Asymptotic shape of small cells
A stationary Poisson line tessellation is considered whose directional distribution is concentrated on two different atoms with some positive weights. The shape of the typical cell of such a tessellation is studied when its area or its perimeter tends to zero. In contrast to known results where the area or the perimeter tends to infinity, it is shown that the asymptotic shape of cells having small area is degenerate. Again in contrast to the case of large cells, the asymptotic shape of cells with small perimeter is not uniquely determined. The results are accompanied by a large scale simulation study.
Keywords. Asymptotic shape, random geometry, random polygon, Poisson process, Poisson line tessellation, random tessellation, small cells, stochastic geometry.
MSC. Primary 60D05; Secondary 60G55, 52A22.
In the preface of the book , D.G. Kendall re-phrased a conjecture about the shape of planar tessellation cells having large area. He considered a stationary and isotropic Poisson line tessellation in the plane and conjectured that the shape of the cell containing the origin is approximately circular if its area is large. First contributions to Kendall’s conjecture are due to Goldmann , Kovalenko  and Miles . The first result for higher dimensions is by Mecke and Osburg  who considered what they call Poisson cuboid tessellations. In a series of papers, Calka , Calka and Schreiber , Hug, Reitzner and Schneider  and Hug and Schneider [7, 8] treated very general higher-dimensional versions and variants of Kendall’s problem for quite general tessellation models (Poisson hyperplanes, Poisson-Voronoi and Poisson-Delaunay tessellations) and size functionals, see also the book chapter  for an overview. The respective results either rely on asymptotic theory for high-density Boolean models or on sharp inequalities of isoperimetric type.
In this paper, we focus on the analysis of the shape of small tessellation cells. So far, we were not able to discover a general principle as the one mentioned above for the large cells behind the asymptotic geometry of small cells. For this reason, we restrict attention to the following simple model and its affine images. Take two independent stationary (homogeneous) Poisson point processes of unit intensity on the two coordinate axes in the plane and draw vertical lines through the points on the -axis and horizontal lines through the points on the -axis; see Figure 1 (left). The collection of these lines (without the two coordinate axes) decomposes the plane into a countable number of non-overlapping rectangles, the collection of which is called a rectangular Poisson line tessellation; see . Of interest in the present paper is the shape of a typical rectangle of the tessellation (the precise definition follows below). Mecke and Osburg  have shown that a typical rectangle tends to be ‘more and more cubical as the area tends to infinity’. In the present paper we are interested in the converse question and ask for the shape of a typical rectangle of small area. We will show that, in contrast to the large area case, the shape of typical rectangles with small area is asymptotically degenerate. Besides rectangles of small area, we also consider rectangles that have small perimeter. For such a situation we obtain a uniform distribution for our parameter measuring the shape of the rectangles (in fact not in general, but at least for the case described above). Again, this result is in contrast to the large perimeter case indicated in  (with proofs given in ). We would like to stress the fact that this is the first paper dealing with the mathematical analysis of small cells in random tessellations although some conjectures together with heuristic arguments have appeared earlier in .
Denote by the space of lines in and by the subspace containing only lines through the origin. We let and be two different lines in and fix . On we define the probability measure by , where stands for the Dirac measure concentrated at , . This is to say, is concentrated on and with weights and , respectively. We also define the translation-invariant measure on by the relation
where stands for the Lebesgue measure on , and is a non-negative measurable function. In other words, is concentrated on two families of lines parallel to and , whereas is an intensity parameter.
Let now be a Poisson point process on with intensity measure as at (1); cf. [16, 17] for definitions. Clearly, the lines of decompose the plane into countably many parallelograms – called cells in the sequel – which have pairwise no interior points in common; see Figure 1 (right). The collection of all cells is denoted by and the intersection point of the two diagonals of a parallelogram is denoted by . We define a probability law on the (measurable) space of parallelograms as follows:
where is a measurable subset of parallelograms and stands for the centered square of area . The definition (2) formalizes the idea of a uniformly selected cell from (regardless of size and shape) and we call a random parallelogram with distribution a typical cell of the tessellation; cf. [16, 17] for background material. Note that by definition the typical cell is centered in the origin, i.e. .
