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TOPICS IN PERCOLATION ALEXANDER DUNLAP Abstract. Percolation is the study of connectedness in a randomly-chosen subset of an infinite graph. Following Grimmett[4], we set up percolation on a square lattice. We establish the existence of a critical edge-density, prove several results about the behavior of percolation systems above and below this critical density, and use these results to find the critical density of percolation on the two-dimensional square lattice. Contents 1. Introduction 2. Measure and probability 3. Graphs 3.1. Definitions and basic theory 3.2. Properties of probability measure on graphs 3.3. Lattices and translation 3.4. Planar Duality 4. Percolation 4.1. Basic theory and the critical probability 4.2. Subcritical percolation: exponential decay 4.3. Supercritical percolation: uniqueness of the infinite cluster 4.4. The critical value in two dimensions Acknowledgments References 1 2 5 5 5 8 10 11 11 13 15 18 20 20 1. Introduction In nature, fluids (such as water) percolate through porous substances (such as earth). The key feature of such a system is the porosity of the medium, which means that the fluid can travel along some, “open,” paths but not along other, “closed,” paths. The exact configuration of the open and closed paths in a given system is of course very complicated, and a specific configuration is not of independent interest. We instead want to study random percolation systems, in which each path is declared open or closed with a given probability. This is analogous to a complex physical system in which it is impossible to predict, say, the permeability of a given particle in the earth, but in which we know the general density of permeable particles in the entire section of earth under consideration. While natural percolation is the inspiration for the subject, the mathematical study of percolation that we describe in this paper does not attempt to model any Date: 20 August 2012. 1 2 ALEXANDER DUNLAP particular physical system. Rather, we think of a percolation system as a mathematical graph and establish results about graphs constructed randomly according to certain parameters. In this paper, we restrict our attention to configurations of the d-dimensional square lattice Ld . We generate such configurations by choosing (as “open”) a random subset of the edges of Ld , which creates a system known as bond-percolation. (In the more general site percolation, subgraphs are created by choosing a subset of the vertices.) Percolation is subject to the density parameter p, which defines the probability of any given edge being chosen open or closed. Given this setup, percolation theory seeks to understand the sizes of “open clusters” in configurations. In particular, we want to know whether these open clusters are of finite or infinite size. Curiously, as the density parameter varies, the probability of the existence of an infinite cluster changes sharply at the critical point pc , which depends on the density of the lattice. The percolation system behaves in qualitatively different ways in the “subcritical” and “supercritical” phases. When p < pc , open clusters are almost surely finite, and we can ask questions about their size (Section 4.2). When p > pc , there is almost surely an infinite open cluster, and we can ask questions about how many such infinite open clusters there are (Section 4.3). Combining results from both phases, we can actually derive the value of pc for d = 2 (Section 4.4), which for general d is very difficult to compute. Chapter 2 sets down some basic concepts of measure theory and probability, and Chapter 3 establishes various definitions and theorems that will be useful for working with graphs and lattices. We apply this built-up theory to percolation systems in Chapter 4. 