Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. (English) Zbl 1098.60012

Nagel, Werner; Weiss, Viola

Adv. Appl. Probab. 37, No. 4, 859-883 (2005).

The authors introduce a new model for random translation invariant tessellations in the \(d\)- dimensional Euclidean space \(\mathbb R^d\), \(d\geq 1\). A mathematical motivation for these tessellations comes from the consideration of iteration (or nesting) of tessellations. The operation of iteration generates a new tessellation \(I(Y_0, \mathcal Y)\) from a “frame” tessellation \(Y_0\) and a sequence \({\mathcal Y}=\{Y_1,Y_2, \dots\}\) of independent, identically distributed tessellations by subdividing the \(i\)th cell of \(Y_0\) by intersecting it with the cells of \(Y_i\), \(i=1,2,\dots\). This operation of iteration can be applied repeatedly, and combined with an appropriate rescaling. The problem arises as to whether there exists a limit tessellation when the number of repetitions goes to infinity. A further question is how such limit tessellations can be described if they exist. A related question was stated by R. Cowan [in: Stochastic geometry, geometric statistics, stereology. Teubner-Texte Math. 65, 64–68 (1984; Zbl 0551.60011)]. The authors [Adv. Appl. Probab. 35, No. 1, 123–138 (2003; Zbl 1023.60015)] showed that a translation invariant tessellation can appear as a limit (in the sense of weak convergence) of repeated rescaled iteration if and only if it is stable with respect to iteration (STIT). The authors (loc. cit.) also listed some properties of STIT tessellations, but the main problem of their existence remained open.
The present paper describes a construction for all translation invariant random tessellations that are STIT. Thus, the class of all stable translation invariant tessellations is fully characterized. Concerning potential applications, these STIT tessellations can be good approximations for real tessellations with “hierarchical” structures, which can be observed in some crack or fracture structures, see W. Nagel and V. Weiss [“A planar crack tessellation which is stable with respect to iteration”, Jenaer Schriften zur Mathematik und Informatik 12/04 (Tech. Rep.), Jena, (2004); available at: http://www.fh-Jena.de/\(\sim\) weiss/doc/prepstit.pdf)].
The paper is organized as follows: The definitions of iteration and of STIT tessellations are recalled in Section 3. In Section 2, the model, referred to as a crack STIT tessellation, is introduced and its construction inside a bounded window is described in detail. Section 4 is devoted to the study of the capacity functional of the random closed set of all boundary points of the cells of the crack STIT tessellation. Finally, in Section 6, the domain attractions of crack STIT tessellations are described as the sets of all translation invariant tessellations in \(\mathbb R^d\) that have the same parameters \(S_V\) and \(\mathcal R\) (where \(S_V\) is the mean total \((d-1)\)-volume of the set of boundary points of cells per unit \(d\)-volume, while \(\mathcal R\) is the directional distribution (“rose of directions”) of the cell boundaries at a typical point [for exact definitions see: R. Schneider and W. Weil, “Stochastische Geometrie”. Leipzig: B. G. Teubner (2000; Zbl 0964.52009)].

Reviewer: Viktor Oganyan (Erevan)

MSC:

60D05	Geometric probability and stochastic geometry
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
52A22	Random convex sets and integral geometry (aspects of convex geometry)

Keywords:

iteration of tessellations; nesting of tessellations; weak convergence; stability

Citations:

Zbl 0551.60011; Zbl 1023.60015; Zbl 0964.52009

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