arXiv subjects
Publications and source records attributed to Luca De Sanctis.
We suggest a purely combinatorial approach to a general problem in system reliability. We show how to determine if a given vector can be the signature of a system, and in the affirmative case exhibit such a system in terms on its structure function. The method employs results from the theory of simplicial sets, and provides a full characterization of signature vectors.
We compute the fluctuations of the magnetization and of the multi-overlaps for the dilute mean field ferromagnet, in the high temperature region. The rescaled magnetization tends to a centered Gaussian variable with variance diverging at the critical line. The rescaled multi-overlaps also tend to centered independent Gaussian variables, but their covariance remain finite at the critical line.
We study the mean field dilute model of a ferromagnet. We find and prove an expression for the free energy density at high temperature, and at temperature zero. We find the critical line of the model, separating the phase with zero magnetization from the phase with symmetry breaking. We also compute exactly the entropy at temperature zero, which is strictly positive. The physical behavior at temperature zero is very interesting and related to infinite dimensional percolation, and suggests possible behaviors at generic low temperatures. Lastly, we provide a complete solution for the annealed model. Our results hold both for the Poisson and the Bernoulli versions of the model.
We provide a strategy to find in few elementary calculations the critical exponents of the overlaps for dilute spin glasses, in absence of external field. Such a strategy is based on the expansion of a suitably perturbed average of the overlaps, which is used in the formulation of the free energy as the difference between a cavity part and the derivative of the free energy itself, considered as a function of the connectivity of the model. We assume the validity of certain reasonable approximations, e.g. that higher powers of overlap monomials are of smaller magnitude near the critical point, of which we do not provide a rigorous proof.
We study the effects of bounded confidence thresholds and of interaction and external noise on Axelrod's model of social influence. Our study is based on a combination of numerical simulations and an integration of the mean-field Master equation describing the system in the thermodynamic limit. We find that interaction thresholds affect the system only quantitatively, but that they do not alter the basic phase structure. The known crossover between an ordered and a disordered state in finite systems subject to external noise persists in models with general confidence threshold. Interaction noise here facilitates the dynamics and reduces relaxation times. We also study Axelrod systems with metric features, and point out similarities and differences compared to models with nominal features. Metric features are used to demonstrate that a small group of extremists can have a significant impact on the opinion dynamics of a population of Axelrod agents.
We prove that the Aizenman-Contucci relations, well known for fully connected spin glasses, hold in diluted spin glasses as well. We also prove more general constraints in the same spirit for multi-overlaps, systematically confirming and expanding previous results. The strategy we employ makes no use of self-averaging, and allows us to generate hierarchically all such relations within the framework of Random Multi-Overlap Structures. The basic idea is to study, for these structures, the consequences of the closely related concepts of stochastic stability, quasi-stationarity under random shifts, factorization of the trial free energy. The very simple technique allows us to prove also the phase transition for the overlap: it remains strictly positive (in average) below the critical temperature if a suitable external field is first applied and then removed in the thermodynamic limit. We also deduce, from a cavity approach, the general form of the constraints on the distribution of multi-overlaps found within Quasi-Stationary Random Multi-Overlap Structures.
We provide a systematic treatment of self-averaging identities for various spin systems. The method is quite general, basically not relying on the nature of the model, and as a special case recovers the Ghirlanda-Guerra and Aizenman-Contucci identities, which are therefore proven, together with their extension, to be valid in a vaste class of spin models. We use the dilute spin glass as a guiding example.
In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More specificly, the average of the overlap, which is naively expected to take values in the whole interval $[-1,+1]$, becomes positive if we ``first'' apply an external field, so to destroy the gauge invariance of the model, and ``then'' remove it in the thermodynamic limit. This phenomenon emerges at the critical point. This first result is weaker that the one obtained by Talagrand (not limited to the average of the overlap), but we show here that, at least in average, the overlap is proven to be non-negative with no use of the Ghirlanda-Guerra identities. The latter are instead needed to obtain the second result, which is the control the behavior of the overlap at the critical point: we find the critical exponents of all the overlap correlation functions.
We investigate overlap fluctuations of the Sherrington-Kirkpatrick mean field spin glass model in the framework of the Random Over- lap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and coworkers, who developed a variational approach to the Sherrington-Kirkpatrick model. We propose here an iterative procedure to show that, in the so-called Boltzmann ROSt, Aizenman-Contucci (AC) polynomials naturally arise for almost all values of the inverse temperature (not in average over some interval only). The same results can be obtained in any ROSt, including therefore the Parisi structure. The AC polynomials impose restric- tions on the overlap fluctuations in agreement with Parisi theory.
Derrida et al. and Schütz and Stinchcombe gave algebraic formulas for the correlation functions of the partially asymmetric simple exclusion process. Here we give a fairly general recipe of how to get these formulas and extend them to the whole time evolution (starting from the generator of the process), for a certain class of interacting systems. We then analyze the algebraic relations obtained to show that the matrix approach does not work with some models such as the voter and the contact processes.
We extend the approach of Aizenman, Sims and Starr for the SK-type models to their spherical versions. Such an extension has already been performed for diluted spin glasses. The factorization property of the optimal structures found by Guerra for the SK model, which holds for diluted models as well, is verified also in the case of spherical systems, with the due modifications. Hence we show that there are some common structural features in various mean field spin models. These similarities seem to be quite paradigmatic, and we summarize the various techniques typically used to prove the structural analogies and to tackle the computation of the free energy per spin in the thermodynamic limit.
We study the learning dynamics of agents who adapt to heterogeneous comfort levels in the context of an El-Farol type game, and show that even an infinitesimal degree of heterogeneity in the resource levels leads to a significant reduction of the fluctuations of the collective action, and removes the phase transition observed in models with homogeneous comfort level. Our analysis is based on dynamical methods of disordered systems theory, in particular on a generating functional approach, and confirmed by numerical experiments. We also report on simulations of a system in which the comfort levels fluctuate in time, and point out crucial differences between models in which the comfort levels of the agents fluctuate collectively and individually respectively. Finally we comment on a possible characterisation of El-Farol and Minority Games according to the presence or absence of ergodicity-breaking phase transitions at infinite integrated response.
We propose an ultrametric broken replica symmetry trial function for the diluted mean field sping glass model in the framework of random multi-overlap structure.
We extend to the random K-SAT and p-XOR-SAT optimization problems the results obtained for the Viana-Bray model of diluted mean field spin glass.
We introduce the concept of Random Multi-Overlap Structures in diluted spin glasses, following the ideas of Aizenman, Sims and Starr for non-diluted models. As a result, we prove the generalized bound and variational principle for the free energy per site, in analogy with the results of Aizenman, Sims and Starr for non-diluted models. We also prove some stability properties of the optimal RMOSt, analogous to the one found by Guerra for non-diluted models.