1 Introduction

The recent awarding of the Nobel Prize for Physics to Giorgio Parisi has turned the spotlight on Italian contributions to complex systems and, more generally, to the Italian school of statistical mechanics. The Rome group, however, is not the only relevant one in Italy, one of the pillars of that school being certainly the group of Neapolitan researchers who currently contribute substantially to various branches of that field of study (see, for example, Refs. [1,2,3]), working both in Italy and abroad, also gaining important international recognition. However, the interesting history of the Naples group is practically unknown, and only sporadic episodes are generally known to insiders, based mainly on personal memories only. The present paper aims to fill such a gap, by resorting to accurate oral interviews with the protagonists of that group and analyzing appropriate archival documents. We will then be able to trace the history of the most important school of statistical mechanics in Italy after that in Rome, from its inception to the beginning of the present century, highlighting its connections with the international community and assessing its impact on the overall development of statistical physics.

Interestingly enough, as with all the branches relating to theoretical physics developed in Naples, this field too had as its founder Eduardo R. Caianiello [4], who in the 1950 s took up the Chair of Theoretical Physics which had been assumed and almost immediately abandoned almost twenty years earlier by Ettore Majorana [5]. He did found the novel Institute of Theoretical Physics in Naples, by relaunching teaching and research activities, and encouraging the participation to and the organization of workshops and conferences, thus allowing Neapolitan physics to quickly (re-)enter the international scientific panorama. In particular, in 1960 Caianiello strongly promoted the formation of a group working on many-body physics within a field-theoretical approach, laying the groundwork just by organizing two dedicated schools in Ravello (on the Amalfi Coast), with a remarkable lecture series on the many-body problems that resulted into a key reference text [6, 7] in the field.

Subsequently, Caianiello assembled a small group of talented graduates to focus on this relatively new domain that opened the way toward the study of the structure of matter, which at that time was dominated in Italy by nuclear and high energy physics. The group was coordinated by Maria Marinaro and Bruno Preziosi, later (1962) assisted by Marco Maturi and two young graduates, Antonio Coniglio and Giuseppe Iadonisi. Caianiello made available to them the beautiful Villa Vergiliana, a historical building located in Cuma, near Naples, where the group started joining regularly for long days of self-taught study. It is just from these meetings that the first original results about superconductivity and superfluidity came out. However, already a few years later this research group took two different directions: Preziosi and Iadonisi turned toward solid state physics, while Marinaro and Coniglio devoted themselves to the study of critical phenomena. Here, we will focus just on the development of this last subgroup, starting from the crucial experience (discussed in the following two sections) that some members of it primarily acquired abroad, and later reported in Italy (Sect. 4), eventually culminating in the actual formation of a well established statistical mechanics group in Naples (Sect. 5).

The basic source of information for the present study relies, as already mentioned above, on oral interviews, complemented by archival documents from the Department of Physics in Naples and, especially, papers and reports written by the protagonists of our story, as cited throughout the paper. In particular, over a period of time of approximately two months we conducted several interview sessions with Antonio Coniglio and Lucilla de Arcangelis, mainly adopting a simple and “open” methodology, that is: start with a more or less general question, let the interviewee talks about everything that question brought to his/her memory, press him/her with questions that allow to detail what he/she reported. In such a way the interviewee is practically the one “conducting the interview,” the interviewer providing no specific pre-mapped path. A chronological and/or thematic reconstruction of the material acquired, obviously required for the present study, was later carried out without the interviewees, even making recourse to published or archival sources, but discussed with the interviewee in subsequent meetings. Coniglio provided most of the crucial information to reconstruct the birth of the statistical mechanics group in Naples, its subsequent development (until his retirement and even beyond), as well as the collaborations with foreign groups, as described below. The latter mentioned collaborations were more detailed by de Arcangelis (not limited to her own direct involvement), and both were able to trace the picture of the social, political and economic dynamics prevailing in Naples (and elsewhere) during the period of time considered here, also confirmed and highlighted by other Neapolitan witnesses, not necessarily connected to the area of statistical mechanics. The conversations with both researchers were crucial for focusing on the technical contributions of the various protagonists mentioned here, including those not strictly linked to Naples. In fact, the interviews performed greatly facilitated not only the appropriate understanding of the articles published (or even unpublished reports) we have used and that we cite in the course of the present work, but in general they have also allowed us to appreciate the relevance of those contributions in the development of statistical mechanics. The information so far acquired by Coniglio and de Arcangelis was then integrated by that provided by younger researchers, such as Mario Nicodemi and Annalisa Fierro, which allowed us to reconstruct the most recent history of the Naples group, and also to better outline its pre-history, of which nevertheless they had been direct or indirect witnesses. For the sake of legibility, however, in what follows we will not explicitly cite the contributions of the given interviewees, although we will never fail to make references to published papers.

2 London experience

As well known, a ferromagnetic system such as a magnet has the property of attracting iron at low temperatures, while, by increasing the temperature, a critical value can be reached, above which this property disappears, and the system becomes paramagnetic. Such phase transitions are described by thermodynamical quantities that diverge when the critical temperature is approached, according to power laws characterized by critical exponents. Interestingly enough, such critical exponents are “universal”; that is, their values are the same irrespective of the given substance experiencing the phase transition, be it a liquid or a ferromagnet, for example: The paramagnetic–ferromagnetic transition, indeed, turns out to have the same properties as those concerning the gas–liquid transition. The modeling of critical phenomena, such as the ferromagnetic transition, attracted a growing interest in the international community during 1960 s [8], and, inspired by the personal example of Caianiello, Marinaro and Coniglio moved abroad to engage with the cutting-edge research in such a new exciting field, both spending one year in London. At Queen Mary College, Marinaro worked with Geoffrey Sewell (and John Valatin) on the characterization of phase transitions in a general class of Ising spin systems [9], while Coniglio joined the group at King’s College led by Cyril Domb, who was among the most successful scholars in the study of critical phenomena and phase transitions, as well as in the study of the recently proposed problem of percolation.

2.1 First interesting results

Still in Naples, building upon the pioneering work of Kadanoff, who first introduced a group theory approach in statistical physics, Marinaro and Coniglio deduced a general form of the scaling law describing the invariance of thermodynamic functions under the group of dilations [10]. When the two young scholars arrived in London, they then delivered a seminar where they showed their result, and which attracted Domb’s attention. Indeed, when the generalized law is adopted, a continuous infinity of critical exponents appears, corresponding to the continuous infinity of possible paths leading to the critical point. Thus, an asymmetry naturally emerges between temperatures below and above the critical point, with different critical exponents in the two cases. Just this particular feature attracted a lot of interest in London, since series expansion calculations performed at that time at King’s College by Sykes’ group exhibited exactly such asymmetry, contrary to the scaling laws prediction of identical critical exponents above and below the critical point. The general form of scaling introduced by Marinaro and Coniglio contained multiscaling as a particular case, which was later proved—by means of a simple change of variables—to be the multifractality property. Noteworthy, Marinaro and Coniglio’s paper appeared well before Mandelbrot’s formal introduction of fractals [11] and, then, of multifractality in physical systems [12].

