Interpolating the Sherrington-Kirkpatrick replica trick

The interpolation techniques have become, in the past decades, a powerful approach to lighten several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented into the cavity field technique, or its variants as the stochastic stability or the random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick. Among the models where these methods have been used (namely, dealing with frustration and complexity), probably the best known is the Sherrington-Kirkpatrick spin glass: In this paper we are pleased to apply the interpolation scheme to the replica trick framework and test it directly to the cited paradigmatic model: interestingly this allows to obtain easily the replica-symmetric control and, synergically with the broken replica bounds, a description of the full RSB scenario, both coupled with several minor theorems. Furthermore, by treating the amount of replicas $n\in(0,1]$ as an interpolating parameter (far from its original interpretation) this can be though of as a quenching temperature close to the one introduce in off-equilibrium approaches and, within this viewpoint, the proof of the attended commutativity of the zero replica and the infinite volume limits can be obtained.Comment: This article is dedicated to David Sherrington on the occasion of his seventieth birthda

Replica symmetry breaking in mean field spin glasses trough Hamilton-Jacobi technique

During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field Sherrington-Kirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite volume limit for the free energy, and its Parisi expression, in terms of a variational principle, involving a functional order parameter. Even the expected property of ultrametricity, for the infinite volume states, seems to be near to a complete proof. The main structural feature of this model, and related models, is the deep phenomenon of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general frame, where replica symmetry breaking is embedded in a kind of mechanical scheme of the Hamilton-Jacobi type. Here, the analog of the "time" variable is a parameter characterizing the strength of the interaction, while the "space" variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing is assumed, or replica symmetry, we build up a progression of dynamical systems, with an increasing number of space variables, which allow to weaken the effect of the potential in the Hamilton-Jacobi equation, as the level of symmetry braking is increased. This new machinery allows to work out mechanically the general K-step RSB solutions, in a different interpretation with respect to the replica trick, and lightens easily their properties as existence or uniqueness.Comment: 24 pages, no figure

Criticality in diluted ferromagnet

We perform a detailed study of the critical behavior of the mean field diluted Ising ferromagnet by analytical and numerical tools. We obtain self-averaging for the magnetization and write down an expansion for the free energy close to the critical line. The scaling of the magnetization is also rigorously obtained and compared with extensive Monte Carlo simulations. We explain the transition from an ergodic region to a non trivial phase by commutativity breaking of the infinite volume limit and a suitable vanishing field. We find full agreement among theory, simulations and previous results.Comment: 23 pages, 3 figure

Sopravvivenza di microrganismi potenzialmente patogeni su superfici porcellanate

È stata valutata la capacità di sopravvivere sulla superficie inerte porcellanata di tazzine da caffè di quattro microrganismi potenzialmente patogeni (Streptococcus pyogenes, Staphylococcus aureus, Pseudomonas aeruginosa, Escherichia coli). Dopo inoculazione è stata dimostrata capacità di sopravvivenza fino a 24 ore di Streptococcus pyogenes, Staphylococcus aureus ma non di Pseudomonas aeruginosa, Escherichia coli. Questi dati suggeriscono il potenziale ruolo patogeno giocato dalle superfici di stoviglie di largo uso nella trasmissione di infezioni da alcuni microrganismi gram-positivi

Equilibrium statistical mechanics on correlated random graphs

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.Comment: 30 pages, 4 figure

Stochastic Stability: a Review and Some Perspectives

A review of the stochastic stability property for the Gaussian spin glass models is presented and some perspectives discussed.Comment: 12 pages, typos corrected, references added. To appear in Journal of Statistical Physics, Special Issue for the 100th Statistical Mechanics Meetin

The mean field Ising model trough interpolating techniques

Aim of this work is not trying to explore a macroscopic behavior of some recent model in statistical mechanics but showing how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfil our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a bridge to a Parisi-like theory is tried and discussed.Comment: 35 pages, no figur

