  Motivated by the rapidly increasing relevance of virtual material design in
the domain of materials science, it has become essential to assess whether
topological properties of stochastic models for a spatial tessellation are in
accordance with a given dataset. Recently, tools from topological data analysis
such as the persistence diagram have allowed to reach profound insights in a
variety of application contexts. In this work, we establish the asymptotic
normality of a variety of test statistics derived from a tessellation-adapted
refinement of the persistence diagram. Since in applications, it is common to
work with tessellation data subject to interactions, we establish our main
results for Voronoi and Laguerre tessellations whose generators form a Gibbs
point process. We elucidate how these conceptual results can be used to derive
goodness of fit tests, and then investigate their power in a simulation study.
Finally, we apply our testing methodology to a tessellation describing real
foam data.
