Bernoulli Percolation on random Tessellations
We generalize the standard site percolation model on the -dimensional lattice to a model on random tessellations of . We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument [BK89], develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processes or certain classes of Gibbs processes introduced in [SY13]. In a second paper we will investigate first passage percolation on random tessellations.
Key words: Percolation, random tessellation, uniqueness of the infinite cluster, non-trivial phase transition
MSC (2010): 60K35, 60D05
The percolation model was introduced by Broadbend and Hammersley on -dimensional lattices in the late fifties. In the meantime it was generalized to transitive or even quasi-transitive graphs and also to the random graph induced by the Poisson-Voronoi tessellation [BR06a]. While the result of Bollobàs and Riordan, that the critical value for face percolation on the -dimensional Poisson-Voronoi tessellation is excellent, it relies very much on the specifics of the Poisson-process and on the geometry of the plane. We want to start a very general investigation of Bernoulli face percolation on random tessellations of . By that, we mean a two-step model, where a tessellation is constructed randomly in the first step while in the second step each cell of this tessellation is colored black independently of each other cell with probability .
The structure of the paper is as follows. Section 2 contains generalizations of three well known theorems on percolation from transitive graphs to graphs with significantly less structure or symmetry. We show first, that , second that in the planar case (using the argument of Zhang) and third the uniqueness of the infinite cluster (adapting the proof of Gandolfi, Grimmett and Russo from [GGR88]). In these theorems the transitivity has been substituted by weaker properties that we will almost surely find in the realisations of our random tessellations. However these theorems are interesting in their own right.
In section 3 we proof the uniqueness of the infinite cluster by adapting the Burton Keane argument [BK89] to our model. This will be possible under the extremely weak assumptions of stationarity and a moment condition on the cell-distribution. As a corollary we will show, that in the planar case. At this point one might hope, that an ergodic or maybe a mixing tessellation also exhibits a non-trivial phase transition. We will sketch a mixing counterexample that shows, that this is not the case in general.
Following this counterexample we will propose two frameworks in section 4 and 5, that imply a non-trivial phase-transition. The first one is a kind of mixing condition. The second one is defined by auxiliary random fields, that encode how often very small or very large cells are observed. In both sections we will show for various classes of point processes, that the Voronoi tessellation induced by them fit into one of the two frameworks. This will include determinantal processes and certain classes of Poisson cluster and Gibbsian point processes introduced in [SY13].
The emphasis in this paper lies on the large generality and basic results.
2 Percolation on non-transitive graphs
The purpose of this section is, to generalize the proofs of well known results from percolation on lattices. We want to do this in such a way, that the assumptions hold a.s. for various types random tessellations. We introduce the necessary notation first.
We work on an abstract probability space . Let be a finite or countably infinite index set and a family of i.i.d. random variables, where , for all . We write for the distribution of . The space with the usual product -algebra is equipped with the canonical partial order , i.e. for any we have iff for all . We call a real-valued function increasing, iff for all . A function is decreasing iff is increasing. An set is increasing (decreasing) iff the indicator function is increasing (decreasing). The well known FKG-inequality states, that for two increasing functions
where is the expectation with respect to . A proof can be found in the book of Grimmett [Gri99] which might as well be the best starting point for an introduction to percolation.
Without further mentioning, we always work on undirected, locally finite graphs without loops or double edges. Let be a graph with vertex set and edge set . We say a set induces the subgraph iff . The site percolation model studies the family of Ber distributed random variables and the random subgraph of induced by the set .
A path , of is a finite or countably infinite sequence of vertices such that for any and if . We call the connected components of the clusters. In a cluster any two vertices are connected by a path in . A cycle in is a finite path where starting and ending vertex are adjacent. We use for the usual graph metric and extend this notion in the way that for . For the ball is defined as , and we write instead of , where is the root of . The outer and inner vertex boundary of a set is defined by and .
Percolation may be understood as the study of a randomly colored graph. To be precise, we call a pair a coloring of , if is a map from to . By a black path (cluster) we mean a path (cluster) of the subgraph of that is induced by the vertices with . We write for the black cluster that contains and remark that this might be the empty set if is white. In most cases we abbreviate this by if and are clear. In this section we will study the randomly colored graph .
is the percolation function and
is the critical value.
An obvious approach to proof results for percolation on random graphs is, to find conditions that hold for a.e. graph and study, what kind of results are implied by them. The most important property for many results in the classical theory on lattices is the transitivity of the underlying graph. This is completely destroyed in our model and we have to find substitutes for it.