The following fact will turn out to be crucial for our further investigations. We state it here for two dimensions, its proof, however, will be presented in Section 3 below for the analogous model in arbitrary space dimension. This extension is applied in Section 2.3.
The edge lengths of the typical cell of a Poisson parallelogram tessellation with intensity measure given by (1) are independent and exponentially distributed random variables with parameters (for the edge parallel to ) and (for the edge parallel to ), where is the intersection angle between and .
2.2 Results for small cells
We consider the typical cell of a Poisson line tessellation as described above. Its random edge lengths are denoted by and and its area by . To measure the shape of the typical cell we introduce two deviation functionals. The first one is
which is a random variable taking values in (this was the reason for the choice for the factor ). We notice that is scale invariant, i.e., does not change if the parallelogram is rescaled by some constant factor. Moreover, we have if exactly one of the edge lengths or is zero, i.e., if the parallelogram degenerates in that it is a line segment of positive length. For single points, i.e. , is not defined. As a second deviation functional we introduce
This is not a scale invariant quantity, but we notice that if and only if the parallelogram is degenerated to a point.
We investigate first the asymptotic behavior of the deviation functionals and under the condition that the typical cell area tends to zero. Our main result in this direction reads as follows.
Let . It holds that
Some comments are in order about the interpretation of Theorem 2. Firstly, (6) shows that the asymptotic shape of a typical cell of small area tends to that of a line segment. On the other hand, (7) shows that, in the limit, this line segment cannot have positive length. This phenomenon is well reflected in the simulation study presented in Section 2.3. We also remark that (5) gives an upper bound for the rate of convergence.
Besides cells of small area, also cells with small perimeter can be considered. In this case the picture is somewhat different from that presented for the small area case in Theorem 2. In what follows we denote by half of the perimeter length of the typical cell (the factor is chosen for simplicity as will become clear in the proof).
Let . If , is uniformly distributed on given that , i.e.,
independently of . If otherwise ,
where and with .
Theorem 3 shows that the conditional deviation functional , given , follows a uniform distribution on its range in the particular case . This means that not only in contrast to the case of large perimeter (see [12, 15]), but also in contrast to the case of small area considered in Theorem 2 above, the asymptotic shape of cells that have small perimeter is not uniquely determined. This phenomenon is well reflected by the simulation study in forthcoming Section 2.3. We also refer to a related short discussion at the beginning of Section 7 in  about the independence of the shape of the zero cell and its perimeter. Such an interpretation becomes less obvious whenever . However, it is easily seen from the precise formula stated in Theorem 3 that the limit relation
is in order.
2.3 Simulation results and outlook to higher space dimensions
To highlight and to underpin the theoretical results in the previous subsection we performed the following simulation study. We chose , and , where and are the two orthogonal coordinate axes. In this case, the edge lengths of the typical cell of the rectangular Poisson line tessellation are independently exponentially distributed with mean ; see Proposition 1. Hence, we simulated independent realizations of the random vector with and i.i.d. Exponential(1). From the collection of cells obtained this way, we selected the cells with smallest area. These are shown in Figure 2 (left). Histograms of the deviation functionals and are shown in Figure 3. Both the line segment shape of the cells with an accumulation around the origin and the peak at zero in the histograms for the deviation functionals are nicely visible.
As discussed above, the area is only one measure of size of a cell. We investigate now the shape of small cells in the sense that their perimeter tends to zero. Therefore, the cells with smallest perimeter were extracted from the sample generated above. These cells together with their deviation functionals are shown in Figure 2 (right) and Figure 4, respectively. In this situation, the maximum of the edge lengths also tends to zero, but Theorem 3 implies that follows a uniform distribution on the interval , which is also nicely visible in the histograms. The minimal and maximal sizes of area and perimeter of the cells included in the statistics described above are given in Table 1.
