Consistency of constructions for cell division processes

Abstract

For a class of cell division processes in the Euclidean space R d , spatial consistency is investigated. This addresses the problem whether the distribution of the generated structures, restricted to a bounded set V , depends on the choice of a larger region W ⊃ V where the construction of the cell division process is performed. This can also be understood as the problem of boundary effects in the cell division procedure. It is known that the STIT tessellations are spatially consistent. In the present paper it is shown that, within a reasonable wide class of cell division processes, the STIT tessellations are the only ones that are consistent.

Tessellation-valued processes that are generated by cell division

“…It was shown in [21] that, for STIT tessellations, the (L- ) (D- ) cell division process can be launched in W without regarding boundary effects, because the STIT model is spatially consistent. And in [19] it was shown that all the other models considered in the present paper are not spatially consistent, which means that for the construction of information outside of W is also needed.…”

“…The STIT tessellation process driven by is a cell division process with (L- ) and (D- ). In [19] it was shown that in the class of cell division processes defined above, only the (L- ) (D- ) model has the property of spatial consistency which is sufficient for its existence. It is therefore of interest to show the existence of further cell division processes without requiring spatial consistency.…”

Parameter Optimization on a Tessellation Model for Crack Pattern Simulation

“…which means that the restriction of Y t,W to W ′ has the same distribution as Y t,W ′ , see [14]. This consistency property yields that, for any t > 0, there exists a spatially stationary (or homogeneous, which means the invariance of the distribution under translations of the Euclidean plane) random tessellation Y t of R 2 such that the restriction of Y t to W has the same distribution as Y t,W for all polygons W ∈ P. Note that this spatial consistency is lost when the STIT model is modified, and hence the distribution of the tessellation generated in a window depends on the choice of this window, see [16].…”

“…The modifications of STIT lose the spatial consistency property (4), see [16]. Therefore, in an arbitrarily given window W , one cannot start the cell division process appropriately, such that (4) is satisfied.…”

A Mecke-type formula and Markov properties for STIT tessellation processes

“…They have been invented in [25] and since their introduction they have stimulated lots of research, cf. [4,11,13,14,15,16,20,23,26,34,35,36,37,38,40,41].…”