If we take a look at the proof that on a lattice, then we see, that the argument follows from an exponential growth condition on the number of paths of length starting in the origin. To be more precise, there is a constant depending on the lattice, such that for all natural number . Apparently this condition won’t hold a.s. for most random graphs stemming from random tessellations, as it is very often possible to observe cells with an arbitrary number of neighbouring cells. However the following trivial Lemma shows that this property can be weakened. This weakened property holds a.s. in framework II introduced in Section 5. Let , be the set of connected subsets of size of that contain .
Let be an infinite connected graph. If there is a such that for all large enough , then
Proof: We have for all and large enough
This implies if .∎
Uniqueness of the infinite cluster
Another well known result is the uniqueness of the infinite cluster. This was first proven by Aizenman et. al. in [AKN87] and simplified by Gandolfi et. al. in [GGR88]. A short time afterwards Burton and Keane gave a very elegant new proof [BK89]. While the first proof is a bit more technical and not as robust as the one of Burton and Keane, it doesn’t rely as much on the transitivity and it can be quantified [Cer15]. We will now show how the transitivity in [GGR88] can be relaxed, though we have to remark, that this generalization is maybe more of theoretical interest as the Burton Keane argument can be extended directly to percolation on random tessellations under extremely weak assumptions.
For any infinite connected graph and let be the event, that is adjacent to two infinite black clusters in .
Let be an infinite connected graph with the following properties:
exists for the functions
If then for all .
There is a such that
Then there is at most one infinite cluster in for any .
Proof: The proof is only a minor generalization of the one in [GGR88] but we do it for the sake of completeness. The statement is trivial for , hence let . If there are two infinite clusters with positive probability, then there is a vertex such that .
We write for the subgraph of induced by and define the sets
This means, that contains the black clusters of restricted to that touch the boundary of , is their union, is the set of vertices that have a neighbour that connects to the boundary of via a black path and is the subset of where vertices have two neighbours in distinct black clusters connecting to the boundary of . If a vertex is contained in a black cluster that touches the boundary of , then is neighbour of such a cluster if we set its color to white. This implies
We want to apply the following large deviation result. For any there is a constant that doesn’t depend on such that
The proof can be found in [GGR88] and holds without any changes for arbitrary graphs. We define the set
and for a fixed the event
For the clusters that are not contained in we have
which tends to zero for due to properties 1. and 2. of .
For the clusters in we have
The first summand tends to for large while the second one tends to zero due to property 2. and the fact that
Putting everything together we see, that
and assumption 1. implies the assertion.∎
The assumptions in Theorem 2.2 resemble some kind of ergodicity on and may in fact be shown for graphs induced by ergodic random tessellations where balls have an a.s. polynomial growth, i.e. if there are constants such that . However it is not trivial to show this polynomial growth for an arbitrary random tessellation. We will address this problem in our second paper on first passage percolation on random tessellations. Note also that property 2. doesn’t depend on the choice of the root .
The planar case
The third result we want to generalize in this section is, that in a planar lattice there can’t be a coexistence of an infinite white and an infinite black cluster. The most basic proof of this result can be found in [Gri99, p. 289] named argument of Zhang. It was later generalized to lattices with a -fold symmetry in [BR08]. We will use some of their arguments, to show, that the uniqueness of the infinite cluster already implies that in the planar case.
Let be an infinite connected planar graph. There is at most one infinite cluster at if and only if for any . Moreover we have
in this case.
The proof of Theorem 2.3 contains some tedious topological details. A background in planar graph theory with a quite rigorous approach can be found in [Die10] or [MT01]. The Jordan Curve Theorem (JCT) can be found in [Hal07] along with some interesting historical remarks.
We call a continuous map from a curve. A curve is called closed if , Jordan curve if it is injective, closed Jordan curve if it is closed and injective when restricted to and polygonal if it is piecewise linear. We will identify any curve with its image in . The JCT states that for any closed Jordan curve the set consists of one bounded (the interior) and one unbounded (the exterior) connected component. Moreover if two points are connected by a Jordan curve that crosses an odd number of times, then one of these points lies in the interior and one lies in the exterior of .
A cut vertex of a connected graph is a vertex such that deleting results in being not connected anymore. A graph is called planar if there is an embedding of in the plane such that all edges are piecewise linear and don’t intersect (edges do not contain their endpoints). Moreover, the embedding has to be locally finite, i.e. any bounded component of is intersected only by a finite number of vertices and line-segments of the embedding.
We will state two Lemmas first, that contain the topological arguments.
Let be an infinite connected planar graph with root . Then for all there is a closed Jordan curve with the following properties:
No edge of intersects .
Each vertex in is either contained in or in its interior.
