Constrained Percolation in Two Dimensions
We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality.
Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.
A central question in percolation and statistical physics models is when there exists an infinite cluster; that is, an infinite connected component of elements having the same state. The onset of infinite clusters as a model parameter is varied may often be taken as a defining characteristic of a phase transition or critical point.
A key problem is to determine whether infinite clusters are present at the critical point. In particular, this question remains open in the archetypal case of independent bond percolation on the hypercubic lattice in dimensions (see [28, 16, 22, 15] for details). Far more is known about certain two-dimensional models, culminating in spectacularly detailed understanding of the critical phase, via Schramm-Loewner Evolution [43, 44, 29, 47, 30]. In many settings, a necessary precondition for analysis of this kind is self-duality. One then expects the phase transition point to coincide with the self-dual point, which is the point where the primal and dual models have equal parameters. However, proving this is often difficult, since one must rule out coexistence of primal and dual infinite clusters. This question was open for 20 years (between  and ) for bond percolation on the square lattice. A further important circle of questions concerns the number of infinite clusters. Uniqueness of the infinite cluster for independent percolation on the hypercubic lattice was also open for many years, before the proof in  (later simplified and generalized in ). The central conjecture in percolation on general transitive graphs is that non-amenability of the graph is equivalent to existence of a non-uniqueness phase .
In this article we address questions of existence, coexistence, and uniqueness of infinite clusters for a class of constrained percolation models in two dimensions. By this we mean that the configuration is restricted to lie in a subset of the sample space where certain hard local constraints are satisfied. Subject to this restriction, the probability measures that we consider are very general. Our main results require only very weak symmetry and ergodicity conditions. In particular, we do not assume stochastic monotonicity or correlation conditions such as the FKG inequality.
Many standard statistical physics models can be interpreted as constrained percolation models. Examples include the dimer model, or perfect matching model (in which a configuration is a subset of edges in which each vertex has exactly one incident edge present ); the 1-2 model (where each vertex has one or two incident present edges ); the 6-vertex model (where configurations are edge orientations on a degree-4 graph in which each vertex has in- and out-degree 2 ); and some general vertex models that can be transformed to dimer models on decorated graphs via the holographic algorithm ([45, 12, 41, 32]). We will give applications of our main results to dimer and XOR Ising models.
Phase transitions of certain constrained percolation problems have been studied extensively. See, for example,  for the dimer model, and  for the 1-2 model. The integrability properties of these constrained percolation problems make it possible to exactly compute finite-dimensional distributions and correlation functions. The critical parameter, i.e. the parameter where discontinuity of a certain correlation function is observed, can often be computed as the solution of an explicit algebraic equation.
Although there are many results describing the phase transitions of constrained percolation problems by a microscopic observable, e.g. spin-spin correlation, up to now, very few papers study the phase transitions of constrained percolation models from a macroscopic perspective, e.g. the existence of an infinite cluster. Even though sometimes we know that a phase transition exists with respect to a macroscopic observable, the exact value of the critical parameter is unknown , and it is not known if the critical parameter in the macroscopic sense coincides with the critical parameter in the microscopic sense, except for very few special models, such as the 2-dimensional Ising model [2, 35, 33]. One difficulty for constrained percolation problems is that there is often no stochastic monotonicity; see also [42, 38, 21].
1.2. Constrained Percolation
In this paper, we study a class of constrained site percolation problems on the 2-dimensional square lattice . The vertex set consists of all points with integer coordinates. Two vertices and are joined by an edge in if and only if .
Each face of is a unit square. We say that two faces of are adjacent if they share an edge. Let be a face of , and let be the coordinate of the vertex at the lower left corner of . We color white if is odd. If is even, we color black.
We consider site percolation on , i.e. the state space is . We call an element of a configuration, and we call the state of . We impose the following constraint on site configurations.
Around each black face, there are 6 allowed configurations , , , , , , where the digits from the left to the right correspond to vertices in clockwise order around the black square, starting from the lower left corner.
Note that in the unconstrained case, around each black square, there are 16 different configurations, only 6 of which are allowed in the constrained case. See Figure 1 for local configurations of the constrained percolation around a black square.
Let be the set of all configurations satisfying the constraint above. Let be a probability measure on . We will be interested in such measures satisfying the following conditions.
is -translation-invariant, where is the subgroup of generated by and .
is -ergodic; i.e., any -translation-invariant event has probability or under .
is symmetric under exchanging and ; i.e. writing for the map defined by for each , the measure is invariant under , that is, for every event .