Back to Italy, Marinaro and Coniglio still worked for some time on Kadanoff’s approach of scaling (see, for example, [13]), but Marinaro gradually changed (or, rather, came back to) her interests in field theory topics, and, after a Visiting Professorship in the John Klauder’s group at the Bell Laboratories in New Jersey, finally moved in 1974 to the University of Salerno, where she shifted her interests to other research fields. Early in the 1970 s, then, Coniglio was practically the only scholar in Naples doing research in this branch of statistical mechanics: This fact could have easily discouraged him from pursuing a career in this field, but it eventually turned out to be quite positive. As we will see shortly, however, it instead pushed him to search for better opportunities abroad and, furthermore, it also paved the way to collaborations with people working in different areas: The young and dynamic Institute of Theoretical Physics in Naples was an ideal environment for that, as described below.

In the 1970 s, the geometric in nature critical phenomenon of percolation became a hot topic in statistical physics due to its versatility [14], and Coniglio started working on it too, driven by his attitude of envisaging simple physical pictures behind complex phenomena. The critical behavior of different (liquid or ferromagnetic) systems was shown to be characterized by the presence of clusters of particles that, as the critical point approaches, coalesce through a cooperative process, till to finally diverge when the critical point is reached. Coniglio was attracted by percolation due to the actual possibility to study in greater detail the properties of clusters near the critical point, with the hope to understand critical phenomena, like spontaneous magnetization, by using more geometrical concepts, such as the size of a cluster near the critical point. The problem then aroused of how defining a cluster in such a way to represent a correlated group of particles. In the Ising model, the most natural way of doing this was to consider clusters made of nearest neighbors spin in the same state, since 2D Monte Carlo simulations suggested that the size of these clusters diverge at the critical point. This fact pointed out that, in such systems, the percolation critical point (where the clusters’ size diverges) and ferromagnetic critical point (where a spontaneous magnetization appears) coincide, so that the two phenomena could be regarded as two connected aspects of one and the same critical behavior. Attracted by this curious circumstance, during his further stays in London Coniglio started working out the problem using an analytical approach.

First, he solved exactly the problem of cluster size distribution and percolation for interacting spins on Bethe lattices, showing that in such cases the coincidence between percolation and ferromagnetic critical points does not occur [15]. Interestingly, this is one of the first instances of correlated percolation, i.e., of a percolation problem for a system where an interaction term between the sites is present. Then, Coniglio devised a heuristic argument based on topology to explain why the 2D Ising model exhibits such a peculiar behavior [16].

2.2 Neapolitan epistemic style

Already from these first results it is possible to appreciate the working style that has always guided Coniglio, that is, his habitual and preferred approach to a given problem, which will later characterize the entire Neapolitan group of statistical mechanics. We think that it can be very useful to the reader if we immediately highlight the key features of this approach, so that he can follow the historical development more easily without getting too lost in some technicalities.

Coniglio’s approach is certainly theory-based: The starting problem is often theoretical, motivated either by the search for a simple and comprehensive picture of different phenomena or by the need to better understand the origin of some experimental result. Coniglio and his followers often provided exact solutions to specific problems, but this came out by a clever use of simplified models, as easily recognizable in Coniglio and Marinaro first work, as well as in subsequent work about the analytical solution of the time-dependent Ginzburg–Landau model when collaborating with Marco Zannetti on growth kinetics, application of frustrated percolation, etc.

However, the true characteristic feature of Coniglio’s approach is no doubt its exquisitely geometric nature, since the role of visualization was central throughout his research. The clearest example of this feature is percolation, which became a tool to visualize critical phenomena, with the key concept of cluster becoming a heuristic tool able to make the description of critical phenomena more intuitive.

Finally, analogy played a prominent role in Coniglio’s and his fellows’ work, allowing a transfer of models and techniques from a field to the other that made Neapolitan research on statistical physics highly interdisciplinary, as clearly testified by the large number of collaborations that Naples’ group established in different fields over time, the London experience being just a first example. In the following sections we will discuss a number of topics that clearly show this feature, and that are not limited to Coniglio’s own work, such as pair-correlation functions, correlated percolation and polymer gelation, Ising and Potts models (leading to the fruitful concept of Coniglio–Klein clusters), dilute ferromagnets and random resistor network, breaking processes and fuse-insulator network (Lucilla de Arcangelis), field theory and kinetic aggregation processes (Concetta Amitrano and Franco di Liberto), granular systems (in collaboration with Hans Herrmann), and so on.

Without, however, further anticipating the most fruitful research topics of the Neapolitan group, let us now resume our story.

2.3 Further workforce in Naples

Once back to Naples from London, Coniglio asked for help to the mathematical physicist Giovanni Gallavotti (who graduated in Rome in 1963 and then taught Mathematical Methods for Physics in Naples from 1972 to 1975) to get a rigorous proof about his heuristic argument about the 2D Ising model, mentioned above. Gallavotti was, however, little responsive, and the challenge was accepted by the younger Lucio Russo, who, graduated at the University of Naples in 1969, first obtained a scholarship in 1970 and then became a professore incaricato at the same University from 1973–4 until 1978, when he moved to Modena. With Russo’s help, Coniglio solved the problem, showing that Ising clusters necessarily percolate at the critical point (that is, percolation and magnetization points coincide), the key element being the topological impossibility for infinite clusters to stay separated in two dimensions. This result and its generalization, with the proof that Ising clusters do not necessarily percolate in higher dimensions, were achieved with the collaboration of Fulvio Peruggi and Chiara Nappi [17, 18].

Notwithstanding their exceptional abilities, which later granted them key recognitions, Gallavotti and Russo contributed little (but remarkably) to the development of the Naples statistical mechanics group, due to their short stays in Naples, as well as to the fact that their research interests—focused on more mathematical aspects—were too far from the main approach pursued by Coniglio and other collaborators. For example, the only result produced by Gallavotti (in collaboration with Russo and Franco Di Liberto), during his Neapolitan period (1972–1975), concerned the problem of the isomorphism between Ising model equilibrium probability distributions in the convergences domain of the cluster expansion and Bernoulli schemes of the same entropy [19]. This isomorphism problem was further studied by Russo and Gabriella Monroy, who obtained a major result in coding theory, by proving the existence of a finitary code between the one-dimensional Ising model with nearest neighbor interaction and the Bernoulli shift with the same entropy [20]. In Naples, however, Russo continued to work also on the mathematics of percolation and disordered systems, laying the foundation for his subsequent work of enduring importance. In particular, in addition to the results obtained with Coniglio and collaborators, he further proved the smoothness of the 2D percolation probability dependence on the occupation probability away from the critical point, and that the mean size of the finite clusters is finite [21].

The problem of how defining clusters in higher dimensions Ising models remained open but, meanwhile, with Umberto De Angelis and Antonio Forlani Coniglio started a novel collaboration by working on a new approach to the percolation problem based on the theory of the pair correlation function used in fluid systems. A theory of the pair connectedness was developed to describe physical clusters in liquids and lattice systems, and the structure of the percolating cluster in gas–liquid transitions was analyzed [22]. All this contributed to the development of random and correlated percolation by developing a general theory able to study continuum and correlated percolation based on Meyer cluster expansion, allowing the extension of many results from fluid theory to percolation.