We have , where is the set of vertices of where an infinite path may start, that intersects only once.
Proof: We will consider a finite connected planar graph first. A closed walk is a cycle in that allows to visit vertices multiple times. Any face of induces a walk along its boundary in a natural way. This is called the facial walk. It is well known (see [Die10]) that has exactly one unbounded face with a facial walk , . The walk has the property that two consecutive edges and , enter in clockwise order without any other edges in between. Moreover, there is a small circular sector enclosed by the ends of the edges and that is contained in . If we fix a starting vertex and require each edge to be traversed at most once, the walk is uniquely determined.
Our first claim is, that any vertex that is contained multiple times in is a cut-vertex of . Assume is contained at least two times in , then there are two circular sectors at that are contained in . We connect and with a polygonal curve starting in , ending in that is contained in (see Figure 1). Extending by line-segments from to and from to we obtain a closed Jordan curve .
Considering the two edges and that enclose we observe that the curve starting in traversing as well as and ending in crosses exactly once at . Hence and lie in different connected components of and the deletion of implies that and are not connected anymore.
Now we look at the finite subgraph of that is induced by . The the facial walk of the unbounded component of is the natural basis for the closed curve . We will construct in three steps. We start with the closed curve which is obtained by traversing along its edges once. We obtain by replacing each part of going from a vertex to with a curve that also connects and but lies in and doesn’t intersect apart from the starting and the ending points. Due to the local finiteness of the embedding of this is even possible in such a way, that the segments in from to doesn’t intersect for different . The curve already fulfills properties 1., 2. and 3. as is a subset of the boundary of . However it is still possible, that a vertex occurs multiple times in . In this case is not injective.
To correct this problem, we modify to get by skipping any points that occur more than once. To skip a point , we delete a small part of that leads to and a small part of that comes from and connect the dangling ends directly without visiting and without intersecting anything else (see Figure 2). It remains to show, that contains no cut-vertex, as in this case we don’t destroy property 3. by the procedure.
Let be a cut-vertex of and let , be the connected components of that emerge after the deletion of . Let be such that . Hence for any we have . However for any vertex we know that . This implies that contains no cut-vertex of which finishes the proof.∎
A Jordan ray is an injective mapping such that . We identify these objects again with their image in .
Let be a closed Jordan curve containing the points in this order. Let be polygonal Jordan rays and let be two polygonal Jordan curves with the following properties:
Then we have
Proof: We chose a radius large enough such that . Let where , be the first intersection point of and . It follows with the help of the JCT that we may traverse in a way, such that are visited in this order. The sets and contain by construction polygonal curves starting in , ending in , being contained in and starting in , ending in , being contained in . Another application of the JCT yields, that and have to intersect, due to the order in which lie on .∎
Proof of Theorem 2.3: If for all , then there is a.s. no infinite cluster which proves one direction.
Let , and let us assume that there is a.s. exactly one infinite black cluster . This implies the a.s. existence of an infinite white cluster . We may choose large enough such that
where is defined as in Lemma 2.4. Let be the closed Jordan curve that exists due to Lemma 2.4 for the chosen . Let , be the points of ordered in the way that is induced if is traversed in clockwise direction.
Now a slight adaptation of the arguments from [BR08] is enough to finish the proof. We state it for the convenience of the reader. For any set we define the decreasing event ”There is no infinite black path starting in that intersects only once.” and the function
Clearly the following properties hold for :
, (due to FKG-inequality),
It follows from (ii), (iii) and (iv) that for any and
The idea is, to separate into four parts that have a low -value. At first we separate into two parts and , . Taking implies and . It follows from (2.8), that if we reduce by one, we increase the value of by a factor of at most two while we decrease by a factor of at most two. Hence there has to be a such that
For such a
In the same way, we separate and another time into the parts and such that
where we define for in same way as except that we use white paths instead of black ones. Due to symmetry we have and by choosing small enough we obtain, that the event
has positive probability. This event describes the case where there are black infinite paths emanating from and and white infinite paths emanating from and . Each of these paths intersects only in its starting point. We assumed that there is only one infinite cluster of each color and hence and has to be connected by some finite black path , while same is true for some white finite path that has to connect and . These paths fulfill by construction exactly the required properties in Lemma 2.5 and hence and have a nonempty intersection, which is a contradiction as it would imply a vertex that has both colors.∎
3 Percolation on random tessellations
In this section we want to define our model in a rigorous way, discuss several reasonable assumptions and show the uniqueness of the infinite cluster by adapting the Burton Keane argument from [BK89].
We will interpret randomly colored random tessellations as independently marked particle process. This leads to the following definitions and notational conventions (a broad introduction to point processes and random tessellations can be found in [SW08]).