We will also consider the following variations of (A1) and (A2) (one weaker and one stronger). Let be a positive integer.
is -ergodic; i.e., any -translation-invariant event has probability or under .
Note for we have
where “” means “implies”.
1.3. Contours and Clusters
Let be a set of vertices in . We say that is connected if it induces a connected subgraph of .
Let . A cluster of is a maximal connected set of vertices of in which every vertex has the same state. If all the vertices in a cluster have the state 0 (resp. 1), we call the cluster a 0-cluster (resp. 1-cluster). A cluster may be finite or infinite. Here is our first main result.
Let be a probability measure on the constrained percolation state space , satisfying . Let be the event that the number of infinite clusters is nonzero and finite. Then
Note that Theorem 1.1 requires no assumptions of stochastic monotonicity, correlation inequalities, or rotation-invariance. See [13, 24, 46] for related results requiring stochastic monotonicity and rotation-invariance.
The conclusion of Theorem 1.1 does not in general hold for unconstrained percolation meausres. Here is an example. Let and be independent families of i.i.d. Bernoulli random variables with parameter . Define a run of (resp. ) to be a maximal nonempty interval of on which the corresponding variables are all equal. Define a run rectangle to be a set of the form where is a run of and is a run of . Note that the run rectangles partition . Call a run rectangle a site rectangle if both and are runs of 1s, and a bond rectangle if exactly one of them is a run of s. Let be the (random) graph whose vertex set is the set of site rectangles, whose edge set is the set of all bond rectangles, and where a vertex and an edge are incident if some site of one is adjacent in to some site of the other. It is easily seen that is isomorphic to a.s. Now take a uniform spanning tree of (conditional on and ). Finally, assign a vertex of the value if it lies in a site rectangle or it lies in a bond rectangle whose edge is included in the spanning tree; otherwise assign value . It is straightforward to check that the resulting random configuration on is -translation-invariant and symmetric. Moreover, a.s. there exist both an infinite 0-cluster and an infinite 1-cluster. Indeed, the measure is ergodic as well. (This can be checked from mixing properties of the uniform spanning tree.)
We also consider contours separating clusters. For this purpose, we introduce two auxiliary square grids, and , whose vertices are located at centers of white faces of the original square grid . The primal (resp. dual) auxiliary square grid (resp. ) has vertices located at points of the plane, in which both and are even (resp. odd). Two vertices , of (resp. ) are joined by an edge of (resp. ) if and only if . Evidently each face of or is a square of side length 2. See Figure 2.
Moreover, each black face of is crossed by an edge of (in the sense that and share a center). Similarly, is crossed by an edge of . The two edges of and that cross the same black face of are perpendicular to each other. Each configuration in corresponds to a configuration in , where (resp. ) is the edge set of (resp. ), as follows. For each black face of , if the configuration around is or , then we set , for the two edges and that cross . If the configuration around is or (so that the two upper vertices have one state, and the two lower vertices have the other), then we let take value 1 on the horizontal edge ( or ), and 0 on the vertical edge. Similarly in the cases and , we set to be 1 on the vertical edge and 0 on the horizontal edge. We say an edge is present (resp. absent) if it has state (resp. 0). See Figure 3.
The present edges of and are boundaries separating the state 0 and the state 1. It is easily verified that each vertex of or has an even number of incident present edges. Moreover, the present edges of and can never cross.
The image of under the map is a subset of . We call elements of this image contour configurations. A configuration in is a contour configuration if and only if
each vertex in or has an even number of incident present edges;
present edges in and present edges in do not cross.
We use to denote the set of all contour configurations. The map is surjective and 2-to-1. Specifically, the configurations and (but no others) have the same image under , where is the exchange map defined in (A3).
Let , and . Each connected component of present edges in is called a contour of . Since present primal edges and dual primal edges do not cross in a contour configuration, either all the edges in a contour are primal edges (edges of ), or all the edges in a contour are dual edges (edges of ). A contour is a primal (resp. dual) contour if it consists of edges of (resp. edges of ). A (primal or dual) contour is called finite (resp. infinite) if it consists of finitely many (resp. infinitely many) edges. Note that a contour may have 4 edges sharing a vertex; see Figure 3.