On composite systems of dilute and dense couplings

Composite systems, where couplings are of two types, a combination of strong dilute and weak dense couplings of Ising spins, are examined through the replica method. The dilute and dense parts are considered to have independent canonical disordered or uniform bond distributions; mixing the models by variation of a parameter $\gamma$ alongside inverse temperature $\beta$ we analyse the respective thermodynamic solutions. We describe the variation in high temperature transitions as mixing occurs; in the vicinity of these transitions we exactly analyse the competing effects of the dense and sparse models. By using the replica symmetric ansatz and population dynamics we described the low temperature behaviour of mixed systems.Comment: 35 pages, 9 figures, submitted to JPhys

Biotic, mineralogical, petrographic, and geomorphological characterization of the Falerno-Domitio shoreline (Campania region, southern Italy), with implication for environmental health studies: preliminary results

We report the first results of an ongoing study related to the project FARO (i.e. the Italian “Fund for original research projects”, granted by the Università di Napoli Federico II and IMI bank partner). This research project aims to the enhancement of the physical and biotic features of the coastal landscape related to the Falerno-Domitio shoreline, located in the mid-north coast of the Campania region (southern Italy), from the Garigliano river and Torregaveta. In the national scenario, this area can be considered as a valuable “natural laboratory”, for its wildlife (i.e. the Natural Reserve Foce Volturno, the Regional Park of Campi Flegrei, etc.), famous archaeological sites (i.e. Cuma excavations), and peculiar geological and volcanological characteristics (i.e. Phlegraean Fields). Unfortunately, it also suffers for a strong pollution and environmental degradation due to human activities. The research consists of a multidisciplinary analysis, mainly based on a bathymetric sensing, sampling of both the sea bottom sediments and the beach sands; it comprises: 1) integrated monitoring of the quality of environmental health through a biological study, 2) geomorphological and sedimentological analyses of the area and of the whole sample sets, with GIS data processing, 3) taxonomic and ecological analyses of selected benthic meiofauna assemblages, 4) mineralogy, petrography and geochemistry of beach sands along the shoreline, as well as of sea bottom samples. A complete sampling work of the beach sands, from the Garigliano estuary to the Cuma site, has been done, and the results of mineralogical, petrographic and chemical features, mainly in relation to major and trace elements data, as well as the granulometric curves, are presented. The ecologic and eco-toxicological studies are also carried out on selected samples, revealing the structure of meiofauna (benthic foraminifers and ostracods) assemblages. Tests on the occurrence of the bio-indicator organism Artemia salina have also performed, showing a relatively low toxicity of the samples analysed up to now. Preliminary bathymetric data are also presented

Impact of gastrointestinal side effects on patients’ reported quality of life trajectories after radiotherapy for prostate cancer: Data from the prospective, observational pros-it CNR study

Radiotherapy (RT) represents an important therapeutic option for the treatment of localized prostate cancer. The aim of the current study is to examine trajectories in patients’ reported quality of life (QoL) aspects related to bowel function and bother, considering data from the PROState cancer monitoring in ITaly from the National Research Council (Pros-IT CNR) study, analyzed with growth mixture models. Data for patients who underwent RT, either associated or not associated with androgen deprivation therapy, were considered. QoL outcomes were assessed over a 2-year period from the diagnosis, using the Italian version of the University of California Los Angeles-Prostate Cancer Index (Italian-UCLA-PCI). Three trajectories were identified for the bowel function; having three or more comorbidities and the use of 3D-CRT technique for RT were associated with the worst trajectory (OR = 3.80, 95% CI 2.04–7.08; OR = 2.17, 95% CI 1.22–3.87, respectively). Two trajectories were identified for the bowel bother scores; diabetes and the non-Image guided RT method were associated with being in the worst bowel bother trajectory group (OR = 1.69, 95% CI 1.06–2.67; OR = 2.57, 95% CI 1.70–3.86, respectively). The findings from this study suggest that the absence of comorbidities and the use of intensity modulated RT techniques with image guidance are related with a better tolerance to RT in terms of bowel side effects