Let be a metric space equipped with the Borel--algebra . We write for the set of locally finite counting measures on and equip it with the -algebra generated by the sets , . A measure is called locally finite iff for all bounded . A measurable mapping is called a point process on and is to be interpreted as a random collection of points in . Each point process permits a representation
where is the Dirac measure and are -valued random variables [SW08, Lemma 3.1.3]. The measure on is called the intensity measure of . In the important special case, that and we call the intensity of ( is the Lebesgue measure). Random tessellations will later be defined by letting be the space of compact and convex subsets of equipped with the Hausdorff metric.
To be able to add a color information to each cell, it is convenient to work with marked point processes. If we have a point process on with representation and an i.i.d. sequence with , that is independent of , we call
the independently marked version of . To indicate which is used, we will write instead of where necessary.
From now on we will only work with point processes on or and their independently marked versions (each marked point process on is also a point process on ). If is equal to either of these spaces, the canonical translation operator , is defined by
We use the same notation for the shift on the marked spaces defined by
This corresponds to the idea that only the points are shifted while each point retains its mark. A point process on is also called a particle process.
We recall, that a point process is stationary iff for all . Let be the -Algebra of translation invariant events, i.e. events with for all . A stationary point process is called ergodic if for all . It can be shown, that if is ergodic, then its independently marked version is ergodic too; see [DVJ07, Proposition 12.3.VI.].
We recall the definition of the Laplace functional of a point process applied to a function
We will extend this definition to -valued functions and remark that the integral or the expectation might not exist in this case. An introduction to point processes and the Laplace functional for random measures can be found in [DVJ03].
After the introduction of point and particle processes, we turn to tessellations. A set with non-empty interior is called a cell. A countable set of cells is called a tessellation (or mosaic) if
each ball in is intersected by at most a finite number of cells of ,
the cells of cover ,
the interiors of any two distinct cells in doesn’t overlap.
The cell of that contains is denoted by (if there is more than one cell containing we chose an arbitrary rule to break ties). The cell is called the zero cell.
Let be the set of tessellations and observe, that any tessellation induces a graph with vertex set . Two cells are adjacent in iff they have a -dimensional intersection, i.e. iff is not contained in any -dimensional hyperplane. The tessellation induces a second graph where any two cells with nonempty intersection are adjacent. The distinction of and will mostly be relevant in the 2-dimensional case. Apart from that, all results will hold for both graphs. The zero cell is the root in and .
We denote by the trace of on and call a measurable mapping a random tessellation. Hence a random tessellation is a point process of convex compact particles that form a tessellation. In the same spirit let be the set of colored tessellations with the -Algebra . For and a random or deterministic -valued sequence we define the marked tessellation . A marked tessellation induces a colored graph where , (the same notations are used for a random tessellations in place of ).
For the rest of this article we will work with a stationary random tessellation and a random sequence . Under the random tessellation is independent of the i.i.d. sequence that has , distributed marginals. Hence is stationary and we have
where denotes the distribution of .
The first thing one might observe is, that unlike in the case of percolation on a fixed graph, in our model the existence of an infinite black cluster can have a probability different from 0 or 1. We could, for instance, create a random tessellation by taking a randomly shifted square lattice with probability 1/2 and a randomly shifted honeycomb lattice otherwise. The resulting random tessellation would exhibit an infinite black cluster with probability 1/2 for between the lattice dependent percolation thresholds (which are known to be different). To rule out this somehow pathological case, we will restrict ourselves to ergodic random tessellations most of the time. As the set of colored tessellations that contain an infinite black cluster is translation invariant, we have for any ergodic random tessellation , since is also ergodic in this case.
Now it is only natural to define the percolation function and threshold for by
By a standard coupling argument we see that is non-decreasing in . It is also clear that there is an infinite black cluster in for any a.s. and that there is no infinite black cluster in for a.s. . The existence of an infinite cluster at is obviously an open and hard problem in most cases.
Uniqueness of the infinite cluster with Burton and Keane
We are now in the position to adapt the Burton Keane argument [BK89] to our model.
Let be a stationary random tessellation and . If
then there is a.s. at most one infinite black cluster in .
Proof: The claim is trivial for , so let for the rest of the proof. First, we assume that is ergodic. We already mentioned that this implies the ergodicity of . For each we define the set
These sets are translation invariant and hence exactly one of the events , will a.s. hold while all others will a.s. not.
Let us assume that holds a.s. for a fixed . In this case lies a.s. in the set
We fix an and define the random variables , , as the number of infinite black clusters in the colored graph which we obtain, if all vertices from in