Let be a cluster of , and let be a contour of . We say is incident to if there exists , and a vertex , such that is at Euclidean distance from the center point of .
Let be a probability measure on . Note that induces a probability measure on contour configurations in , via the map . We consider a random configuration in with law . Then the image is an associated contour configuration with law .
Let (resp. ) be the corresponding marginal distribution of on bond configurations of (resp. ). Let (resp. ) be the state space consisting of bond configurations of (resp. ) satisfying the condition that each vertex has an even number of incident present edges. In some cases, we may wish to assume the following.
has finite energy in the following sense: let be a face of , and be the set of four sides of the square . Let . Define to be the configuration obtained by switching the states of each element of , i.e. if , and otherwise; see Figure 4. Let be an event, and define
Then whenever .
Note that, for each , the corresponding defined in Assumption (A4) is still in .
Let be a probability measure on , and consider the corresponding contour configurations as defined above. Under Assumptions –:
-a.s. there are neither infinite primal contours nor infinite dual contours.
-a.s. there are no infinite clusters.
Let be the (0- or 1-) cluster including the origin in a random constrained percolation configuration. We define the mean cluster size as follows
where is the number of vertices in the cluster . By Proposition 1 of , in any translation-invariant measure on that has no infinite 0- or 1-clusters satisfies . Therefore Theorem 1.2 implies that for any probability measure on satisfying (A1)-(A4).
Note that Assumption (A4) is important for the conclusion of Theorem 1.2. In fact, if Assumption (A4) does not hold, it is possible that there exists more than one infinite cluster with positive probability. Consider, for example, a distribution of constrained percolation configurations on , such that each row of is either all 0’s with probability , or all 1’s with probability , and the configurations on different rows are independent. This distribution satisfies Assumptions (A1)–(A3), but not (A4) (and it is not ergodic under the group of horizontal translations). With probability 1 there exist infinitely many infinite clusters (indeed, -clusters) under such a distribution.
Under the same Assumptions (A1)–(A4), we have a stronger conclusion. In order to state the conclusion, let (resp. ) be a contour configuration on (resp. ). Let (resp. ) be the graph obtained from by removing every edge that is crossed by a present edge of (resp. ).
Let be a probability measure on satisfying the Assumptions –. Let be the corresponding marginal distribution on bond configurations in . Let be the union of all primal contours. Then -a.s. has no infinite components.
We also have some results on contours without the symmetry assumption (A3).
Let be a probability measure on the constrained percolation state space , satisfying Assumptions ,,, for some positive integer . Then -a.s. there is at most one infinite primal contour.
Finally, a random contour configuration induces a random site configuration on as follows. Fix a vertex , and assume that takes value 1 with probability , and takes value with probability , independent of . For any two adjacent vertices , let if and only if the edge crosses a present edge in . Let be the measure on , induced by in the way described above. We introduce the following new assumption.
Let be a probability measure on the constrained percolation state space , satisfying , , , , for some positive integer . Then -a.s. there are no infinite primal contours.
1.4. Examples of Constrained Percolation Measures
We present some applications of Theorems 1.1–1.5 to perfect matchings on the square-octagon lattice, as well as the XOR Ising model on the square grid. We will obtain results about infinite clusters and infinite contours in these well-known models.
Consider perfect matchings on the square-octagon lattice. See Figure 5 for a picture of the square-octagon lattice. Each perfect matching (or dimer configuration) on the square-octagon lattice is a subset of edges such that each vertex is incident to exactly one edge in the subset. There are two types of edges in the lattice: Type-I edges are edges of the square faces, and Type-II edges are edges of the octagonal faces but not of the square faces. (Type-II edges are diagonal lines in Figure 5).
We now connect perfect matchings on the square-octagon lattice with constrained percolation configurations on . Recall that is the square grid, whose faces are unit squares. We place a vertex of at the midpoint of each Type-II edge. A face of is constructed from the midpoints of four Type-II edges around a square or the midpoints of four Type-II edges around an octagon; see Figure 5. If a face of encloses a square face of the square-octagon lattice, we color it black. If a face of is enclosed by an octagon face, we color it white.