A first nucleus of a statistical mechanics’ group was, then, apparently forming in Naples, later including also Di Liberto, Monroy and Rodolfo Figari, which began an intense scientific activity on all these issues. However, due to various reasons not strictly scientific in nature, but rather generally related to the local political situation and ambient conditions at the Institute of Theoretical Physics in Naples, the time was not yet ripe for the effective formation of a stable group, and novel connections appeared on the scientific horizon, once again far from Naples.

3 Naples–Boston connections

Following a suggestion by Eugene Saletan, who at that time was in Naples collaborating with Giuseppe Marmo in Mathematical Physics, around the mid-1970 s Coniglio wrote to Harry Eugene Stanley at the Boston University (BU) to explore the possibility of spending there a sabbatical period. The two had already exchanged some correspondence on topics of mutual interest, so that Stanley accepted with some enthusiasm that proposal, and in 1977 Coniglio then moved to Boston as a Visiting Assistant Professor, where he spent about three years. It was a clever choice. At that time, indeed, Boston was a world-leading hub for statistical physics, with weekly seminars delivered by outstanding scientists and, especially, with an uncommon open-mindedness toward new ideas, making it the best place for a young and ambitious researcher. Many important ideas were developed during this sabbatical period, which marked the beginning of a long-term collaboration between Naples and Boston, proving extremely important for the development of Naples’ group, as we will see.

3.1 Polymers, ferromagnets and correlated percolation

The first problem Coniglio attacked in Boston at the Center for Polymer Studies was that of the sol–gel transition [23], a critical phenomenon occurring in some polymer solutions where a continuous inorganic lattice containing an interconnected liquid phase undergoes a transition from liquid to solid below a critical temperature. Toyoichi Tanaka from MIT—who discovered “smart” gels able to largely swell or contract in volume in response to the smallest changes in temperature, light, chemistry or other agents—held a seminar at BU illustrating some puzzling experimental results that indicated the presence of a percolation line near the critical point. It appeared that neighboring monomers can interact, therefore forming permanent bonds, and, by varying some control parameters, it was possible to increase the number of bonds until reaching a percolation threshold. The formation of a percolating network is just responsible for the sol–gel transition, where the viscosity diverges (structural arrest). These results could not be explained within the current Flory–Stockmayer theory, and this triggered Coniglio’s attention: Polymer solutions were, indeed, clear examples of interacting systems to which the methods of correlated percolation mentioned above could be naturally applied. In collaboration with Stanley and another Research Associate at BU, William Klein, he built a new model for polymer gelation able to predict new effects and especially to account for Tanaka’s results [24]. This was the first of several works Coniglio made about polymer gelation, which over time became one of his main lines of research [25].

A second major breakthrough was the solution to the above-mentioned puzzle of how defining clusters in higher dimensional Ising models, which led to the introduction of the so-called Coniglio–Klein clusters. During the Boston years, Coniglio found that the q-state Potts model—a generalization of the Ising model—could be further generalized to describe random correlated percolation, by means of the introduction of an additional parameter \(p_{B}\) describing the probability to establish a bond connecting parallel spins, with the sole purpose of defining a new kind of cluster. Indeed, by attending a seminar by Nihat Berker, where an identical generalized q-state Potts model was used in a completely different context, Coniglio realized that such Potts model describing the generalized random correlated percolation coincides with the Ising model, for the particular value \(p_{B} = 1 - \textrm{e}^{-2J/kT}\) (where J is the interaction term of the Ising model, k is the Boltzmann constant and T is the temperature) [26]. In other words, he showed that the clusters with parallel spins connected by links exhibiting such particular value of \(p_B\) diverge at the critical point with the same exponents of the Ising model. This was an impressive result, which received great attention by the community, given the geometric insight it provided into the formation of physical clusters. The geometric characterization of the Ising critical point in terms of the Coniglio–Klein clusters, having the properties of percolating with the Ising exponents, later led to the Swendsen–Wang cluster dynamics [27]. The work did have a strong impact on multiple fields of statistical physics (including QCD [28] and cluster fragmentation in nuclear matter [29]), its relevance finally leading in 2017 Journal of Physics A to classify it as one of the 50 most influential papers published in the previous 50 years [30].

In 1981 Coniglio got his permanent position as Full Professor in Statistical Mechanics at the University of Naples, but before coming back to Italy he completed another high-impact work about dilute ferromagnets [31]. These systems are ferromagnets whose magnetic properties are weakened by the addition of impurities, so that it is possible to study the onset of critical behavior as a function of both temperature T and magnetic spin concentration p. Experiments performed at Brookhaven in the 1970 s showed the existence of a multicritical point as \(T \rightarrow 0\) for some value \(p=p_{c}\), for which two different correlation lengths could be defined: \(\xi _T = (T - T_c)^{-\nu _T}\) and \(\xi _p = (p - p_c)^{-\nu _p}\). Information about the structure of the percolating cluster could be obtained from the measurement of the so-called crossover exponent \(\phi = {\nu _p}/{\nu _T}\). Initial experiments performed on Heisenberg-like materials, where spins could align along any spatial directions, gave a value \(\phi \simeq 0.73\), in good agreement with the self-avoiding walk model developed for such materials by Stanley and others. In this model, the crucial simplifying assumption was made that the backbone of the infinite cluster at the critical point is made only of links, while blobs would be irrelevant. However, the model provided the same predictions also for Ising-like materials, for which a crossover exponent \(\phi \simeq 1\) was measured, and this discrepancy was not theoretically understood.

On the other hand, by starting from the consideration that in numerical simulations only the presence of blobs was observed and not also of links, Aharony, Mandelbrot and Kirkpatrick proposed an alternative model with a backbone described by a Sierpinski gasket with no links. At some point, Mandelbrot and Aharony held a seminar at BU about dilute ferromagnets, where they showed that their model coincidentally has a fractal dimension close to that of the percolating cluster of dilute ferromagnets. The results they obtained were not in agreement with experiments, but this seminar suggested to Coniglio a new approach to the problem, by avoiding any a priori assumption about the structure of the percolating cluster, namely about the relative presence of links and blobs at the critical point. In those days he was working out the problem by following this line of reasoning, and, during a concert at Boston esplanade, he had the key intuition: Although the backbone consists mainly of bonds belonging to blobs, nevertheless at \(T=0\) every blob behaves like a single spin, so that, in the propagation of correlation in Ising-like materials, links played the main role and could not be excluded from the analysis. This was not the case for Heisenberg-like ferromagnets, and just such a difference was at the origin of the different values of \(\phi \). The crossover critical exponents of the dilute Ising model and Heisenberg model were then proved to be related to the fractal dimension of the singly connected bonds and the resistivity exponent, respectively, since the structure of the percolating cluster in the Heisenberg model turned out to be analogous to that of a random resistor network. He obtained predictions in perfect agreement with the measurements performed on dilute ferromagnets by Birgeneau, thus providing a geometrical interpretation of why the two crossover exponents were different. Such a work led to the description of the percolating cluster in terms of links and blobs, where links (or “red bonds,” according to Stanley’s terminology) have a fractal dimension equal to the thermal exponent, while it was already known that the fractal dimension of the whole cluster coincided with the field exponent. This model finally became the standard one to describe the percolating cluster at the critical point and, as a consequence, Coniglio’s theory has been used over the years in understanding a variety of transport phenomena: spread of epidemics; diffusion, flow and propagation of correlations in random media; percolation properties of complex networks, etc.

