Silvio Franz

Replica Bounds for Optimization Problems and Diluted Spin Systems

In this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by Guerra (cond-mat/0205123) to prove that the replica method generates variational bounds for disordered systems. We analyze a family of models that includes the Viana–Bray model, the diluted p-spin model or random XOR-SAT problem, and the random K-SAT problem, showing that the replica/cavity method, at the various levels of approximation, provides systematic schemes to obtain lower bounds of the free-energy at all temperatures and of the ground state energy. In the case of K-SAT and XOR-SAT it thus gives upper bounds of the satisfiability threshold. Our analysis underlines deep connections with the cavity method which are not evident in the long range case.

THEORY OF NON-LINEAR SUSCEPTIBILITY AND CORRELATION LENGTH IN GLASSES AND LIQUIDS

Abstract Within the framework of the effective potential theory of the structural glass transition, we calculate for the p-spin model and for a hard sphere liquid in the hypernetted chain approximation a static non-linear susceptibility related to a four-point density correlation function, and show that it diverges in mean field with exponent γ=1/2 as the critical temperature Tc is approached from below. When Tc is approached from above, we calculate for the p-spin model a time dependent non-linear susceptibility and show that there is a characteristic time where this susceptibility has a maximum, and that this time grows with decreasing T. This susceptibility diverges as Tc is approached from above, and has key features in common with a generalized susceptibility related to particle displacements, previously introduced to measure correlated particle motion in simulations of glass-forming liquids.

On non-linear susceptibility in supercooled liquids

In this paper, we discuss theoretically the behaviour of the four-point non-linear susceptibility and its associated correlation length for supercooled liquids close to the mode-coupling instability temperature Tc. We work in the theoretical framework of the glass transition as described by mean-field theory of disordered systems, and the hypernetted-chain approximation. Our results give an interpretation framework for recent numerical findings on heterogeneities in supercooled liquid dynamics.

A minimal stochastic model for influenza evolution

We introduce and discuss a minimal individual based model for influenza dynamics. The model takes into account the effects of specific immunization against viral strains, but also infectivity randomness and the presence of a short lived strain-transcending immunity recently suggested in the literature. We show by simulations that the resulting model exhibits substitution of viral strains along the years, but that their divergence remains bounded. We also show that dropping any of these features results in a drastically different behaviour, leading either to the extinction of the disease, to the proliferation of the viral strains or to their divergence.

Replica bounds for diluted non-Poissonian spin systems

In this paper we extend replica bounds and free energy subadditivity arguments to diluted spin-glass models on graphs with arbitrary, non-Poissonian degree distribution. The new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method. As a byproduct we obtain self-averaging identities that generalize the Ghirlanda–Guerra ones to the multi-overlap case.

The Response of Glassy Systems to Random Perturbations: A Bridge Between Equilibrium and Off-Equilibrium

We discuss the response of aging systems with short-range interactions to a class of random perturbations. Although these systems are out of equilibrium, the limit value of the free energy at long times is equal to the equilibrium free energy. By exploiting this fact, we define a new order parameter function, and we relate it to the ratio between response and fluctuation, which is in principle measurable in an aging experiment. For a class of systems possessing stochastic stability, we show that this new order parameter function is intimately related to the static order parameter function, describing the distribution of overlaps between clustering states. The same method is applied to investigate the geometrical organization of pure states. We show that the ultrametric organization in the dynamics implies static ultrametricity, and we relate these properties to static separability, i.e., the property that the measure of the overlap between pure states is essentially unique. Our results, especially relevant for spin glasses, pave the way to an experimental determination of the order parameter function.

Dynamic phase transition for decoding algorithms.

The state-of-the-art error correcting codes are based on large random constructions (random graphs, random permutations, etc.) and are decoded by linear-time iterative algorithms. Because of these features, they are remarkable examples of diluted mean-field spin glasses, both from the static and dynamic points of view. We analyze the behavior of decoding algorithms by mapping them onto statistical-physics models. This allows us to understand the intrinsic (i.e., algorithm independent) features of this behavior.

A ferromagnet with a glass transition

We introduce a finite-connectivity ferromagnetic model with a three-spin interaction which has a crystalline (ferromagnetic) phase as well as a glass phase. The model is not frustrated, it has a ferromagnetic equilibrium phase at low temperature which is not reached dynamically in a quench from the high-temperature phase. Instead it shows a glass transition which can be studied in detail by a one-step replica-symmetry-broken calculation. This spin model exhibits the main properties of the structural glass transition at a solvable mean-field level.-1

Constrained Boltzmann-Gibbs measures and effective potential for glasses in hypernetted chain approximation and numerical simulations

By means of an effective potential associated with a constrained equilibrium measure and apt to study frozen systems, we investigate glassy freezing in simple liquids in the hypernetted chain (HNC) approximation. Unlike other classical approximations of liquid theory, freezing is naturally embedded in the HNC approximation. We give a detailed description of the freezing transition that is analogous to the one given in a large class of mean-field long range spin glass. We compare our findings with Monte Carlo simulations of the same system and conclude that many of the qualitative features of the transition are captured by the approximated theory.

Exact solutions for diluted spin glasses and optimization problems

We study the low temperature properties of p-spin glass models with finite connectivity and of some optimization problems. Using a one-step functional replica symmetry breaking ansatz we can solve exactly the saddle-point equations for graphs with uniform connectivity. The resulting ground state energy is in perfect agreement with numerical simulations. For fluctuating connectivity graphs, the same ansatz can be used in a variational way: For p-spin models (known as p-XOR-SAT in computer science) it provides the exact configurational entropy together with the dynamical and static critical connectivities (for p = 3, gamma(d) = 0.818, and gamma(s) = 0.918), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds ( gamma(var)(c) = 4.396 and gamma(var)(c) = 2.149).