Given a perfect matching of the square-octagon lattice, we may consider its restriction to the set of all Type-II edges. There is a bijective correspondence between such restrictions and site percolation configurations on in . Specifically, a vertex of at the midpoint of a Type-II edge has state 1 in the constrained site configuration in if and only if the Type-II edge is present in the perfect matching of the square-octagon lattice. It is easy to verify that this is indeed a bijection. A present Type-II cluster (resp. absent Type-II cluster) of a dimer configuration on the square-octagon lattice is a set of present (resp. absent) Type-II edges such that their midpoints form a 1-cluster (resp. 0-cluster) of the constrained percolation configuration, given by the above bijection. Equivalently, Type-II clusters may be defined by considering two Type-II edges to be adjacent if some endpoint of one is adjacent to some endpoint of the other.
In order to define a probability measure for perfect matchings on the square-octagon lattice, we introduce edge weights. We assign weight 1 to each Type-II edge, and weight to the Type-I edge . Assume that the edge weights of the square-octagon lattice satisfy the following conditions.
The edge weights are -translation-invariant.
If , are two Type-I edges around the same square face, such that both and are horizontal, or both of them are vertical, then .
If , are two Type-I edges around the same square face, such that exactly one of , is horizontal and the other is vertical, then .
The reason we assume (B1) is to define a -translation-invariant measure. The reason we assume (B2) and (B3) is to define a measure for dimer configurations of the square-octagon lattice, which, under the connection described above to constrained percolation configurations in , will induce a probability measure on satisfying the symmetry assumption (A4).
Under (B1)–(B3), the edge weights are described by two independent parameters. We may sometimes assume the stronger translation-invariance condition below, which reduces the parameters to one.
The edge weights are -translation-invariant.
In , the authors define a probability measure for any bi-periodic, bipartite, 2-dimensional lattice. Specializing to our case, let be the probability measure of dimer configurations on a toroidal square-octagon lattice (see  for details). Let be the set of all perfect matchings on , and let be dimer configuration, then
where is the weight of the edge . As , converges weakly to a translation-invariant measure (see ).
Contours separating present Type-II clusters and absent Type-II clusters are be defined to be the contours in the corresponding constrained percolation model.
For given edge weights satisfying –, -a.s. there are no infinite present Type-II clusters or infinite absent Type-II clusters; morevoer there are no infinite contours.
Without (B4), we have a weaker conclusion.
For given edge weights satisfying –, -a.s. there is at most one infinite contour.
Next, we discuss the XOR Ising model. Consider an Ising model with spins located on vertices of the dual square grid . Assume each edge of has coupling constant . In order to make the connection to the dimer model, note that crosses exactly one square face of the square-octagon lattice; see Figure 5. Let be either of the two sides of parallel to . Assume the coupling constant and the edge weight satisfy the following identity
When , there is a unique satisfying identity (4).
The XOR Ising model (see ) is a random spin configuration on given by
where , are two independent Ising models on vertices of , taking values in . Assume both and have coupling constants given by (4), and both and are sampled according to the law of the Ising model obtained as the weak limit under periodic boundary conditions; see . A “”-cluster (resp. “”-cluster) of an XOR Ising configuration, whose spins are located on vertices of , is a maximal connected set of vertices of in which every spin has state “” (resp. “”) in . Similarly we can define an XOR Ising model with spins located on vertices of .
A contour configuration for an XOR Ising configuration, , defined on (resp. ), is a subset of (resp. ), whose state-1-edges (present edges) are edges of (resp. ) separating neighboring vertices of (resp. ) with different states of . (Recall that and are planar duals of each other.) Contour configurations of the XOR Ising model were first studied in , in which the scaling limits of contours of the critical XOR Ising model are conjectured to be level lines of Gaussian free field. It is proved in  that the contours of the XOR Ising model on the square grid correspond to level lines of height functions of the dimer model on the square-octagon lattice, inspired by the correspondence between Ising model and bipartite dimer model in . We will study the percolation properties of the XOR Ising model, as an application of the main theorems proved in this paper for the general constrained percolation process.
Before stating the results on the percolation properties of the XOR Ising model, we identify the critical and non-critical phases for the family of XOR Ising models under consideration. Consider an Ising model, with spins located on vertices of and coupling constants obtained from dimer edge weights of the square-octagon lattice by (4), such that the dimer edge weights satisfy Assumptions (B1)–(B3). Under the translation-invariance assumption (B1), the Ising model obtained above has the same coupling constant on all the horizontal edges (denoted by ), and the same coupling constant on all the vertical edges (denoted by ).