The connection with Boston did not end there: Just the same year that Coniglio got his professorship in Italy, another researcher arrived in Boston from Naples to further obtain interesting results.

3.2 Fruitful interdisciplinarity for multifractality

Lucilla de Arcangelis graduated at the University of Naples in 1980, under the supervision of Iadonisi, with a thesis in Solid State Physics. However, her training in this specific field had been quite limited, and she was eager to expand her knowledge to have a more solid background before devoting herself to original research: At that time, PhD courses were not yet available in Italy, and so she sent several applications to US universities. Many of them gave her a positive feedback, but she finally opted for Boston University, where she would spend five fruitful years, from the end of 1981 to 1986.

Her initial experience in Boston was though: She followed lectures and worked as Teaching Assistant, but, like many women in a male-predominant academic environment, she struggled with gender issues. It took quite some time and extra effort to be recognized as a talent, which actually happened when she passed the painstaking comprehensive exam with an outstanding 4.04/5—the highest rank in the whole Physics Department. This achievement brought her multiple PhD thesis offers, starting with a project about graphite intercalation compounds under the guidance of George Kirczenow and in collaboration with the celebrated Mildred Dresselhaus from MIT. However, when Kirczenow got a tenure position at Simon Fraser University, de Arcangelis decided to stay in Boston and asked Coniglio’s advice about how to pursue her career. He proposed her to switch from solid state physics to statistical mechanics, also considering the promising prospects that Boston offered in this field: It turned out to be a wise advice.

de Arcangelis got her PhD in 1986 with Sidney Redner, with an extremely successful thesis entitled Multifractality in percolation: the voltage distribution, with results displayed into three high-impact papers published reflecting once again the dynamic environment of BU in those years.

In the first paper [32], a percolation model was used to analyze the voltage distribution of a random resistor network. It was solved analytically, and predicted the existence of an infinite set of independent exponents able to describe the voltage moments near the percolation threshold: This was the first-ever explicit appearance of multifractality in the study of critical phenomena. Here, the dominant idea was that they were described by only two independent critical exponents (thermal and field exponents), so that, since the percolation problem also falls into the class of critical phenomena (of a geometric type), it was difficult to accept the idea of the presence of an infinite number of critical exponents, which was instead typical of multifractality (despite the fact that multifractality had already been introduced in the context of turbulence a few years earlier [33, 34]). de Arcangelis’s results aroused some suspicion in the community (also causing a considerable delay in the publication of the paper), the importance of which was recognized only after the explosion of studies about multifractality.

The first mesoscopic model for breaking processes in heterogeneous media was developed, instead, in the second paper mentioned [35]. The key idea was to build a mapping between the finite elements of the given material and a fuse-insulator network, which was then studied using percolation methods. The breaking of this network under an external voltage was analyzed as a critical phenomenon, thus providing the first scalar model for the breaking of an elastic material under a tension–compression stress—a problem usually solved by much more complex vector models. The most interesting feature of this method was that scaling laws may hold near the critical point independently of the particular model used, according to their universality property, then opening a new field of research about mesoscopic approaches to these problems.

Finally, the third paper [36] was devoted to the propagation of a neutral tracer (e.g., an ink drop) in a fluid flowing within a porous medium, obtaining exact results along with numerical algorithms for computing the motion of the tracer in the medium.

de Arcangelis’ interdisciplinary research involved both new theoretical ideas and practical applications from the very beginning, and the impactful results she was able to get earned her several proposals for postdoctoral positions. The inviting proposal from Dietrich Stauffer, one of the founding fathers of percolation theory, to join his group in Cologne could not be refused, and then in 1986 she finally moved to Germany.

4 Back to Europe

The situation found by Coniglio upon his return to Naples in 1981 was practically the same as the one he had left for his sabbatical period abroad, that is, a lack of appropriate attention toward frontier research. Although he did act to lay the foundations for a group that actively worked on statistical mechanics problems, he nevertheless kept alive the already existing international collaborations and established new ones, so that throughout the 1980 s he kept spending long periods abroad, both in Boston and elsewhere in France (Saclay), Brasil (Rio de Janeiro) and Canada (Nova Scotia). All this had a twofold consequence: The formation of a stable group was obviously slowed down, but the strengthening of international connections later proved essential for its future growth.

4.1 Cluster dynamics and high-\(T_c\) superconductivity

An interesting study related to the breakdown of hyperscaling in high-dimensional systems dates back to this period [37]. It was known from the theory of the renormalization group that, beyond a given critical dimension \(d_c\), the critical exponents do not depend anymore on dimensionality, their values being given by the mean field theory; for example, in Ising models this happens for \(d_c = 4\), while \(d_c = 6\) in percolation models. According to his attitude, Coniglio tried to understand the mechanism underlying such thresholds by devising a physical-geometrical explanation of this occurrence based on intuitive arguments. He finally found that the breakdown of hyperscaling was related to the presence of infinitely many critical clusters, each of them with a fractal dimension equal to 4 in any dimension greater than \(d_c\). As a consequence, the important results followed that the critical exponents in dimensions above the critical one are the same in any dimension.

Clusters dynamics was also the topic of another work [38], where the Coniglio–Klein droplets mentioned above, giving a geometrical description of the fluctuations in the q-state Potts model, were shown to have a fractal structure at criticality made of links and blobs as in percolation. Coniglio provided exact values in 2D for the fractal dimension of the red bonds for any q, by using the mapping from the Potts model to the Coulomb gas, while for \(q=0\) he obtained the fractal dimension of the red bonds in the spanning tree. The Mandelbrot–Given model for percolation clusters was then proved to correctly describe the fractal structure of the Potts clusters.

High critical temperature superconductivity was, instead, a novel topic on which Coniglio gave an as well relevant contribution during a long research period in BostonFootnote 1 [39]. In 1986 Georg Bednorz and Alex Müller in Zurich made the groundbreaking discovery that a ceramic material (lanthanum barium copper oxide) has a very high critical temperature of \(35^{o}\)K, later independently confirmed also for other materials by Skoji Tanaka in Tokyo and Paul Chu in Houston. This discovery instantly made high-\(T_c\) superconductivity the hottest topic in statistical mechanics, starting a race to explain it theoretically. Coniglio and his collaborators in Boston and Tel Aviv managed to deduce the existence of electron–phonon interactions using a spin glass model that was able to elucidate the phase diagram of those materials displaying high-\(T_c\) superconductivity. Indeed, by studying the temperature-concentration phase diagram of the relevant lanthanum cuprate, they found that the magnetic interactions of the hole spins with the copper spins yielded frustration. This explained the fast decrease in the Ne’el temperature and yielded a new spin glass phase, the same interactions providing a strong hole–hole potential, potentially leading to pairing and superconductivity [40]. This microscopic spin glass model had so wide resonance that in 1988 the corresponding paper was the most cited one in the Physical Review Letters.

