On Some fundamental aspects of Polyominoes
on Random Voronoi Tilings
Abstract.
Consider a Voronoi tiling of based on a realization of a inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected union of Voronoi tiles. In this paper we provide tail bounds for the number of boxes that are intersected by a Voronoi polyomino, and viceversa. These results will be crucial to analyze selfavoiding paths, greedy polyominoes and firstpassage percolation models on Voronoi Tilings and on the dual graph, named the Delaunay triangulation [5, 6].
Key words and phrases:
1. Introduction
To any locally finite subset of one can associate a partition of the plane as follows. To each point corresponds a polygonal region , the Voronoi tile at , consisting of the set of points of which are closer to than to any other . Closer is understood here in the euclidean sense, and the partition is not a real one, but the set of points which belong to more than one Voronoi tile has Lebesgue measure 0. From now on, is understood to be distributed like a Poisson random set on with intensity measure . We shall always assume that is comparable to Lebesgue’s measure on , , in the sense that there exists a positive constant such that for every Lebesguemeasurable subset of :
(1.1) 
Notice that, with probability one, when two Voronoi tiles are connected, they share a  dimensional face. The collection is called the Voronoi tiling (or tessellation) of the plane based on .
The study of Voronoi tilings has a very long history. The terminology is in honour of Voronoi [7], who used these tilings to study quadratic forms. Our aim is to study some fundamental aspects of finite and connected union of Voronoi tiles. These objects are called Voronoi polyominoes (Figure 1). Polyominoes were first introduced in periodic tilings of the plane. The Voronoi setting is rather different from the periodic one since we are now considering a random environment induced by the underlying Poisson random set.
The main results of this article, that will be stated in Section 2, will provide tail bounds for the maximum and minimum number of square boxes intersected by a Voronoi polyomino that contains the origin and has size . They will be important tools to analyze selfavoiding paths, greedy polyominoes and firstpassage percolation models on the Voronoi random setting [5, 6]. The idea to prove them is to combine block arguments with standard results for greedy lattice animal and site percolation models. The block argument is to consider a large box in so that it contains with “high probability” some configuration of points which prevent a Voronoi tile to cross it completely. The “high probability” alluded to is some percolation probability: we need that the “bad boxes” (those who can be crossed) do not percolate. In Section 3 we will prove some technical lemmas concerning greedy lattice animals and site percolation models that will be combine with the block argument to prove the theorems in Section 4. In Section 5 a slightly different random set up is introduced and we will draw an outline of how the results can be extended to this new set up.
2. Polyominoes on Voronoi Tilings
Let denote the usual cardinality of a set , and for each subset of let . For each natural number a Voronoi polyomino of size is a connected union of Voronoi tiles. Let denote the collection of all polyominoes such that the origin and , and let be the collection of all polyominoes such that and . Let denote the dimensional integer lattice. For each let
and for each connected set let
Theorem 1.
There exists a constant such that if then
(2.2) 
Further, there exist constants such that if then
(2.3) 
In the same polyomino model one could consider and of over all polyominoes of size touching (). The same method to prove Theorem 1 can be extended to this situation, yielding to similar large deviations bounds (2.2) and (2.3).
2.1. Selfavoiding paths on the Delaunay Triangulation
An important graph for the study of a Voronoi tiling is its facial dual, the Delaunay graph based on . This graph, denoted by is an unoriented graph embedded in which has vertex set and edges every time and share a dimensional face. We remark that, for our Poisson random set, a.s. no points are on the same hyperplane and no points are on the same hypersphere and that makes the Delaunay graph a well defined triangulation (Figure 2). This triangulation divides into bounded simplexes called Delaunay cells. For each Delaunay cell no point in is inside the circumhypersphere of . Polyominoes on the Voronoi tiling correspond to connected (in the graph topology) subsets of the Delaunay graph.
Let be the nearest point to , and let (resp., ) be the collection of all selfavoiding paths starting at and of size (number of vertices) (resp., ). Recall that polyominoes on the Voronoi tiling correspond to connected (in the graph topology) subsets of the Delaunay graph. Therefore, for each corresponds a unique polyomino and, analogously, for each corresponds a unique polyomino . Let . Thus, the following corollary is a straightforward consequence of Theorem 1.
Corollary 2.
There exists a constant such that if then
(2.4) 
Further, there exist constants such that if then
(2.5) 
2.2. The inverse problem
Until now we have been concerned with the size of a lattice covering of a Voronoi polyomino of size . It is also natural to consider the inverse problem, i.e., the number of Voronoi tiles that one needs to cover a connected set composed by lattice boxes. Precisely, for each connected set (in the nearestneighbor sense) let
A connected subset of is also called a lattice animal. We denote the collection of all lattice animals such that and . A Voronoi covering is defined by taking
Theorem 3.
There exist constants such that if then
(2.6) 
One important consequence of Theorem 3 concerns the following construction: Let and consider the Polyomino generated by all Voronoi tiles that intersect the line segment . Then one can always find a selfavoiding path with vertices in and that connects to . Clearly,
and hence, by Theorem 3, we have the following corollary:
Corollary 4.
There exist constants such that if then
(2.7) 
where denotes the euclidean norm.
3. Technical Lemmas
3.1. A greedy lattice animal model with Poisson weights
Let
Then is a collection of independent Poisson random variables. By (1.1),
