arXiv subjects

Filipe Mussini

Publications and source records attributed to Filipe Mussini.

3 recordsLinked to original sources

arXiv2020

The fractal cylinder process: existence and connectivity phase transition

We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension $d\geq 2,$ and a connectivity phase transition whenever $d\geq 4.$ We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point. A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results. In addition we also determine the almost sure Hausdorff dimension of the fractal set.

math.PR↗

arXiv2018

Random cover times using the Poisson cylinder process

In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are "raining from the sky" at unit rate. Our main results concerns the asymptotic of this cover time as the set $A$ grows. If the set $A$ is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead $A$ has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.

math.PR↗

arXiv2015

Invariance under quasi-isometries of subcritical and supercritical behaviour in the Boolean model of percolation

In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

math.PR↗