The Poisson line tessellations considered in this paper have natural analogues in higher space dimensions, the Poisson cuboid tessellations for which we refer to  and to Section 3 below. Also for these tessellations the question about the shape of small cells can be asked. Natural candidates are the typical cell of small volume, small surface area or small total edge length. Unfortunately, we were not able to derive the higher-dimensional pendants to Theorem 2 or Theorem 3 in full generality. This is mainly due to technical complications that arise for space dimensions and cause that the Abelian-type theorem used in the proof of Theorem 2 can no more be applied. For this reason we carried out a simulation study concerning small cells in . For this purpose, independent realizations of the random vector with i.i.d. Exponential(1) were generated representing the random edge lengths of the typical cell (this is justified by Proposition 4 below). From the sample of cells the cells with smallest volume, surface area and total edge length were extracted. Histograms for the deviation functionals (again the factor 3 yields a value of 1 for a cube and the range ) and for these cells are shown in Figure 5. The minimal and maximal sizes of the cells included in the statistics above are summarized in Table 2.
|volume 3D||3.20 e-15||4.97e-13|
|surface area 3D||1.74e-8||3.58e-7|
|total edge length 3D||6.80e-4||3.88e-3|
3 Proof of Proposition 1 and its higher-dimensional extension
As announced in Section 2, we will formulate and prove here a higher-dimensional version of Proposition 1. To state the result, we first need to introduce the higher-dimensional model. So, fix a space dimension , let be linearly independent unit vectors in and fix weights such that . We define linear hyperplanes by
and the probability measure on the space of hyperplanes through the origin by putting . The translation-invariant measure on the space of (affine) hyperplanes in induced by is given by
where is a fixed constant and where is non-negative and measurable. Note that taking and we get back the set-up described in Section 2 for the planar case .
Now, let be a Poisson point process on with intensity measure as defined above. By abuse of notation, we will identify with the random closed set in , which is induced by the union of all hyperplanes in . They decompose into a countable set of random parallelepipeds and the distribution of the typical cell (parallelepiped) of this tessellation is defined similarly as in (2). For let be the line spanned by . Then the discussion around [14, Equation (6.3)] together with [16, Theorem 4.4.7] shows that is a homogeneous Poisson point process on of intensity
where is the angle between and (). In the particular planar case we have and with . We can now state the higher-dimensional version of Proposition 1.
The edge lengths of a typical cell of a Poisson cuboid tessellation induced by are independent and exponentially distributed random variables with parameters , respectively.
As a first step let us describe an alternative construction for the random set , which in the planar case has already been considered in the introduction. Recall the definition of the lines from above and let for each , be a homogeneous Poisson point process on with intensity . We assume that are independent. Now, for each , place hyperplanes through the Poisson points on orthogonal to . The collection (or union) of all hyperplanes constructed this way has the same distribution as .
As a next step we describe a construction of the typical cell. To carry this out, we denote by the almost surely uniquely determined -dimensional parallelepiped of the tessellation induced by that contains the origin. Then is divided by the hyperplanes into smaller parallelepipeds meeting at the origin. With probability one, exactly one of these parallelepipeds, say, has the property that for all its corners the first coordinate is non-negative. Now, Theorem 10.4.7 in  implies that (up to translations) has distribution defined by (the higher-dimensional analogue of) (2). In other words, has (again up to translations) the same distribution as the typical cell of the Poisson hyperplane tessellation induced by , see also Section 4 in . Note in particular that the random parallelepiped has one of its corners at the origin. In view of the construction of described at the beginning of the proof, this implies that the edge lengths of are the distances from the origin of independent and homogeneous Poisson point processes on with intensities , respectively, to their next point on the left or right (depending on the position of within ). Thus, standard properties of such point processes allow us to conclude that the edge-lengths of the typical parallelepiped are independent and exponentially distributed with parameters . ∎
4 Proof of Theorem 2
We claim that without loss of generality we can restrict the proof of Theorem 2 to the case , and , where and are the two orthogonal coordinate axes. To show this, let us denote such a tessellation by and a tessellation with general parameters , and by . We notice now that due to our assumptions on and there exists a non-degenerate linear transformation such that after application of has the same distribution as . We notice further that the images under of a parallelogram, a line segment and a point are again a parallelogram, a line segment and a point, respectively. Thus, the statement of Theorem 2 is invariant under non-degenerate linear transformations of the underlying Poisson line tessellation. This implies that it is sufficient to establish the statement for one special choice of , and , viz. .