4.2 Different directions: cellular automata and the filtering of aluminum cans

A bit earlier, as said above, de Arcangelis arrived at the Institute for Theoretical Physics in Cologne for a postdoc one year collaboration with Stauffer, with whom she worked mainly on Kauffman cellular automata, used for example to explain the genetic differentiation. The 2D random Boolean networks introduced by Kauffman were shown to have a transition to chaos, and she calculated by Monte Carlo simulations the period distribution of the limit cycles for the square lattice, both at the transition point at away from it, showing that the triangular lattice interestingly has the same fractal dimensions as the square lattice [41]. Later, she also studied (by computer simulations) the Kauffman random networks of automata on a simple cubic lattice, obtaining an observed transition between the frozen and chaotic phase and measuring the fractal dimension of the asymptotic actual damage at the phase transition [42].

Although de Arcangelis developed an excellent working relationship with Stauffer (who was her main scientific mentor, together with Coniglio), after just one year in Cologne she won the first fellowship from the European Community, and then moved to the Centre d’Ètudes Nuclèaires in Saclay, where worked for three years starting from 1987. In Saclay she collaborated with Hans Herrmann’s group on different topics, including cellular automata and percolation, with a focus on numerical simulations aimed to generalize her previous PhD work on percolation and breaking processes. Such phenomena caught the attention of statistical physicists not only for the potential technological interest, but also for the intriguing patterns and scaling laws those processes exhibit, so that standard techniques used in the study of disordered systems were applied to fracture. As already mentioned above, de Arcangelis already focused her attention in Boston on the simplified system of a network of electrical fuses, for which different theoretical models existed. In collaboration with Herrmann, she developed a numerical simulation of a random fuse network with random breaking thresholds, obtaining peculiar breaking characteristics for the system [43]. Indeed, the system considered was shown to exhibit two different regimes, with a multifractal structure at the very late stage of breaking, whereas it behaved as a homogeneous system in the first regime, with results remarkably stable with respect to variations of the quenched disorder in the thresholds. In a subsequent work [44], universal scaling laws for the breaking characteristics were also deduced for the first time describing the dependence of the fracture on the system size. Such scaling laws between external force, total displacement, number of bonds cut and the size of the system were shown to hold for three different models, corresponding to three different physical situations of disordered media (scalar, central force, and beam model).

Remarkably, despite numerical methods were not much appreciated in the French scientific community at that time, the impactful work carried out by de Arcangelis earned her a CR1 research position at the Laboratoire de Physique et Mécanique des Milieux Hétérogènes, of CNRS in Paris. She would spend here three more years, during which her attention turned to applied problems solved using methods drawn from the study of critical phenomena, such as corrosion, breaking processes and filtration in porous media. For example, the Parisian Laboratoire obtained substantial funding from the French industrial company Pechiney that produced aluminum cans for a well-known soft drink, for which good filtering was essential, otherwise the cans would explode at the slightest stress in the presence of impurities inside them. Filtration processes are especially difficult with aluminum and far from efficient, due to the easy clogging of the filters themselves, and de Arcangelis and her collaborators tried to optimize this process by studying in detail the dynamics of filtration, a very difficult problem in fluid physics [45]. The experimental and numerical studies about deep bed filtration of small non-Brownian particles in suspension by a random packing of larger spheres revealed a transition in particle capture. It was analyzed as a phase transition phenomenon in analogy with percolation, showing that the critical behavior of the penetration length differed from the known results for random and directed percolation. Furthermore, when interactions between particles were considered [46], the team discovered intriguing collective effects attributed to hydrodynamic phenomena, for which packets of particles penetrates further than the same number of particles released one at a time.

4.3 Multiscaling, multifractality and frustrated percolation in Naples

Toward the end of the 1980 s, Marco Zannetti arrived in Naples as an Associate Professor; he graduated in Rome as early as 1962 and got his PhD on the field theory for continuous phase transitions at the Brandeis University, supervised by Daniel J. Amit. Zannetti already held an equivalent position at the University of Salerno since 1983, but in 1987 he decided to move to Naples, where he established a fruitful collaboration with Coniglio. The two scholars contributed to the development of phase-ordering processes by providing for the first time an exact analytical solution for the time-dependent Ginzburg–Landau model, in the limit in which the number of components of the order parameter goes to infinity [47, 48], finding a multiscaling behavior in growth kinetics. A similar multiscaling behavior was also later found, in collaboration with Concetta Amitrano and Paul Meakin, in the density profile in diffusion-limited aggregation in two dimensions, by using large scale computer simulations [49,50,51]

Interestingly, Coniglio was among the first to contribute with his coworkers to the development of multifractality and multiscaling even in diffusion-limited aggregation (DLA) model, in addition to his contributions with de Arcangelis and Redner in percolation, already introduced above (further works related to random resistor and random superconducting networks, as well as multifractal structure in the incipient finite cluster, are in Refs. [52, 53]). Indeed, the later work with Zannetti just discussed was part of a line of research already open in Naples, to which Amitrano and Di Liberto contributed for some time. For example, as early as 1986, Amitrano, Coniglio and Di Liberto applied the standard Green’s function technique to calculate the growth probability distribution in kinetic aggregation processes, finding that a hierarchy of independent exponents is required to describe the critical behavior [54]. In other words, they already discovered that, in both DLA and related models continuously depending on a parameter, a non-conventional scaling for the growth-site probability distribution emerges, which is different from a pure power law.

With several collaborators, including Amitrano, Di Liberto, Monroy, Zannetti and the newcomer Vittorio Cataudella, in Naples Coniglio developed in those years the model of frustrated percolation [55]. This concept was actually introduced by Coniglio, and taken up by few others outside Naples, which over time has had several applications in various branches. In such a model, two competing interactions acting on the given statistical system are introduced, one tending to align spins with each other, and the other tending to anti-align them, for example. The resulting dynamics is thus very complex, since the formation of a cluster requires a rearrangement of the whole system. Coniglio and his collaborators applied this model in order to explain phase transitions in systems more complex than the Ising model, as already mentioned above. With Di Liberto, Monroy and Peruggi, he proved that the frustrated percolation model exactly maps the spin glass model into a geometrical model—the frustrated percolation model is related to the Ising spin glass model just as standard percolation is related to the Ising model—and tried to generalize the idea of Coniglio–Klein clusters to such systems [56]. Despite this problem was only partially solved, the model of frustrated percolation was later fruitfully applied to a large number of problems.

5 And finally a group was born!

From what discussed above, it is clear that a non-negligible number of people worked in Naples on statistical mechanics topics during 1980 s, essentially having Coniglio as their reference person. However, contrary to naive expectations, this fact cannot be assumed as a fundamental characteristic for the formation of a stable group for several different reasons.