2. Measure and probability We briefly formulate basic probability concepts using measure theory. Notation 2.1 (Set-theoretic notations). We write |A| to denote the cardinality of a set A and P(A) to denote the power set of A. Given sets A and B, we define the symmetric difference A 4 B as the set of elements that are members of either A or B but not both. In symbols, we have A 4 B = (A \ B) ∪ (B \ A). Definition 2.2. Given a set Ω, an algebra on Ω is a collection F ⊆ P(Ω) such that 1. ∅, Ω ∈ F. 2. A ∈ F =⇒ Ac ∈ F. S 3. If (Ai )i is a finite collection of elements of F, then i Ai ∈ F. Definition 2.3. An algebra F on S Ω is a σ-algebra if, given a countable collection (Ai )i of elements of F, we have i Ai ∈ F. Fact 2.4. By de Morgan’s Laws, an algebra is closed under finite intersections and a σ-algebra is closed under countable intersections. Proposition S 2.5. If Fi ⊆ F2 ⊆ F3 ⊆ · · · is an increasing sequence of σ-algebras, then F = i Fi is an algebra. Proof. Clearly, ∅, Ω ∈ F. If A ∈ F, then A ∈ Fi forSsome i, so Ac ∈ Fi ⊆ F. If n A1 , . . . , An ∈ F, then A1 , . . . , An ∈ Fi for some i, so j=1 Aj ∈ Fi ⊆ F. Definition 2.6. Given a set Ω and a σ-algebra F on Ω, a function P : F → [0, 1] is a probability measure on F if the following conditions hold: TOPICS IN PERCOLATION 1. 2. 3. 3 P(∅) = 0. P(Ω) = 1. S If (Ai )i is a countable collection of disjoint elements of F, then µ( i Ai ) = P i µ(Ai ). Definition 2.7. A probability space is a triple (Ω, F, P) of a set Ω (called the sample space), a σ-algebra F on Ω (whose members are called events), and a probability measure P on F. If F and P are clear from the context, we will often speak of the probability space Ω to refer to the probability space (Ω, F, P). Definition 2.8. We say that an event A is almost sure, or that A happens almost always, if P(A) = 1, and that it happens almost never if P(A) = 0. Facts 2.9. The following are basic properties of probability measures: 1. (Monotonicity.) If A ⊆ B, then µ(A) ≤ µ(B). 2. (Countable subadditivity.) If (Ai )i S is a countable collection of elements of P F, not necessarily disjoint, then µ( i Ai ) ≤ i µ(Ai ). S 3. (Continuity from below.) If A1 ⊆ A2 ⊆ · · · , and A = i Ai , then µ(A) = limn→∞ µ(Ai ). T 4. (Continuity from above.) If A1 ⊇ A2 ⊇ · · · , and A = i Ai , then µ(A) = limn→∞ µ(Ai ). Definition 2.10. Two events A and B are independent with respect to a measure P if P(A ∩ B) = P(A)P(B). Definition 2.11. Given events A, B ∈ F such that P(B) > 0, we define the conditional probability P(A | B) = P(A ∩ B)/P(B). Fact 2.12. If A and B are independent, then P(A | B) = P(A). Definition 2.13. 1. Suppose that (Ω, F, P) is a probability space and X is a real-valued function on Ω such that X(Ω) is finite or countable. We say that X is F-measurable, and that X is a discrete random variable, if {ω | X(ω) = x} ∈ F for each x ∈ X(Ω). 2. If A ∈ F, we define the indicator function of A to be the discrete random variable given by ( 1 if ω ∈ A; 1A (ω) = 0 otherwise. Notation 2.14. We will write P(X = x) to denote P {ω ∈ Ω | X(ω) = x}. Definition 2.15. Let Ω be a sample space. 1. If A is a collection of subsets of Ω, then the σ-algebra generated by A, denoted σ(A), is the intersection of all σ-algebras F of Ω such that A ⊆ F. 2. If X is a collection of discrete random variables X : Ω → R, then the σalgebra generated by X, denoted σ(X), is the intersection of all σ-algebras F of Ω such that X is F-measurable for all X ∈ X. Fact 2.16. If X is as in Definition 2.15.2, then σ(X) = σ X −1 (x) X ∈ X, x ∈ X(Ω) . It will be useful to approximate events in a σ-algebra generated by an algebra by events in the original algebra. 