An Ising model on the square grid with coupling constants on each horizontal edge and on each vertical edge is said to be critical if
The Ising model is said to be in the low temperature state if
The Ising model is said to be in the high temperature state if
It is known that in the high temperature state, the Ising model has a unique Gibbs measure, and the spontaneous magnetization vanishes; while in the low temperature state, the Gibbs measures are not unique and the spontaneous magnetization is strictly positive under the “”-boundary condition. See [31, 1, 33].
We claim that if the dimer edge weights also satisfy (B4), then the Ising model has critical coupling constants; otherwise the Ising model has non-critical coupling constants. See . It is straightforward to check that given (4) and (B1)–(B3), (6) is equivalent to (B4).
We define the critical XOR Ising model (resp. non-critical XOR Ising model) to be one obtained from the product of two independent critical Ising models on a square grid such that
each Ising model has coupling constants on horizontal edges, and on vertical edges, such that , satisfy (resp. do not satisfy) (6);
each Ising model has a probability measure that is the weak limit of measures on finite graphs with periodic boundary conditions.
For the critical XOR-Ising model as defined above,
almost surely there are no infinite “”-clusters, and no infinite “”-clusters;
almost surely there are no infinite contours.
Now let us turn to the non-critical XOR Ising model.
The non-critical XOR Ising model, as defined above, almost surely has at most one infinite contour.
An XOR Ising model is said to be in the low temperature state (resp. high temperature state) if both and are in the low temperature state (resp. high temperature state). Recall that both and have the same parameters and .
In the low temperature XOR Ising model, almost surely there are no infinite contours.
For the high-temperature XOR Ising model, we can prove the existence of a unique infinite contour in sufficiently high temperature as follows.
Let be the critical probability for the i.i.d Bernoulli site percolation on the square grid. Note that . Let satisfy
Consider a high-temperature XOR Ising model on the square grid, in which each horizontal edge has coupling constant , and each vertical edge has coupling constant satisfying (8). If , then almost surely
there are no infinite “”-clusters or infinite “”-clusters;
there exists exactly one infinite contour.
Moreover, let , be obtained from , by
If we assign the coupling constant to each horizontal edge of the square grid, and to each vertical edge, then we obtain a low-temperature XOR Ising model in which the total number of infinite “”-clusters and “”-clusters is exactly one almost surely.
The paper is organized as follows. In Section 2, we prove combinatorial results regarding configurations of contours and clusters. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.4. In Section 5, we prove Theorem 1.2. In Section 6, we prove a few combinatorial and probabilistic results in preparation to prove the remaining main theorems of the paper. In Section 7, we discuss similar combinatorial results in unconstrained percolation. In Section 8, we prove Theorem 1.3. In Section 9, we prove Theorem 1.5. In Section 10, we prove Theorems 1.11, 1.10, 1.9, 1.8, 1.7 and 1.6. In Appendix A, we prove a combinatorial lemma required for the proof of Theorem 1.4.
2. Contours and Clusters
In this section, we prove some combinatorial and probabilistic results regarding infinite contours and infinite clusters in constrained percolation processes, in preparation for the proofs of Theorems 1.4, 1.3, 1.2 and 1.1. It will frequently be convenient to consider graphs embedded into the plane in the usual way; we identify and edge with closed line segment joining its the endpoints.
We begin with the following elementary lemma.
Consider the four vertices of a white face of . If in a constrained configuration in , all the four vertices have state 0, then flipping the states of all vertices to 1, and preserving the states of all the other vertices of , we obtain another constrained configuration in . Similarly, we may change the states of all the four vertices of one white face of from 1 to 0, and obtain another configuration in .
It is easy to verify that each of the four adjacent black faces has configurations satisfying the constraint. ∎
We introduce an augmented square grid , whose vertices are either vertices of , centers of faces of , or midpoints of edges of . Two vertices of are joined by an edge of if and only if .
Note that is a square grid with edge length . Let be the planar dual lattice of , which is also a square grid with edge length .
For any edge , consider the rectangle in consisting of all the points within distance at most of the line segment joining its endpoints. For a contour , let . The topological boundary of in is precisely a union of line segments corresponding to a set of edges of . The interface of is this set of edges . See Figure 6.