(3.8) 
It is a standard result in combinatorics^{1}^{1}1To see this, notice that for each lattice animal one can (injectively) associate an “exploration” nearest neighbor path such that and for each . Thus and will do. that
(3.9) 
for a finite .
Lemma 5.
If then
3.2. Site percolation schemes
Throughout this section will denote an i.i.d. site percolation scheme (or random field) with parameter . If we say that is open. Otherwise, we say that it is closed.
Lemma 6.
If and then
where denotes the set of all connected sets such that and .
Proof of Lemma 6. Let denote the binomial coefficient. If and , by taking a connected subset of of size we may assume that has size exactly . Then there exists some subset of exactly sites of with . By (3.9), this shows that
(3.10)  
whenever and .
A closed cluster is a maximal connected set of closed vertices of . Let denote the closed cluster that contains (it is empty if is open). For a finite subset of let denote the collection of all closed clusters (with respect to ) intersecting .
Lemma 7.
If is an increasing function from to then
Proof of Lemma 7. The proof of Lemma 7 is due to Raphaël Rossignol and the author is grateful for his help. This inequality is also a crucial tool for proving the results in [6]. Let us recall Reimer’s inequality (see Grimmett’s book [1] p.39). Let be a positive integer, let and define . For and , define the cylinder event generated by on by:
If and are two subsets of , define their disjoint intersection as follows:
Reimer’s inequality states that:
Remark that is a commutative and associative operation, and that, for any subsets of ,
Now take large enough so that . Let , and order the elements of . For any in , and any , let be the closed cluster (for the configuration in containing , which is empty if . Define:
Let be nonnegative integers and:
Then, we claim that:
Indeed, let . Then, for every , . Furthermore, . This shows that , and proves the claim above. Therefore, using Reimer’s inequality,
(3.11) 
Now, let be an increasing function from to . Define as follows:
Denote by the range of . Define:
and
By convention, set and define . We can write:
Define as the set of nonempty components , for . Since , we can write:
where the first inequality follows from (3.11). Finally, we may let tend to infinity, and then use the Lebesgue’s monotone convergence theorem for the right hand side and Fatou’s lemma for the left hand side.
Let denote the lattice boundary of with respect to the norm, and define
Lemma 8.
If and then
3.3. Domination by product measures
Let be a collection of random variables that take values and and which satisfy the following conditions: (i) for each pair such that all sites in are at distance greater than from all sites in (in the supnorm sense), the collections of random variables and are independent; (ii) . In this case we say that is a dependent random field whose marginals are at least , and denote the class of all such fields.
Let and be two random fields. We say that dominates from below if there is a coupling (joint realization) between and such that for all . We refer to Liggett’s book [2] for more details in stochastic domination and couplings. Theorem 0.0 of Liggett, Schonmann and Stacey [3] states that when is close to , the random fields in are dominated from below by an i.i.d. random field with density . Further, one can make arbitrarily close to by taking close enough to .
Lemma 9.
Let . There exists such that for all if then
Further, there exists a constant such that if then
Proof of Lemma 9. We note that if dominates from below then
for any lattice animal . Hence, Lemma 9 follows by combining Theorem 0.0 in [3] together with Lemma 6 and Lemma 8.
The proof of the first part of Lemma 9 (in the dependent set up) could be done directly without using [3]. One needs to notice that given any set of boxes, we can pick a subset of independent boxes of size , and then use the same argument as before. However, the proof of the second part is more delicate and it is not clear (for the author) that it could be easily adapted to the dependent situation.
3.4. The block argument
For each , and let
Given a locally finite set , we say that a (square) box is a full box if cutting it regularly into subboxes, each one of these boxes contains at least one point of the set . Let
Lemma 10.
Let be a finite and connected subset of and assume that is a full box for all . If and then .
Proof of Lemma 10. Assume that but . Then there will exist such that
On the other hand, is a full box for all . By picking and in the (euclidean) boundary of and , respectively, this implies that there exist such that
(the right hand side of the inequality is the length of the diagonal of a subsquare). However,
and hence
which yields to a contradiction since .
Lemma 11.
Under (1.1),
4. Proof of the Theorems
Proof of (2.2)
Proof of (2.3)
For each let
and recall that . Then, for any ,
(4.13) 
Consider the nonhomogeneous dependent percolation scheme in defined by
( denotes the indicator function of an event.) If we say that is a good box. By Lemma 11, where
By Lemma 9, if we pick such that
then and
(4.14) 
whenever . Now, let
Notice that there exists at least one set such that for all and . Now, write . By Lemma 10, if and and then , and thus
By (4.13), this shows that
whenever .
Proof of (2.6)
5. A two dimensional modified Poisson model
In [5] a random set is constructed from a realization of a twodimensional homogeneous Poisson random set as follows. Order the points of in some arbitrary fashion, say . Fix and . For each let
Divide into subboxes (as before) of the same length , say . Now we construct the modified random set (whose distribution will also depend on and ) by changing the original Poisson random set inside each , as follows.

If then set ;

If then set by uniformly selecting points from .

If then set by adding an extra point uniformly distributed on .
In a few words, we tile the plane into boxes of size and we insist that each tile is a full box, and that no tile contains more than Poisson points. We make the convention and denote by the Delaunay Triangulation based on . It is clear that method can be applied in this set up for each fixed . However, we want to emphasize that it allows us to do so simultaneously for all sufficiently large.