4 ALEXANDER DUNLAP Theorem 2.17. Suppose that P is a probability measure, F̃ is an algebra on Ω, and F = σ(F̃). For every A ∈ F and ε > 0, there is an à ∈ F̃ so that P(A 4 Ã) < ε. o n Proof. Let G = A ∈ F ∀ε > 0, ∃à ∈ F̃ so that P(A 4 Ã) < ε . It is clear that F̃ ⊆ G. We claim that G is a σ-algebra, which will imply that F = σ(F̃ ) ⊆ G and thus prove the theorem. We clearly have ∅, Ω ∈ G. Suppose A ∈ G and ε > 0. Then there is an à ∈ F̃ so that P(A 4 Ã) < ε. But since A 4 à = Ac 4 (Ã)c , and (Ã)c ∈ F̃, this means that Ac ∈ G. S Suppose that (Ai )i∈N is a countable collection of elements of G, let A = i∈N Ai , Sn and let Bn = i=1 Ai . Let ε > 0. By continuity of the measure P, there is an n ∈ N so that |P(Bn ) − P(A)| < ε, which implies that P(Bn 4 A) = P(A \ Bn ) < ε. For each i = 1, . . . , n, there is an Âi ∈ F̃ such that P(Âi 4 Ai ) < ε/n, which implies Sn Sn that P( i=1 Âi 4 Bn ) < ε. Therefore, P( i=1 Âi 4 A) < 2ε, so A ∈ G since Sn i=1 Âi ∈ F̃ by the definition of algebra. Definition 2.18. Given a countable Q collection of probability spaces {(Ω Q i , Fi , Pi )}, respectively, we say that A ⊆ Ω = i Ωi is a cylinder set if A = i Ai , where Ai ∈ Fi for all i and Ai = Ωi for all but finitely many i. We can then define the product σ-algebra F on Ω as the σ-algebra Q generated by the set of all cylinder sets. A product (probability) measure P = i Pi is a (probability) measure on F such Q Q that P ( i Ai ) = i P(Ai ). The actual construction of product measure, and thus the proof of the following theorem, is omitted. Theorem 2.19. For a countable product of probability spaces, a unique product measure exists. Q Notation 2.20. If Ω = i Ωi , we have the usual projection functions pi : Ω → Ωi . Proposition 2.21. If {Ωi } is a countable collection of probability spaces and Ω = Q Ω is endowed with the product σ-algebra F, then F = σ ({pi }). i i Notation 2.22. Let p ∈ [0, 1]. We can define a probability measure µp on P({0, 1}) by µp ({1}) = p. Given a finite or countable set E, we will write Pp for the measure on {0, 1}E given by Pp = (µp )E . Definition 2.23. Given a probability space (Ω, F, P) and a discrete random variable X, we define the expected value of X by Z X E[X] = X dP = xP(X = x). Ω x∈X(Ω) Facts 2.24. The following are basic properties of expected value: 1. (Linearity.) Given two random variables X and Y and a real number a, we have E[aX + Y ] = aE[X] + E[Y ]. 2. (Monotonicity.) Given two random variables X and Y such that X(ω) ≤ Y (ω) for almost all ω ∈ Ω, we have E[X] ≤ E[Y ]. 3. The expected value of an indicator function of an event is the probability of the event. Symbolically, E[1A ] = P(A). Notation 2.25. With setup as in Notation 2.22, we will write Ep to denote the expected value operator with respect to the product measure Pp . TOPICS IN PERCOLATION 5 3. Graphs 3.1. Definitions and basic theory. Definition 3.1. An undirected graph G is a pair (V, E) consisting of a set V and a set E of unordered pairs of elements in V . The elements of V are called vertices and the elements of E are called edges. If e is an edge between vertices v1 , v2 , we will write e = hv1 , v2 i = hv2 , v1 i. In this paper, the term graph will always refer to an undirected graph. Definition 3.2. A configuration of a graph G = (V, E) is a function ω : E → {0, 1}, or equivalently an element w ∈ {0, 1}E . An edge e is called open in ω if ω(e) = 1 and closed if ω(e) = 0. We will often wish to speak of configurations derived from other configurations by opening or closing certain subsets of edges, and will use the following notation frequently. Notation 3.3. Let G = (V, E) be a graph and let ω be a configuration of G. Suppose that F ⊆ E is a subset of the edges of G. Define the configurations ω F and ωF by ( ( 1 if e ∈ F ; 0 if e ∈ F ; F ω (e) = ωF (e) = ω(e) otherwise; ω(e) otherwise. In other words, ω F and ωF are the configurations obtained from ω by taking all edges of F to be open and closed, respectively. Definition 3.4. Let G = (V, E) be a graph, ω be a configuration on G, and x, y ∈ V . We say that x and y are connected in ω, and write x ↔ y, if there is a finite sequence x = z1 , z2 , . . . , zn−1 , zn = y such that (zi , zi+1 ) is an open edge in ω for all i = 1, . . . , n − 1. Fact 3.5. The connectedness relation ↔ is an equivalence relation. Definition 3.6. A cluster in a configuration ω is an equivalence class under ↔. We will write Cx (ω) for the cluster containing the node x; in particular, C0 (ω) will denote the cluster containing the origin. 