In particular, each component of the interface of the contour is either a self-avoiding cycle or a doubly-infinite self-avoiding path, consisting of edges of , and each vertex of is incident to 0 or 2 edges in the interface. Here by self-avoiding cycle we mean a finite connected component of edges of in which each vertex of has two incident edges.
The interface of a contour configuration is the union of interfaces of all contours in . Each component of the interface of is a self-avoiding cycle or doubly-infinite self-avoiding path in . See Figure 6 for an example of the interface. See also .
For any contour configuration , contours can never intersect interfaces, when interpreted as subsets of .
Any intersection must lie in the interior of either a black square or a white square. It is straightforward to check the two cases separately. In the case of a black square, we use the fact that primal contours and dual contours cannot cross each other. ∎
Throughout this section, we let be a constrained percolation configuration, and let be the corresponding contour configuration in .
For any component of the interface of , Let be the set consisting of all the vertices of whose distance to is . Then all the vertices in lie in a single cluster, and is the vertex set of a doubly-infinite self-avoiding path (resp. self-avoiding cycle) if is a doubly-infinite self-avoiding path (resp. self-avoiding cycle).
First of all, note that is a connected set of vertices in . Now, if not all the vertices in are in the same infinite cluster, then there exist a pair of adjacent vertices , such that the edge of crosses a contour. Then the contour crossing must cross the interface as well, but this is a contradiction to Lemma 2.2.
Next we show that is the vertex set of a doubly infinite self-avoiding path or a self-avoiding cycle. Let be the set of edges of whose distance to is . It is straightforward to check that . The fact that has degree 2 follows by local case analysis.
See Figure 6 for an example of such a part of a cluster, represented by black points. ∎
Let be a nonempty collection of contours. Any two vertices in a connected component of are connected by a path in that lies in the same component of .
For each , let be the unit square centered at . Consider a path in from to . Let
Then we claim that is a connected set in . Indeed, when the path enters a new square, it must do so either by crossing an edge of (the dual graph of ), or by passing through a vertex of . In the former case, the edge of does not lie in a contour of ; in the latter case, the vertex of does not lie in a contour of . ∎
In the following lemma, contours may be primal or dual as usual. We say a cluster is incident to a contour, if there exists a vertex of in the cluster and an edge of or in the contour, such that the Euclidean distance of the vertex and the contour is .
Consider any configuration and associated contour configuration .
If there exist at least two infinite contours, then there exists an infinite 0-cluster or an infinite 1-cluster.
If and are two infinite contours, then there exists an infinite cluster incident to .
If is a cluster incident to two infinite contours, then is an infinite cluster.
Suppose that is an infinite contour and is a nonempty collection of infinite contours, such that . Let be the unbounded component of containing . Then there is an infinite cluster in .
We first prove II., which immediately implies I.. If there exist at least two infinite contours, then we can find two distinct infinite contours and , two points and (midpoints of edges of ), and a self-avoiding path , consisting of edges of and two half-edges, one starting at and the other ending at , and connecting and , such that does not intersect any infinite contours except at and at . Indeed, we may take any path intersecting two distinct infinite contours, and then take a minimal subpath with this property.
Let be the first vertex along starting from . Let be the midpoint of the line segment . Then lies on the interface of . Let be the connected component of the interface of containing . Then is either a doubly-infinite self-avoiding path or a self-avoiding cycle consisting of edges of .
We consider these two cases separately. Firstly, if is a doubly-infinite self-avoiding path, then we claim that is in an infinite (0 or 1-)cluster of the constrained site configuration on . Indeed, by Lemma 2.3, all the vertices in are in the same cluster and is a doubly infinite self-avoiding path in .
Secondly, if is a self-avoiding cycle, then considering as a union of line segments in , has two components, and , where is the component including . Since is a cycle, exactly one of and is bounded, and the other is unbounded. Using Lemma 2.2, implies . Since , and is an infinite contour, we deduce that is unbounded, and is bounded. Note that by Lemma 2.2, so either , or . If , then any path consisting of edges of and one half-edge incident to and connecting and must intersect , and therefore must intersect also. In particular, intersects not only at , but also at some point other than . This contradicts the definition of . Hence . By Lemma 2.2, this implies . But is impossible since is infinite and is bounded. Hence this second case is impossible.
Therefore we conclude that there exists an infinite (0 or 1)-cluster incident to . This establishes II., and hence I..
