## Stochastic mass fields

A complete statistical characterization of the the two and three point correlation properties of stochastic mass field has been done [L1] by our group in collaboration with Prof. Michael Joyce. This includes both a characterization in real space than in Fourier modes space of systems ranging from highly uniform mass distributions to fractals and multifractals.

**Superhomogeneous stichastic mass density fields:**

This work consists in the study of the spatial distribution of dark matter in the Universe as predicted by the main cosmological models (see also PrimordialDensityFields) [35][39][40][42][44]. The main results concern the statistical meaning of the large scale two-point correlation properties of the spatial fluctuations of the matter density field predicted by these models. In particular we have found that such mass fluctuations have a *sub-poissonian* behavior at large scales being similar to the ones found in stochastic particle distribution with a high degree of order such as some compactified glass-like or slightly perturbed lattices systems (e.g., the so-called *One Component Plasma*) which are for this reason called *superhomogeneous* (or hyperuniform).

**Causality constraints in the primordial cosmological stochastic mass fluctuations:**

We have developed a theoretical study about which would be the effects of causality constraints of the physics underlying the generation of the primordial spectrum of stochastic mass fluctuations in the Universe on the large scale spectrum of fluctuations itself. In particular, starting from the fact that quantum physics should be applied to the generation and primordial evolution of this kind of fluctuations, and that it implies a stochastic nature of mass fluctuations, we have studied the changes introduced by a weak stochasticity on the conclusions found by a popularly believed argument due to Zeldovich about the large scale behavior of the correlations of these primordial fluctuations, finding in this way new and interesting results [43].

## Point processes

In this field, we have developed a general mathematical study on the effects of a stochastic displacement field (with and without internal correlations) on the one and two-point spatial correlation properties of any initial point process (i.e., spatial particle distribution) [48][L1]. This has permitted also to study the statistics of the Voronoi volume and voids in superhomogeneous stochastic particle distributions [47], which are point processes with *sub-poissonian* number fluctuations on large scales (i.e., increasing with the scale slower than a Poisson point process with the same average number density).

— AndreaGabrielli – 10 Nov 2005