3.2. Properties of probability measure on graphs. We review several important theorems for considering probability measure on graphs. While these theorems are not specific to probability on graphs, graphs provide both an application for the theorems and a set of examples for understanding them. In this section, we will work with a discrete set E, the sample space Ω = {0, 1}E , the product σ-algebra F, and the probability measure and expectation operators Pp and Ep defined in Notations 2.22 and 2.25, respectively. For example, E might be the set of edges on a graph G = (V, E) and Ω the set of configurations of G; in this case, Pp would be the probability measure that independently takes each edge to be open with probability p. It is clear that Ω is partially ordered under the usual partial order for functions: ω ≤ ω 0 if ω(e) ≤ ω 0 (e) for all e ∈ E. In our example of graphs and edges, we have ω ≤ ω 0 if all of the edges that are open in ω are also open in ω 0 . Since Ω is partially ordered, we thus also have a notion of an increasing random variable on Ω. 6 ALEXANDER DUNLAP Theorem 3.7. If f : Ω → R is a nondecreasing function, and p ≤ r, then Ep [f ] ≤ Er [f ]. Proof. Consider the sample space Ξ = [0, 1]E with probability measure given by the product of uniform measures on each copy of the unit interval. For p ∈ [0, 1] and ξ ∈ Ξ, define a random configuration ωp (ξ) of Ω by (ωp (ξ))(e) = 1[0,p] (ξ(e)). Note that ωp is distributed according to the product measure Pp . If p ≤ r, it is clear that ωp ≤ ωr . Therefore, we have f (ωp ) ≤ f (ωr ). But this means, since the expected value operator is order-preserving, that Ep [f ] = E(f (ωp )) ≤ E(f (ωr )) = Er [f ]. Definition 3.8. An event A ∈ F is increasing if ω ∈ A, ω 0 ∈ Ω, ω 0 ≥ ω implies ω 0 ∈ A. An event B ∈ F is decreasing if B c is increasing. Examples 3.9. Given a graph G = (V, E), we have the following examples of increasing events. 1. Let E 0 ⊆ E. Then the event A = {ω ∈ Ω | ω(E 0 ) = 1} is increasing. 2. Let v1 , v2 ∈ V . Then the event A = {ω ∈ Ω | v1 and v2 are connected in ω} is increasing. We state the following theorem, a special case of what is called the FKG inequality, with the proof[4] omitted. Theorem 3.10. Suppose that Ω is as above, with E finite, and suppose that P is a product measure on Ω. If A and B are increasing events on Ω, then P(A ∩ B) ≥ P(A)P(B). Corollary 3.11. By symmetry, the (unmodified) conclusion of Theorem 3.10 also holds under the hypothesis that A and B are decreasing: if A and B are decreasing events on Ω, then P(A ∩ B) ≥ P(A)P(B). Remark 3.12. The intuition behind Theorem 3.10 (in the setup of Examples 3.9) is that an increasing event is one that requires edges to be open and that only becomes more likely when it is given that more edges are open. Heuristically, then, the occurrence of one increasing event increases the likelihood of another increasing event. Definition 3.13. Suppose that A and B are increasing subsets of Ω. Define A ◦ B = ω ∈ Ω there is an F ⊆ E such that ωE\F ∈ A and ωF ∈ B , using the notation from 3.3. Remark 3.14. Intuitively, ω ∈ A ◦ B if there is an F ⊆ E so that we only need to examine the edges of F to know that ω ∈ A, and we only need to examine the edges outside of F to know that ω ∈ B. Proposition 3.15. If A and B are increasing subsets of Ω, then A◦B is increasing. Proof. If ω ∈ A ◦ B, then there is an F ⊆ E so that ωE\F ∈ A and ωF ∈ B. If 0 0 ω 0 ≥ ω, then ωE\F ≥ ωE\F and thus ωE\F ∈ A since A is increasing. Similarly, 0 0 ωF ∈ B. Therefore, ω ∈ A ◦ B, so A ◦ B is increasing. TOPICS