replica symmetry breaking. This recipe states that the complexity is given by the Legendre transform of the static free energy, where the Legendre parameter is the largest replica symmetry breaking point of the overlap matrix.

cond-mat.stat-mech↗

arXiv2003

Supersymmetric quenched complexity in the Sherrington-Kirkpatrick model

By using the BRST supersymmetry we compute the quenched complexity of the TAP states in the SK model. We prove that the BRST complexity is equal to the Legendre transform of the static free energy with respect to the largest replica symmetry breaking point of its overlap matrix.

cond-mat.stat-mech↗

arXiv2003

Euclidean random matrices, the glass transition and the Boson peak

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 translation invariant matrices one generically finds a phase transition between a phonon phase and a 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.

cond-mat.soft↗

arXiv2003

Some remarks on the survey decimation algorithm for K-satisfiability

In 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 phenomena appear near the boundary of convergence.

cs.CC↗

arXiv2003

On the origine of the Boson peak

We 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. The Boson peak observed in glasses at low temperature is a remanent of this transition.

cond-mat.soft↗

arXiv2003

Glasses, replicas and all that

In 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 near the dynamic transition (mode coupling transition) and near the equilibrium transition (Kauzman transition) that is present in fragile glasses. The replica method may be used to solve simple mean field models, providing explicit examples of systems that may be studied analytically in great details and behave similarly to the experiments. Finally, using the replica formalism, it is possible to do analytic explicit computations of the properties of realistic models of glasses and the results are in reasonable agreement with numerical simulations.

cond-mat.soft↗

arXiv2002

On the survey-propagation equations for the random K-satisfiability problem

In 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 given instance can be approximated by the solution of the corresponding equations on an infinite tree. We conjecture (and we bring numerical evidence) that the survey-propagation equations on the infinite tree have an unique solution in the suitable range of parameters.

cs.CC↗

arXiv2002

Local overlaps, heterogeneities and the local fluctuation dissipation relations

In 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 relations: using the general principle of stochastic stability these local fluctuation dissipation relations can be proved in a way that is very similar to the usual proof of the fluctuation dissipation relations for intensive quantities. The local overlap and its probability distribution play a crucial role in this proof. Similar arguments can be used to prove that all sites in an aging experiment stay at the same effective temperature at the same time.

cond-mat.dis-nn↗

arXiv2002

On the formal equivalence of the TAP and thermodynamic methods in the SK model

We revisit two classic Thouless-Anderson-Palmer (TAP) studies of the Sherrington-Kirkpatrick model [Bray A J and Moore M A 1980 J. Phys. C 13, L469; De Dominicis C and Young A P, 1983 J. Phys. A 16, 2063]. By using the Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry, we prove the general equivalence of TAP and replica partition functions, and show that the annealed calculation of the TAP complexity is formally identical to the quenched thermodynamic calculation of the free energy at one step level of replica symmetry breaking. The complexity we obtain by means of the BRST symmetry turns out to be considerably smaller than the previous non-symmetric value.

cond-mat.dis-nn↗

arXiv2002

The cavity method at zero temperature

In 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 energy of the global ground state, as well as the complexity of equilibrium states at a given energy. Full results are presented for a Bethe lattice with connectivity equal to three.

cond-mat.dis-nn↗

arXiv2002

Local fluctuation dissipation relation

In 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 for intensive quantities. Similar arguments can be used to prove that all sites in an aging experiment stay at the same effective temperature at the same time.

cond-mat.dis-nn↗

arXiv2002

Two spaces looking for a geometer

In 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 interested to study the limit n going to zero). These two spaces are not known to most of the mathematicians, but they are widely used by physicists. It is possible that, by extending the notion of space, they can become well defined mathematical objects.

cond-mat.dis-nn↗

arXiv2002

Scale Invariance in disordered systems: the example of the Random Field Ising Model

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.

cond-mat↗