Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering, and life sciences. In this work, we investigate the statistical properties of d-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case d=3. We first analyze the behavior of the key features of these stochastic geometries as a function of the dimension d and the linear size L of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two labels with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster, and the average cluster size.

Eigenvalue problems for neutron transport in random geometries are key for many applications, ranging from reactor design to criticality safety. In this work we examine the behaviour of the reactivity and of the kinetics parameters (the effective delayed neutron fraction and the effective neutron generation time) for three-dimensional UOX and MOX assembly configurations where a portion of the fuel pins has been randomly fragmented by using various mixing statistics. For this purpose, we have selected stochastic tessellations of the Poisson, Voronoi and Box type, which provide convenient models for the random partitioning of space, and we have generated an ensemble of assembly realizations; for each geometry realization, criticality calculations have been performed by using the Monte Carlo code TRIPOLI-4 R , developed at CEA. We have then examined the evolution of the ensemble-averaged observables of interest as a function of the average chord length of the random geometries, which is roughly proportional to the correlation length of the fuel fragmentation. The methodology proposed in this work is fairly general and could be applied, e.g., to the assessment of re-criticality probability following severe accidents.