IN PERCOLATION 7 Theorem 3.16 (BK inequality). Let Ω be defined as above and endowed with a product measure P. Suppose that the edge-set E is finite, and suppose moreover that A and B are increasing subsets of Ω. Then P(A ◦ B) ≤ P(A)P(B). Proof. Let 1, . . . , N be an ordering of E. Let Ω × Ω0 be the product of two copies of Ω and let P̂ = P × P be the product measure on Ω × Ω0 . If ω ∈ Ω and ω 0 ∈ Ω0 , let Hj (ω, ω 0 ) = (ω 0 (1), . . . , ω 0 (j − 1), ω(j), . . . , ω(N )). Clearly, ω1 = ω and ωN +1 = ω 0 . For each j = 1, . . . , N , define Âj = {(ω, ω 0 ) | Hj (ω, ω 0 ) ∈ A} B̂ = {(ω, ω 0 ) | ω ∈ B} = B × Ω. It is clear that B̂ and all of the Âj s are increasing since A and B are increasing. Moreover, Â1 = A × Ω0 , so P̂(Â1 ◦ B̂) = P̂((A × Ω0 ) ◦ (B × Ω0 )) = P̂(A ◦ B). Also, ÂN +1 = Ω × A, so P̂(ÂN +1 ◦B̂) = P̂((Ω×A)◦(B×Ω0 )) = P̂ {(ω, ω 0 ) | ω ∈ B and ω 0 ∈ A} = P(A)P(B) by the definition of the product measure P̂. The rest of the proof will establish the chain of inequalities P̂(A ◦ B) = P̂(Â1 ◦ B) ≤ P̂(Â2 ◦ B) ≤ · · · ≤ P̂(ÂN +1 ◦ B) = P(A)P(B), which will imply the desired result. Fix 1 ≤ j ≤ N . We want to show that (3.17) P̂(Âj ◦ B̂) ≤ P̂(Âj+1 ◦ B̂). We do this by conditioning on the value of ω(i), ω 0 (i) for all i 6= j. Let E = E \ {j} and let G = (V, E). We will use the notation ω|E to mean the restriction of ω (as a function E → {0, 1}) to E. Let ω and ω 0 be configurations of G. Define the event C ⊆ Ω × Ω0 by C = ω|E = ω and ω 0 |E = ω 0 . The goal is then to show that (3.18) P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C ≤ P̂ (ω, ω 0 ) ∈ Âj+1 ◦ B̂ C ; if (3.18) holds for all C, then (3.17) clearly follows. Let ω j , ω 0j , ω j , ω 0j be configurations of G that agree on E \ {j} with ω and ω 0 , respectively, and have edge j open and closed, respectively. Consider three cases: 1. (ω j , ω 0j ) 6∈ Âj ◦ B̂. Then P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C = 0 ≤ P̂ (ω, ω 0 ) ∈ Âj+1 ◦ B̂ C , with the inequality following trivially from the definition of measure. 2. (ω j , ω 0j ) ∈ Âj ◦ B̂, so (ω j , ω 0j ) ∈ Âj+1 ◦ B̂. Then P̂ (ω, ω 0 ) ∈ Âj+1 ◦ B̂ C = 1 ≥ P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C . 3. Neither of the above, so (ω j , ω 0j ) ∈ Âj ◦ B̂ and (ω j , ω 0j ) 6∈ Âj ◦ B̂. Membership of a pair of configurations (ω, ω 0 ) in Âj ◦ B̂ does not depend on the value of ω 0 (j), so this implies (ω j , ω 0j ) ∈ Âj ◦ B̂ (ω j , ω 0j ) 6∈ Âj ◦ B̂. 8 ALEXANDER DUNLAP Since Âj ◦ B̂ is increasing, this means that the conditional probability that (ω, ω 0 ) is in Âj ◦ B̂ exactly the probability that ω(j) = 1. More formally, we have P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C = P(ω(j) = 1). Since (ω j , ω 0j ) ∈ Âj ◦ B̂, there are sets F, F 0 ⊆ E so that we have ((ω j )E\F , (ω 0j )E\F 0 )) ∈ Âj and ((ω j )F , (ω 0j )F 0 )) ∈ B̂. The latter membership is equivalent to (ω j )F ∈ B. Consider two cases. (a) j ∈ F , so ω j (j) “contributes” to Âj . The value of w0 (j) does not affect whether or not B̂ occurs, and the value of w(j) does not affect whether or not Âj+1 occurs, so we can certainly remove j from F and add j to F 0 when considering membership in Âj+1 ◦ B̂. With this arrangement 0 of F and F 0 (and conditioning on C), we have (ωE\F 0 , ωE\F 0 ) ∈ Âj+1 0 if and only if ω (j) = 1. There may be other arrangements of F and F 0 that increase the membership of of Âj+1 ◦ B̂, so, in general, P̂ (ω, ω 0 ) ∈ Âj+1 ◦ B̂ C ≥ P(ω 0 (j) = 1) = P(ω(j) = 1) = P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C . (b) j ∈ E \ F . Now ω(j) “contributes” to B̂. There is no effect from w0 (j) on membership in B̂, so we can add j to F when considering membership of a given configuration in Âj+1 ◦ B̂. This may make the probability of membership in Âj+1 ◦ B̂ higher than that in Âj ◦ B̂, but certainly cannot make it lower. Therefore, P̂ (ω, ω 0 ) ∈ Âj+1 ◦ B̂ C ≥ P̂ (ω, ω 0 ) ∈ Âj ◦ B̂ C . In all cases, (3.18) holds. This applies for all fixed configurations ω, ω 0 , so (3.17) holds, which implies the desired result in the manner indicated above. 