Abstract:
Gravitational clustering of a random distribution of point masses is dominated by the effective short-range interactions due to large-scale isotropy. We introduce a one-dimensional cellular automaton to reproduce this effect in the most schematic way: at each time particles move towards their nearest neighbours with whom they coalesce on collision. This model shows an extremely rich phenomenology with features of scale-invariant dynamics leading to a tree-like structure in space-time whose topological self-similarity are characterised with universal exponents. Our model suggests a simple interpretation of the non-analytic hierarchical clustering and can reproduce some of the self-similar features of gravitational N-body simulations.

Abstract:
The debate on the correlation properties of galaxy structures has having an increasing interest during the last year. In this lecture we discuss the claims of different authors who have criticized our approach and results. In order to have a clear cut of the situation, we focus mainly on galaxy distribution in the intermediate range of distances ~ 100 - 200 Mpc/h. In particular we discuss: (i) the validity of the actual data and the concept of "fair sample", (ii) the shift of r_0 with sample depth and luminosity bias, (iii) the value of the fractal dimension, (iv) the problem of the counts from a single point and the case of ESP, (iv) uniformity of angular catalogs. The detection of fractal behavior up to ~ 100 - 200 Mpc/h is enough to rise serious problems to the usual statistical methods used for the characterization of galaxy correlations, standard interpretation of galaxy distribution and theoretical models develoed. The clarification of the intermediate scale behavior is very instructive for the subsequent interpretation of the very large scale galaxy distribution.

Abstract:
The recent observations of galaxy and dark matter clumpy distributions have provided new elements to the understating of the problem of cosmological structure formation. The strong clumpiness characterizing galaxy structures seems to be present in the overall mass distribution and its relation to the highly isotropic Cosmic Microwave Background Radiation represents a fundamental problem. The extension of structures, the formation of power-law correlations characterizing the strongly clustered regime and the relation between dark and visible matter are the key problems both from an observational and a theoretical point of view. We discuss recent progresses in the studies of structure formation by using concepts and methods of statistical physics.

Abstract:
Galaxies and clusters distributions show two major properties: (i) the positions of galaxies and clusters are characterized by a power law distribution indicating properties with respect to their positions. (ii) The distribution of masses is also characterized by a power law corresponding to self similarity of different nature. These two properties are naturally unified by the concept of multifractality. This concept naturally arises if the distribution of matter, as given by both positions and masses, has self similar properties. We discuss the experimental situation in this respect and we introduce a simple stochastic model based of the aggregation process of particles The aim of this model is to understand which characteristic properties of the aggregation probability can gives rise to multifractal distribution In particular we find that a crucial element in this respect is that the aggregation probability should depend on the environment of the aggregation region.

Abstract:
Models of structure formation in the universe postulate that matter distributions observed today in galaxy catalogs arise, through a complex non-linear dynamics, by gravitational evolution from a very uniform initial state. Dark matter plays the central role of providing the primordial density seeds which will govern the dynamics of structure formation. We critically examine the role of cosmological dark matter by considering three different and related issues: Basic statistical properties of theoretical initial density fields, several elements of the gravitational many-body dynamics and key correlation features of the observed galaxy distributions are discussed, stressing some useful analogies with known systems in modern statistical physics.

Abstract:
We discuss how luminosity and space distribution of galaxies are naturally linked in view of their multifractal properties. In particular we show that the mass (luminosity) function corresponding to a multifractal distribution in a given observed volume, consists of a power law followed by an exponential cut-off. This implies that the amplitude $\phi^*$ of the Schechter function scales with the sample depth, as confirmed by various observational data. This effect is analogous to the scaling of the space density due to the fractal nature of the space distribution. This has the important consequence that the luminosity function can be properly defined only in a volume limited sample. Also the so-called "luminosity segregation" and the concept of bias correspond to a natural consequence of multifractality. This implies however that they should be considered from a different perspective with respect to the usual one. Such a concept allows us to unify the space and luminosity distributions as being shaped by a single cause:multifractality which should therefore claim a central stage in theoretical investigations.

Abstract:
In the debate about galaxy correlation there are different questions which can be addressed separately: Which are the statistical methods able to properly detect scale invariance and describe, in general, the properties of irregular and regular distributions? Which are the implications for cosmology of the fractal behavior of galactic structures, up to a certain scale lambda_0? Which is the homogeneity scale lambda_0, i.e. the scale beyond which galaxy distribution has an eventual crossover to homogeneity? These are three different, but related, problems, which must be considered in different steps, from the point of view of data analysis as well as from the theoretical perspective.

Abstract:
Galaxy structures are certainly fractal up to a certain crossover scale \lambda_0. A clear determination of such a scale is still missing. Usually, the conceptual and practical implications of this property are neglected and the structures are only discussed in terms of their global amplitude. Here we present a compact summary of these implications. First, we discuss the problem of the identification of the crossover scale \lambda_0 and the proper characterization of the scaling. We then consider the implications of these properties with respect to various physical phenomena and to the corresponding characteristic values, i.e. r_0, \sigma_8, \Omega, etc. These implications crucially depend on the value of \lambda_0, but they are still important for a relatively small value, say \lambda_0 \approx 50 \hmp. Finally we consider the main theoretical consequences of these results.

Abstract:
The question of the nature of galaxy clustering and the possible homogeneity of galaxy distribution is one of the fundamental problem of cosmology. It is well established that galaxy structures are characterized, up to a certain scale, by fractal properties. The possible crossover to homogeneity is instead still matter of debate. However, independently on the specific value of the homogeneity scale, the fractal nature of galaxy clustering requires new methods and theoretical concepts developed in the area of statistical physics and complexity. In this lecture we discuss a new perspective on the problem of cosmological structures formation, both from the experimental and the theoretical points of view. This new perspective leads to a very interesting and constructive interaction between the fields of cosmic structures, statistical physics and complexity with very challenging open problems which we also discuss.

Abstract:
We introduce a statistical agent based model to describe the phenomenon of drug abuse and its dynamical evolution at the individual and global level. The agents are heterogeneous with respect to their intrinsic inclination to drugs, to their budget attitude and social environment. The various levels of drug use were inspired by the professional description of the phenomenon and this permits a direct comparison with all available data. We show that certain elements have a great importance to start the use of drugs, for example the rare events in the personal experiences which permit to overcame the barrier of drug use occasionally. The analysis of how the system reacts to perturbations is very important to understand its key elements and it provides strategies for effective policy making. The present model represents the first step of a realistic description of this phenomenon and can be easily generalized in various directions.