Coniglio’s hiring as a Full Professor in 1981 was potentially the first key event favorable to the birth of a group in Naples but, as a matter of fact, he only found himself working with somewhat scattered people for more than a decade. This was not due to a lack of willpower or a fallback on Coniglio’s part, but rather to factors typical of the Neapolitan environment of the time. First of all, the 1970 s saw the presence of a number of young scholars open to new ideas, also due to the fact that they did not yet belong to well-crystallized groups, so that a certain “fluidity” from one scientific interest to another could be easily experienced. The situation tightened in the 1980 s, when the economic situation worsened, and everyone began to cultivate only their own specific interests. The crystallization of several research groups resulted then into some opposition, both from possible competitors and from the leaders of the previous generation of physicists, that Coniglio experienced when asking for fellowships to form a research group.

Also, a somewhat strong class and generational struggle was raging in Naples in that period, whereby the young generation fiercely opposed the real or presumed manifestations of power of the older generation who, more accustomed to the monocratic situations of the past in the management of the university institutes, could not easily digest the “exuberances” of the new generation of physicists, to which Coniglio certainly belonged. As a matter of fact, even the Physics Institute of Naples began to experience a brain drain: Graduates looked for positions abroad, also indirectly due to the Italian laws against precarious employment, which did not allow the provision of short-term research grants (the establishment of PhD positions only occurred in 1983).

Old practices of the past manifested themselves, more or less indirectly, even in a more purely scientific context. At that time there was no effective evaluation of research activity by central institutions, so that there was not even a certain (healthy) pressure to publish results: Throughout a period between the late 1970 s and early 1980 s, many people in Naples almost boasted of not doing scientific research, and certainly not publishing it. Also for this reason, for example, several of Coniglio’s collaborators in Naples slowly began to give up, or even to dedicate themselves to other interests.

Furthermore, Coniglio’s attitude, already highlighted above, to always keep the international collaborations he had personally established alive and active, for which he took short or long sabbatical periods abroad on an approximately annual basis, certainly also hindered the formation of a statistical mechanics group. However, Coniglio never had any doubts about leaving Naples: when he won the professorship in Naples, he was also offered a position in Boston (where he had a wife and children), but he did not accept. Such a decision was also facilitated by his feeling that, in addition to doing research, in the USA a good part of the time was used to sell that research in order to obtain adequate funding, a clear example being that of the Grant Monitor, the research evaluator who periodically needed to be convinced when Coniglio was in Boston. Naples was, therefore, a much preferable venue in his eyes.

Starting from the beginning of the 1990 s, however, the situation changed significantly. Coniglio was able to secure the necessary funding through national and European grants, and finally it was possible to form a stable group with young researchers working full time on statistical mechanics and Complex Systems. The main characters of this nascent group were Mario Nicodemi, Antonio De Candia and Annalisa Fierro, who were joined by Giancarlo Franzese (now in Barcelona), Antonio Scala (now in Rome), Emanuela Del Gado (now in Washington) and later by Marco Tarzia (now in Paris), Massimo Pica Ciamarra (now in Singapore) and several others. Thanks also to this army of new forces, the study topics also expanded, always suggested by the international relations which continued to be a constant characteristic of the group.

5.1 Glasses and spin glasses

Spin glasses are solid materials where every atom has random interactions (both ferromagnetic and antiferromagnetic) between spins, so that contrary to ferromagnets they exhibit a disordered magnetic state, where spins are aligned randomly. Such disorder induces frustration, since no configuration exists where each spin simultaneously minimizes all its interactions: Ferromagnetic and antiferromagnetic interactions are distributed in such a manner that not all the spins can satisfy all the interactions simultaneously. Instead, (structural) glasses are liquid materials cooled in a way avoiding their crystallization (something very common in molecular or polymeric liquids), so that their atomic bond structure is highly irregular, contrary to what happens in a solid crystal. Also glasses exhibit frustration due to their disordered structure, and the Naples group was motivated to study even such systems by the hope of extending the methods already applied to describe the frustrated percolation in spin glasses also to them [57].

The group has given a relevant contribution to the understanding of the glassy transition from a liquid to a disordered solid by elaborating analytic models, mean field techniques and numerical simulations of different classes of such systems.

For example, their numerical study of the dynamical properties of fully frustrated systems in 2D and 3D showed that the percolation transition marks the appearing of the large scale effects of frustration below the percolation threshold. The percolation transition of the Coniglio–Klein clusters corresponds to the onset of stretched exponential autocorrelation functions in systems without disorder [58].

Also, an effort was undertaken to understand the problem of universality in glassy systems, i.e., the fact that many physical systems and models exhibit a glass transition, notwithstanding the different microscopic structures. They introduced a peculiar lattice model having properties closely related to those of p-spin glasses in the mean field formalism, but constituted of diffusing particles, so that it was well suited for studying quantities usually important in the study of liquids [59]. The intriguing result was that, from finite dimensions numerical Monte Carlo simulations in 3D, the frustrated lattice gas model did exhibit qualitatively the properties of glass-forming liquids.

Glass transition is not the only example where a structural arrest appears (i.e., viscosity diverges), and several other disordered systems exhibit a slow dynamics followed by a structural arrest driven by suitable control parameters, such as (chemical and colloidal) gels and granular materials. Coniglio and his group contributed significantly to the understanding of all of these peculiar materials.

5.2 Granular materials

The new line of research about such systems was inaugurated by a collaboration of Coniglio with Hans J. Herrmann in Paris, one of the leading experts on the field, and inspired by the Nobel Laureate P.-G. de Gennes [60], later reinforced by one year’s visit of Nicodemi in Paris for his PhD work. Granular matter is composed of particles for which thermal agitation is negligible; Brownian motion is not present, and hence a standard thermodynamical approach cannot be applied. It is a new type of condensed matter, showing both a fluid-like state and a solid-like one: Similarly to sol–gel transitions, granular systems under compression exhibit a transition from a fluid-like state to an amorphous solid, this latter being called a jamming state. They are particularly interesting, since they play a relevant role in phenomena such as avalanches and earthquakes, and are even related to jamming problems like ischemia and heart attacks.

As it can be easily expected, the description of such systems was not an easy task, and a novel formulation was required based on a generalized statistical mechanics approach (initiated by Samuel Edwards): The Naples group was one of the main (and first) ones in developing such a theoretical approach [61], as well as in its application to interpret the experimental results. They showed that the peculiar transition exhibited by granular materials may be studied as a non-thermal critical phenomenon with the density (rather than temperature) playing the role of the control parameter, to which the methods of frustrated percolation could be usefully applied [62]. The analysis of such out-of-equilibrium transition revealed the presence of a mixed behavior between a continuous and discontinuous transition: By employing an Ising lattice gas model with steric frustration to describe the contact network of granular packing, two different transitions were identified, that is, a spin glass transition at the onset of Reynolds dilatancy and a percolation transition at lower densities. Subsequent works further highlighted profound connections of granular materials to spin glasses (and glass-forming liquids), describing different aspects of the known phenomenology, such as the compaction of granular systems in the presence of vibrations and gravity [63].