4.2 Proof for
Proof of (5).
To simplify the calculations we work from now on with and translate the result afterwards to the original deviation functional . To start with the calculation, we write the conditional probability as
In what follows, we consider the numerator and the denominator separately. To deal with the numerator we have to consider the event and . Without loss of generality we can assume that . The condition then becomes and implies . Therefore has to apply to retain as the minimum. Taking the second condition into account, leads to as the upper bound for . Considering the lower bound for , that is , we get the upper bound for from the condition . Thus, the numerator can be written as
The last expression can be determined by a straight forward integration procedure, which yields that asymptotically, as , it behaves like times a constant depending on (the two logarithmic terms cancel out).
Using the substitution and we can write the denominator as
Applying the substitution and in the next step, we get
Now we split this double integral and calculate the resulting integrals directly as far as possible:
Even if the integral in (9) looks rather innocent, its asymptotic behavior as turns out to be not accessible with elementary methods as above. To overcome this difficulty we make use of a theorem of Abelian type. So, let and write
which arises by a shift , and use now a relation between the Laplace transformation and the Laplace-Stieltjes transformation [18, Theorem 2.3a] to conclude from the Theorem of Abelian type [9, Theorem 8.5.2] (or more precisely from Corollary 8.5.1 ibidem) that
Putting together (9) and (10) we see with a Taylor expansion of the exponential function that behaves asymptotically like in the limit as . Combining this with the asymptotic behavior of the numerator implies (5) for as well as for the original deviation functional .
Proof of (7).
We start by re-writing the conditional probability as
The denominator is the same as in the proof of (5) and behaves like as . Next, we consider the numerator and write
Since numerator and denominator both tend to as , we may apply l’Hospital’s rule. For the numerator we find
where stands for the lower incomplete -function.
Now we split this integral into two parts and apply the shift . This leads to
Rearrangement then results in
The term has already been discussed in the proof of (5) above. Thus, we may concentrate on and write to indicate its dependence on . Defining we have that
and can thus apply [9, Corollary 8.5.1] again, which gives that behaves asymptotically like , as . Putting things together we see that the numerator, after differentiation, is bounded, whereas the denominator, again after differentiation, tends to as , implying that
This completes the proof of (7).
5 Proof of Theorem 3
Applying the reduction step as in the proof of Theorem 2 is not possible here since the problem involving a small perimeter is not invariant under non-degenerate linear transformation. However, we can use Proposition 1 to give a direct proof. It implies that the random edge lengths and of the typical cell (parallelogram) are independent and identically distributed according to an exponential distribution with mean and . If it follows that the half perimeter length is Erlang distributed with parameters and . In fact this was the reason for the factor in the definition of . Thus,
If (without loss of generality) , has distribution function
Without loss of generality assume that and have a closer look at the event . If , then may range between and , and if then ranges between and (the remaining case contradicts and ). Thus,
If , evaluation of these integrals yields
and in view of (11) the exact distributional result If otherwise , one shows that
which reduces to the separately treated uniform distribution if . It remains to notice that in view of Proposition 1, if and only if . This completes the proof.
We would like to point out that the first result of Theorem 3 has a well-known background. Namely, let and be two independent and exponentially distributed random variables. Then (or ), given that for some fixed , is uniformly distributed on .
We would like to thank two anonymous referees for their hints and suggestions. They were very helpful for us to improve the text.
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