Poisson - Voronoi Aggregate Tessellations

Tchoumatchenko, Konstantin (Paris, France)
Konstantin.Tchoumatchenko@ens.fr

(joint work with S.Zuyev)

Let $\Theta_0$ and $\Theta_1$ be two tessellations of $\mathbb R^d$. The operation of aggregation consists in merging the cells of $\Theta_1$ whose nuclei fall into the same cell of $\Theta_0$. Recursively applying this operation, we construct from a sequence of tessellations $\{\Theta_i, i=1,2,\ldots\}$ a sequence of the aggregate tessellations of higher order.

Our main results concern the class of aggregate tessellations constructed from a sequence of stationary Poisson - Voronoi diagrams. We study the evolution of the aggregate cells as the order of the tessellation tends to infinity and provide sufficient conditions for the existence of the limit. When the limit tessellation exists, its boundary is a random fractal. We analyze the asymptotics of the boundary contact distribution and provide an upper bound for the Hausdorff dimension of the boundary.