With the consolidation and expansion of the group, in the first years of the new millennium a number of works appeared, where the development of their approach to the study of granular systems was deepened, allowing the treatment of many problems of those systems just by using standard tools of the statistical mechanics formalism [64,65,66,67]. For example, the jamming transition occurring in granular materials was shown to have, at mean field level, the same static and dynamic properties of structural glasses [68,69,70], while the segregation phenomena taking place in granular mixtures was described in terms of an effective free energy, with the notable prediction of large density fluctuations associated to the presence of a critical point in vibrated granular mixtures [71,72,73]. Numerical simulations have confirmed many of these results, which were suitable of experimental test.

5.3 Old and new problems: colloidal gels and chemotaxis

Notwithstanding the exciting new fields studied at the onset of 2000 s, as recalled above, Coniglio’s early interest in polymer gelation processes never waned, also due to the fact that the sol–gel transition shows similarities to the transition in spin glasses. Colloidal gels, for instance, are an example of reversible gels, where the bonds between molecules are not permanent and can be broken by increasing the temperature (or even by applying a shear force), while at low temperatures their behavior is similar to that of chemical gels. Corroborated by his group in Naples, several important contributions were given in discovering the important role in colloidal gels played by the finite lifetime bonds in cluster formation [74]. By means of 3D Monte Carlo simulations of a lattice model for gelling systems, the dynamics of gelation phenomena was studied, showing a connection between classical gelation and the phenomenology of colloidal systems [75]. In attractive colloids, a hard sphere glass transition was known to occur at high temperatures and at a given volume fraction, while, at low temperature and increasing the volume fraction, a much more complex situation is exhibited, with a cluster phase followed by a kinetic arrest showing viscoelastic properties similar to those found in polymer gelation.

Coniglio suggested that, in the mechanism for structure formation giving rise to colloidal gelation, percolation phenomena play a crucial role. Indeed, when an interaction for colloidal suspensions is introduced in the model above, due to the competition of attraction and repulsion, nearest neighbors are favored and next nearest neighbors are not. The relaxation time at low temperatures was shown to first increase critically as a power law in the volume fraction, while deviations from such critical behavior were observed at larger fractions due to the finite lifetime of the bonded structures. They found the important result (in agreement with experimental tests) that, at low volume fraction and low temperature, the formation of a spanning long living large cluster strongly influences the dynamical behavior of the system, with the dynamical arrest taking place through the growth of elongated structures that aggregate to form a connected network of gelation [76]. It was also shown that, near and above the gelation threshold, the disordered spanning network slowly evolves to form an ordered crystalline columnar structure, while, at higher volume fraction, the stable thermodynamical phase is instead a lamellar one, with even a longer ordering time [3].

All such results turned out to be very interesting also in industry and biology, since blood, proteins in water, milk or paint are just colloidal suspensions, and—for example—controlling the process of aggregation in paint and paper industries, or favoring the protein crystallization in the production of pharmaceuticals and photonic crystals, is crucial in many applications.

From combined efforts of physicists and biologists, completely novel works then arose concerning chemotaxis. The development of complex eukaryotic organisms was known to be governed by the ability of their cells to navigate along spatial gradients of extracellular guidance cues offered by chemoattractants, but its physical origin was unknown. Sensing spatial gradients of chemoattractant factors is nevertheless crucial for understanding embryonic development, tissue regeneration and cancer progression, so that having a working model for chemotaxis would be highly desirable.

In this framework, the Naples group proposed a realistic model [77, 78] according to which directional sensing is the consequence of a phase-ordering process, where an effective enzyme–enzyme interaction induced by catalysis and diffusion introduces a system instability driving the system toward a phase separation, taking place at the plasma membrane upon chemoattractant stimulation. This mechanism was able to self-tune the cell to an equilibrium state of phase coexistence, which amplifies a shallow gradient of chemical attractants, by phase separating, thus triggering directional mobility of eukaryotic cells. Noteworthy, the proposed phase-separation scenario provided a unified framework to different aspects of directed cell motility, with a number of properties arising naturally in the model, such as large reversible amplification of shallow chemotactic gradients, selective localization of chemical factors, macroscopic response timescales and spontaneous polarization.

5.4 Non-grouped collaborations: earthquakes and neural systems

The long-time collaborator de Arcangelis participated—with her first graduate, and then PhD, student E. Del Gado—to some of the works on gelation phenomena performed by the Coniglio group and discussed above. As previously recalled, at the beginning of the 1990 s she was still working with satisfaction in France, but felt a certain discomfort, as Paris was essentially a place of transit, where it was difficult to build lasting relationships. Then, when the chance to return to Italy arose, she jumped at this opportunity and in 1993 got a position as an associate professor at the University of L’Aquila. This event gave her the possibility to join, though at a distance, the group in Naples to work on sol–gel transition, and such collaboration continued when, in 1996, she moved to the Second University of Naples (at the Faculty of Engineering, the only one offering a Physics position).

However, for several reasons, this fact did not then translate into an effective strengthening of the group at the University of Naples. In 2002, indeed, de Arcangelis was involved in some work on the physics of earthquakes by Paolo Gasparini, who held the chair of Terrestrial Physics at the same university. Gasparini well knew her brilliant papers on fracture [44], and de Arcangelis took this opportunity to somewhat emancipate herself from Coniglio and build a more autonomous research profile. This actually happened and, after this work, she began a new line of successful research focused on the application of statistical mechanics to the study of the brain, also in collaboration with several international personalities, including Herrmann and Dietmar Plenz at the National Institute of Mental Health in Bethesda. As a matter of fact, a second, different group was about to start, substantially contributing over the years on several other topics in addition to seismic [79, 80] and neural [81] systems, such as social percolation models [82], self-organized criticality [1], solar flares [83], etc., with exchange of students and collaborators from time to time with the Coniglio group.

6 Summary and concluding remarks

Research studies in statistical physics were introduced in Naples in early 1960 s, thanks to the far-sighted promotional action carried out by Caianiello, the founder of the Institute of Theoretical Physics at the University of Naples. In the present work, we have presented the interesting case of the formation of a formal group at that University, devoted just to pursue research on statistical mechanics and its applications, given its notable past and present relevance in the international scientific panorama, as testified for example by the successful reception in the literature of the results obtained.

Until the end of the 1970 s, the relevant activity was to acquire appropriate skills and knowledge mainly abroad, with London and Boston being two key places, and the first favorable opportunity offered in 1981 by the appointment of Coniglio as Full Professor in Statistical Mechanics at the University of Naples did not actually trigger the formation of a group. As we have seen, this was primarily due to Coniglio’s repeated absences from Naples for short or long sabbatical periods, as well as to an effective lack of state incentives to adequately train young researchers, that was unable to prevent their search for success abroad. This last fact is blatantly evident in the training of de Arcangelis who, after graduated in Naples in 1980, successfully worked in Boston and then Cologne and Paris for more than a decade.