We now turn to III.. Assume that is a cluster incident to two distinct infinite contours and . We can find a path , as above, such that every vertex of along is in . Then is infinite since the interface is infinite. This establishes III..
Consider Part IV. of the lemma. We say a contour is incident to , if there exists an edge of or in the contour and a vertex of in , such that the Euclidean distance of and is . We claim that there exists at least one infinite contour in incident to . Recall that is a connected component of . Indeed, if there is no infinite contour in incident to , then . But this is impossible since is nonempty.
Let be an infinite contour in incident to . Since contains at least one infinite contour, we can find an infinite contour such that there exists a path connecting a point and , consisting of a half-edge starting from , a half-edge starting from and edges of , such that crosses no infinite contours except at and at , and all the vertices of along are in by Lemma 2.4. Following the same procedure as above we can find an infinite cluster in adjacent to and . ∎
Lemma 2.5 has the following straightforward corollary.
Let be a probability measure on satisfying Assumptions –. If -a.s. there are no infinite clusters, then -a.s. there are no infinite contours.
Let (resp. ) be the event that there exists at least one infinite primal (resp. dual) contour. There is a bijection between configurations in and configurations in ; specifically, we translate each configuration in by , and obtain a configuration in , and vice versa. By Assumption (A3), we have .
Moreover, since and are translation invariant events, by Assumption (A2), we have either , or .
Suppose that . Since primal contours and dual contours are distinct, -a.s. there exist at least two distinct infinite contours. By Lemma 2.5 I., -a.s. there exists an infinite cluster. ∎
If is a contour, we write for the subgraph obtained from by removing all the edges of crossed by edges of .
Let be an infinite contour. Then each infinite component of contains an infinite cluster that is incident to .
Let be an infinite component of . Let be the midpoint of an edge of , and let be a vertex of , such that the Euclidean distance of and is . Let be the midpoint of the line segment . Then lies on the interface of . Let be the component of the interface of containing .
We claim that is infinite. Suppose that is finite. Then is a self-avoiding cycle. Let (resp. ) be the component of containing (resp. ). Then exactly one of and is bounded, and the other is unbounded. Note that by Lemma 2.2.
We claim that . To see why that is true, note that since is connected and , if is not a subset of , there exist a pair of adjacent vertices , such that and . Then the edge of crosses the interface , and therefore crosses the contour as well. But this is impossible since is an infinite component of .
Since it is impossible that and both and are infinite, we infer that is infinite.
According to Lemma 2.3, all the vertices in lie in an infinite cluster incident to . ∎
Let be in an infinite 0-cluster, let be in an infinite 1-cluster, and let be a path, consisting of edges of and connecting and . Then has an odd number of crossings with infinite contours in total.
In particular, if there exist both an infinite 0-cluster and an infinite 1-cluster, then there exists an infinite contour.
Moving along , two neighboring vertices of have different states if and only if the edge crosses a contour. Since the states of and are different, moving along , the states of vertices must change an odd number of times. Therefore crosses (primal and dual) contours an odd number of times.
It remains to show that the total number of crossings of with finite contours is even. Since crosses finitely many finite contours in total, let be all the finite contours intersecting , where is a nonnegative integer.
Let be the subgraph obtained from by removing all the edges of crossed by the ’s. Since all the ’s are finite, has exactly one infinite component. We claim that both and lie in the infinite connected component of . Indeed, if is in a finite component of , then it is a contradiction to the fact that is in an infinite 0-cluster, because the infinite 0-cluster including cannot be a subset of a finite component of . Similarly is also in an infinite component of . Since has a unique infinite component, we infer that both and are in the same infinite component of .
Since both and lie in the infinite connected component of , we can find a path connecting and , using edges of , such that the path does not intersect at all. Moreover, each vertex of or has an even number of incident edges in . We can transform to using a finite sequence of moves; in each move, the path only changes along the boundary of a single face of . Since the face contains either no vertex of , or a single vertex of even degree in , it is easy to verify that the parity of the total number of crossings is preserved. This implies that must cross infinite contours an odd number of times, because crosses (infinite and finite) contours an odd number of times in total, and crosses finite contours an even number of times. ∎
Assume that is an infinite cluster, and is an infinite contour. Assume that is a vertex of in , and let be the midpoint of an edge of . Assume that there exists a path connecting and , consisting of edges of and a half-edge incident to , such that crosses no infinite contours except at . Let be the first vertex of along starting from . Then .
