Abstract:
We present a detailed description of the Voronoi Tessellation (VT) cluster finder algorithm in 2+1 dimensions, which improves on past implementations of this technique. The need for cluster finder algorithms able to produce reliable cluster catalogs up to redshift 1 or beyond and down to $10^{13.5}$ solar masses is paramount especially in light of upcoming surveys aiming at cosmological constraints from galaxy cluster number counts. We build the VT in photometric redshift shells and use the two-point correlation function of the galaxies in the field to both determine the density threshold for detection of cluster candidates and to establish their significance. This allows us to detect clusters in a self consistent way without any assumptions about their astrophysical properties. We apply the VT to mock catalogs which extend to redshift 1.4 reproducing the $\Lambda$CDM cosmology and the clustering properties observed in the SDSS data. An objective estimate of the cluster selection function in terms of the completeness and purity as a function of mass and redshift is as important as having a reliable cluster finder. We measure these quantities by matching the VT cluster catalog with the mock truth table. We show that the VT can produce a cluster catalog with completeness and purity $>80%$ for the redshift range up to $\sim 1$ and mass range down to $\sim 10^{13.5}$ solar masses.

Abstract:
We prove that the Palm measure of the Ginibre process is obtained by removing a Gaussian distributed point from the process and adding the origin. We obtain also precise formulas describing the law of the typical cell of Ginibre--Voronoi tessellation. We show that near the germs of the cells a more important part of the area is captured in the Ginibre--Voronoi tessellation than in the Poisson--Voronoi tessellation. Moment areas of corresponding subdomains of the cells are explicitly evaluated.

Abstract:
In this paper, the parallel set ΞR of the facets ((d 1)-faces) of a stationary Poisson-Voronoi tessellation in 2 and 3 is investigated. An analytical formula for the spherical contact distribution function of the tessellation allows for the derivation of formulae for the volume density and the specific surface area of ΞR. The densities of the remaining intrinsic volumes are studied by simulation. The results are used for fitting a dilated Poisson-Voronoi tessellation to the microstructure of a closed-cell foam.

Abstract:
We present an objective and automated procedure for detecting clusters of galaxies in imaging galaxy surveys. Our Voronoi Galaxy Cluster Finder (VGCF) uses galaxy positions and magnitudes to find clusters and determine their main features: size, richness and contrast above the background. The VGCF uses the Voronoi tessellation to evaluate the local density and to identify clusters as significative density fluctuations above the background. The significance threshold needs to be set by the user, but experimenting with different choices is very easy since it does not require a whole new run of the algorithm. The VGCF is non-parametric and does not smooth the data. As a consequence, clusters are identified irrispective of their shape and their identification is only slightly affected by border effects and by holes in the galaxy distribution on the sky. The algorithm is fast, and automatically assigns members to structures.

Abstract:
We show that random walk on a stationary random graph with positive anchored expansion and exponential volume growth has positive speed. We also show that two families of random triangulations of the hyperbolic plane, the hyperbolic Poisson Voronoi tessellation and the hyperbolic Poisson Delaunay triangulation, have 1-skeletons with positive anchored expansion. As a consequence, we show that the simple random walks on these graphs have positive speed. We include a section of open problems and conjectures on the topics of stationary geometric random graphs and the hyperbolic Poisson Voronoi tessellation.

Abstract:
We introduce a dynamical system based on the vertices of Voronoi tessellations. This dynamical system acts on finite or discrete point sets in the plane, taking a point set to the vertex set of its Voronoi tessellation. We explore the behavior of this system for small point sets, then prove a general result quantifying the growth of the sizes of the point sets under iteration. We conclude by giving the most interesting open problems.

Abstract:
We present and test TesseRACt, a non-parametric technique for recovering the concentration of simulated dark matter halos using Voronoi tessellation. TesseRACt is tested on idealized N-body halos that are axisymmetric, triaxial, and contain substructure and compared to traditional least-squares fitting as well as two non-parametric techniques that assume spherical symmetry. TesseRACt recovers halo concentrations within 0.3% of the true value regardless of whether the halo is spherical, axisymmetric, or triaxial. Traditional fitting and non-parametric techniques that assume spherical symmetry can return concentrations that are systematically off by as much as 10% from the true value for non-spherical halos. TesseRACt also performs significantly better when there is substructure present outside $0.5R_{200}$. Given that cosmological halos are rarely spherical and often contain substructure, we discuss implications for studies of halo concentration in cosmological N-body simulations including how choice of technique for measuring concentration might bias scaling relations.

Abstract:
A homogeneous Poisson-Voronoi tessellation of intensity $\gamma$ is observed in a convex body $W$. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in $W$. We prove that when $\gamma\rightarrow\infty$, these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between $W$ and its so-called Poisson-Voronoi approximation.

Abstract:
The Sloan Digital Sky Survey has obtained 450 square degrees of photometric scan data, in five bands (u',g',r',i',z'), which we use to identify clusters of galaxies. We illustrate how we do star-galaxy separation, and present a simple and elegant method of detecting overdensities in the galaxy distribution, using the Voronoi Tessellation.

Abstract:
We investigate vapor bubble nucleation in metastable TIP4P/2005 water at negative pressure via the Mean First Passage Time (MFPT) method using the volume of the largest bubble as a local order parameter. We identify the bubbles in the system by means of a Voronoi-based analysis of the Molecular Dynamics trajectories. By comparing the features of the tessellation of liquid water at ambient conditions to those of the same system with an empty cavity we are able to discriminate vapor (or interfacial) molecules from the bulk ones. This information is used to follow the time evolution of the largest bubble until the system cavitates at 280 K above and below the spinodal line. At the pressure above the spinodal line, the MFPT curve shows the expected shape for a moderately metastable liquid from which we estimate the bubble nucleation rate and the size of the critical cluster. The nucleation rate estimated using Classical Nucleation Theory turns out to be about 8 order of magnitude lower than the one we compute by means of MFPT. The behavior at the pressure below the spinodal line, where the liquid is thermodynamically unstable, is remarkably different, the MFPT curve being a monotonous function without any inflection point.