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The fact that galaxy distribution exhibits fractal properties is well established since twenty years. Nowadays, the controversy concerns the range of the fractal regime, the value of the fractal dimension and the eventual presence of a…
Over the past two decades or so, it has become increasingly appparent that, out to quite large distances, galaxies are distributed in a quasi-fractal fashion with fractal dimension $D \approx 2$. Whether or not this behaviour continues onto…
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…
We consider cosmological models based on the spectral action formulation of (modified) gravity. We analyze the coupled effects, in this model, of the presence of nontrivial cosmic topology and of fractality in the large scale structure of…
The functional features of spatial networks depend upon a non-trivial relationship between the topological and physical structure. Here, we explore that relationship for spatial networks with radial symmetry and disordered fractal…
The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the…
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
In this paper, we analyze the effects of expansion on large scale structure formation in our Universe. We do that by incorporating a cosmological constant term in the gravitational partition function. This gravitational partition function…
We analyze the fragmentation behavior of random clusters on the lattice under a process where bonds between neighboring sites are successively broken. Modeling such structures by configurations of a generalized Potts or random-cluster model…
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…
In the last years there has been a growing interest in the understanding a vast variety of scale invariant and critical phenomena occurring in nature. Experiments and observations indeed suggest that many physical systems develop…
We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. Angular correlation function was used as a general instrument for the structure analysis. It was realized later…
We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what…
We investigate the fractal structure of $2d$ quantum gravity coupled to matter by measuring the distributions of so-called baby universes. We demonstrate that the method works well as long as $c \leq 1$. For $c >1$ it is not clear what…
Fractal structures and non-Gaussian velocity distributions are characteristic properties commonly observed in virialized self-gravitating systems such as galaxies or interstellar molecular clouds. We study the origin of these properties…
A numerical study of the transfer across random fractal surfaces shows that their responses are very close to the response of deterministic model geometries with the same fractal dimension. The simulations of several interfaces with…
Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter…
The Minkowski functionals are a mathematical tool to quantify morphological features of patterns. Some applications to the matter distribution in galaxy catalogues and N-body simulations are reviewed, with an emphasis on the effects of…
We study fractal measures on Euclidean space through the dynamics of "zooming in" on typical points. The resulting family of measures (the "scenery"), can be interpreted as an orbit in an appropriate dynamical system which often…