Since crosses no infinite contours except at , let be all the finite contours crossing . We claim that has a unique unbounded component, which contains both and . Indeed, since and ; neither the infinite cluster nor the infinite contour can lie in a bounded component of .
Let be the intersection of the interface of with the unique unbounded component of . Since each , , is a finite contour, each component of the interface of is finite. In particular, consists of finitely many disjoint self-avoiding cycles, denoted by . For , has exactly one unbounded component, and one bounded component.
By Lemma 2.3, each form a self-avoiding cycle of , and all the vertices in , for each fixed , are in the same cluster. Note that each time crosses , it must intersect at a vertex of . We claim that all the vertices in , are also in the same cluster. Indeed, is divided by crossings with the interfaces () into nonoverlapping segments; the interior of each segment is either in a bounded component of , or in the unbounded component of . See Figure 7.
For each segment of whose interior lies in an unbounded component of , all the vertices of on the segment are in the same cluster (since the segment crosses no contours), including two vertices in and , for some . Since for any , , and can be connected by finitely many such steps as described above, we conclude that all the vertices in are in the same infinite cluster. Similarly, is in the same cluster as , for some , and is in the same cluster as , for some . Therefore we have . ∎
Using the same arguments as in Lemma 2.9, we can prove the following.
Assume that , are two infinite contours. Assume that , be the midpoints of edges of . Assume that there exists a path connecting and , consisting of edges of and two half-edges incident to and , such that crosses no infinite contours except at and . Let (resp. ) be the first (resp. last) vertex of along starting from . Then are in the same cluster.
3. Non-existence of finitely many infinite clusters
In this section, we prove Theorem 1.1. Throughout this section, let and let be the associated contour configuration.
The number of ends of a connected graph is the supremum over its finite subgraphs of the number of infinite components that remain after removing the subgraph.
Let be a probability measure on satisfying (Ak1). Then -a.s. no contour has more than two ends.
Let be an infinite contour with at most 2 ends. Then has at most 2 unbounded components.
Assume that has three unbounded components , and ; we will obtain a contradiction.
We can find three points , and such that there exist three semi-infinite paths , and starting from , and , respectively. Moreover, there exists a simply connected domain containing . We can choose the domain such that has exactly 3 unbounded components, denoted by , and , such that (resp. , ) is incident to and (resp. and , and ). Since , and has at most two infinite components (this follows from the fact that has at most two ends), at least one of , , does not include an infinite component of . Without loss of generality, assume that does not include an infinite component of . Then we can find a path connecting and , such that . Hence and are the same component of . But this is impossible.
Therefore we conclude that if is an infinite contour with 2 ends, has at most 2 unbounded components. ∎
Let (resp. , ) be the set of infinite contours such that has exactly 1 (resp. 2,0) unbounded components.
If , then .
If , i.e. contains at least one contour , then is the only infinite contour, since every other contour lies in a component of , and has only finite components. But this is impossible since . ∎
Assume that every infinite contour in has at most 2 ends. Assume that . Let (resp. ) be the number of unbounded components in (resp. ). Then (where possibly both are .)
To prove Lemma 3.4, we first need a fact about metric spaces.
If and are two points of a locally connected metric space , and is a closed set. We say separates from , if and are in two distinct components of .
Let and be two points of a connected and locally arcwise connected metric space , and let be a countable collection of closed sets such that
the common part of every pair of elements of is the closed set (which may be empty);
if and are two arcs from to that lie in , then lies in a compact set which is simply connected in the weak sense and whose closure contains no point of ;
is locally compact.
If no element of separates from in , then does not separate from in .
See Theorem 3 of . ∎
Proof of Lemma 3.4. Since every infinite contour in has at most 2 ends, by Lemma 3.2, the complement of every infinite contour in has at most 2 unbounded components. In other words contains all the infinite contours in . Since , by Lemma 3.3, . Therefore contains all the infinite contours in .
Let (resp. ) be the set of unbounded components in (resp. ). To prove the lemma, it suffices to construct a bijection from to .
For each , define to be the unbounded component of . For each , define
We claim that is a bijection from to . To see why that is true, note that for each , lies in an unbounded component of . For each , the following cases might occur:
contains no infinite contours in