As a matter of fact, Coniglio continued to keep alive and fully active the international collaborations he had previously established (and created new others), but at the same time he actually worked with young colleagues or students in Naples (or with peoples from Naples but working abroad, such as de Arcangelis). This did not result into the effective establishment of a group with Coniglio as the key character (thanks to his expertise and international relationships), since his collaborations with local researchers were sparse and occasional, and often not institutionalized in Naples with fellowships or similar.Footnote 2 This situation did not change even when in 1987 the Roman Zannetti moved from the University of Salerno—where he was Associate Professor since 1983—to Naples, where he fruitfully collaborated with Coniglio: Having failed to secure a Full Professorship in Naples, Zannetti later moved to Salerno in 1990.

At the beginning of the 1990 s, however, a real change of pace occurred, and Coniglio was able to obtain Italian and European funding to guarantee adequate training of young researchers who could successfully work in Naples. The group, then, gradually expanded over the years and strengthened with de Arcangelis return to Italy (though at the University of L’Aquila) in 1993, with whom a closer collaboration began in that period, subsequently evolved into a parallel working group when she moved at the Second University of Naples in 1996.

The Neapolitan group of statistical mechanics effectively inherited the research approach followed from the beginning by Coniglio, with its distinguishing characteristics of being theory-based, geometric and analogical. These characteristics are manifestly evident not only in the method of studying the different problems, but also in the choice of study topics. Above all, the group contributed decisively to that transfer of tools and methods that is so characteristic and fascinating of statistical mechanics, thanks also to the dense network of important international collaborations established over time.

Starting in the 1970 s with the project of understanding critical phenomena using geometrical concepts, Coniglio and collaborators provided a rigorous proof that Ising clusters percolate at the Ising critical point in 2D (but not necessarily in 3D), then posing the problem of a new definition of clusters that would have the properties of Fishers droplet model. This problem was actually later solved through the introduction of the so-called Coniglio–Klein droplets percolating at the Ising critical point with the Ising exponents, obtaining fundamental results in percolation theory and providing a geometrical picture for the universality of critical exponents. The cluster structure of the percolating cluster was then studied in detail, showing a fractal structure made of nodes, links and blobs. As a matter of fact, Coniglio’s theory has been used in understanding a number of transport phenomena, such as the diffusion, flow, propagation of correlations in random media, the spread of epidemics and the percolation properties of complex systems. The Coniglio–Klein droplets of the Ising model were later extended to the q-state Potts model [84], obtaining several interesting exact results [38] that have been used also in complex networks and in the abelian sand pile model. Coniglio and de Arcangelis were the first (in collaboration with Redner) to discover multifractality in percolation and, soon after, in diffusion-limited aggregation, while Coniglio and Zannetti contributed to the development of phase-ordering processes, providing for the first time an analytical solution for the time-dependent Ginzburg–Landau model in the limit of an infinite number of components of the order parameter. Finally, the Naples group significantly contributed in applying the techniques of statistical physics to complex systems biology and soft matter, with the introduction of the concept of correlated percolation—which revealed all its potentialities in the development of models of polymer gelation—and the study of granular systems and glass transition.

Here, we have focused on the group directly or indirectly generated by Coniglio, who effectively originated and established the research work in Naples on statistical physics and its application but, for the sake of completeness and as evidence of the complexity of the Neapolitan situation, another important scholar on the same field settled in Naples in the late 1980 s, at about the same time when Zannetti joined Coniglio in Naples.

Luca Peliti graduated at the University of Rome in 1971 and then got his PhD in 1974 at the already mentioned Queen Mary College of the University of London. His first works were made on critical phenomena in collaboration with Giorgio Parisi, with whom he continued to collaborate occasionally on different problems with some success [85, 86]. Later on, his interests converged on a variety of topics, including fluctuation phenomena [87, 88], systems with slow dynamics and aging [89], genetics [90, 91] and, in general, on the statistical physics of biological systems. Important results on amphiphilic membranes (that is, two-dimensional structures made of molecules formed of a hydrophilic and a hydrophobic moiety) and DNA, whose behavior is dominated by thermal fluctuations that determine complex static and dynamic effects, were obtained by Peliti when he was in Naples. He was able to clarify several properties of those systems, and finally solved the puzzle of the first-order nature of the denaturation transition of DNA [90]. Also, he studied the statistical aspects of Darwinian evolution [92, 93], when a strict analogy was highlighted between its dynamics and that of some disordered systems, obtaining several properties from the distribution of genotype sequences in populations undergoing neutral evolution. Such an approach originated a subfield of statistical mechanics, with relevant applications to the dynamics of infectious diseases.

Notwithstanding these important results, as well as the relevant network of international collaborations established, Peliti did not really form a group of his own as de Arcangelis did (his only “heir” student, who collaborated with him for a certain, short period in Naples, was A. Imparato [94, 95], who later left Naples). Also, although fruitful discussions and some exchange of students between Peliti and Coniglio’s group continuously occurred, no actual and formal merging took place, given the very different research lines within the common field.

In 2014, Peliti retired and left Naples, while Coniglio became Professor Emeritus at the University of Naples “Federico II” since its retiring three years earlier. The group originated by him presently has evolved into different, independent branches at that University, as well as at the University of Campania “Luigi Vanvitelli” (formerly Second University of Naples). de Arcangelis is now Full Professor at the Vanvitelli University and continues to work on stochastic phenomena (also applied to earthquakes, etc.) and neuronal networks (still in collaboration with Herrmann and Plentz), as well as on a number of other topics including granular materials and COVID studies [96,97,98]. Since 2021 she is Chair of the C3 Commission of the IUPAP, while in 2022 she was elected APS Fellow and Member at large of the Division of Computational Physics Executive Committee of the American Physical Society. Nicodemi, instead, is Full Professor of Theoretical Physics at Federico II University and Einstein Visiting Professor at the Max Delbrück Center in Berlin, working at the frontier between the physics of complex systems and molecular biology. He contributed to clarify the complex 3D architecture of the human genome and its connection with the regulation of gene activity, also developing a new technology for genome-scale mapping of interactions chromosomal [99, 100]. For his remarkable results, in 2016 he received the Einstein BIH Fellowship of the Einstein Foundation Berlin, while in 2022 he was awarded the Giuseppe Occhialini Medal and Prize by the Italian Physical Society and UK Institute of Physics. Finally, De Candia is Associate Professor at Federico II University, while Fierro is Researcher at the SPIN Institute of the Italian National Research Council (CNR) in Naples, both working on the traditional topics of former Coniglio’s group (including the dynamics of physical systems undergoing a structural arrest such as glasses, gel and colloids), as well as on the critical dynamics of neural networks and on several different problems relevant in biology and medicine [101, 102], also in collaboration with Coniglio, de Arcangelis and Zannetti [103,104,105,106,107,108].

A number of other first generation researchers emerged from original Coniglio’s group now hold prestigious positions around the World, including for example Pica Ciamarra, a researcher at the Italian National Research Council (CNR) in Naples and Associate Professor at the Nanyang Technological University in Singapore, leading the disordered systems group at this university and researching at the intersection between statistical physics, soft-matter science and complex systems. A similar fate befell young second generation researchers too, all this highlighting the importance of the scientific results achieved by the Naples group, well recognized at an international level.