3.3. Lattices and translation. We single out a special class of graphs, called lattices, for the study of percolation. We will write Zd for the Cartesian product of d copies of the integers. Pd Definition 3.19. The `1 norm on Zd is given by kxk1 = i=1 |xi |. The `∞ norm is given by kxk∞ = max1≤i≤d |xi |. Definition 3.20. The d-dimensional square lattice is the (infinite) graph Ld = (Zd , Ed ), with the edge set Edconsisting of pairs of (non-diagonally) adjacent nodes. More precisely, we set Ed = hx, yi x, y ∈ Zd and kx − yk1 = 1 . For the remainder of the paper, (Ω, F, Pp ) will be the probability space of configurations of Ld with the product measure Pp . Definition 3.21. A box B in Ld is a subgraph of Ld with vertex set V = [a1 , b1 ] × [a2 , b2 ] × · · · × [ad , bd ] and edge set consisting of all edges in Ed with both endpoints TOPICS IN PERCOLATION 9 in V (B). The boundary of B, denoted ∂B, is (as its name suggests) the subgraph of Ld with vertex set V (∂B) = {x = (x1 , . . . .xn ) ∈ V (∂B) | there is an i such that xi ∈ {ai , bi }} and edge set consisting of all edges in Ed with both endpoints in V (∂B). Notation 3.22. Given n ∈ N, we define Λ(n) to be the box generated by [−n, n]d . Definition 3.23. For each edge e ∈ Ed , we have as in Notation 2.20 the projection function pe , which we can consider as a random variable on Ω. Define L(n) = σ ({pe | e ∈ Λ(n)}), the σ-algebra ofSevents that depend only on the configurations of the edges inside Λ(n). Let L = n∈N L(n). Fact 3.24. The set of events L is an algebra, and F = σ(L). The extra structure provided by the lattice lets us consider translations. Note that the lattice itself is invariant under translation: none of the points of the lattice are singled out. Definition 3.25. Let z ∈ Zd . 1. If G is a subgraph of Ld , the translation of G by z is the subgraph G + z of Ld with vertex set V (G + z) = x + z x ∈ Zd and edge set consisting of the edges of G translated similarly. 2. Let ω be a configuration of Ld . The translation of ω by z is the configuration given by (ω + z)(e) = ω(e − z) for all e ∈ Ed . d 3. Suppose that A is an event in the probability space (Ω = {0, 1}E , F = P(Ω), Pp ). The translation of A by z is the event A + z = {ω | ω − z ∈ A}. We say that A is translation-invariant if for all z ∈ Zd , we have A + z = A. 4. A measure P is translation-invariant if P(A) = P(A + z) for all events A and all z ∈ Zd . Examples 3.26. The following events are translation-invariant: 1. The event A = {all edges are open in ω}. 2. The event A = {there is an infinite open cluster in ω}. The event B = {there is an infinite open cluster containing the origin in ω}, however, is not translation-invariant, since translating a configuration in which the origin is part of an infinite open cluster may move the cluster away from the origin. Fact 3.27. Product measure Pp is translation-invariant by definition. Theorem 3.28 (Zero-one law). If A is a translation-invariant event, then either Pp (A) = 0 or Pp (A) = 1. Proof. Given e ∈ Ed , define the random variable e∗∗ : Ω → {0, 1} by e∗∗ (ω) = ω(e). Note that the σ-algebra Ln = σ ({pe | e ∈ Λ(n)}) is the σ-algebra of events that only depend on the edges in Λ(n). S Fix ε > 0. By Proposition 2.5, we know that n∈N Ln is an algebra. Furthermore, because any event can be expressed as a countable union or intersection of events only depending on the edges in finite boxes (see Proposition 2.21), we have S that A ∈ σ n∈N Ln . Therefore, by Theorem 2.17, there is an n ∈ N and an event U ∈ Ln so that Pp (A 4 U ) < ε, which implies |Pp (A) − Pp (U )| < ε and (3.29) |Pp (A)2 − Pp (U )2 | = |Pp (A) + Pp (U )| · |Pp (A) − Pp (U )| < 2ε. 10 ALEXANDER DUNLAP Let z = (n + 1, 0, 0, . . . , 0) ∈ Zd and let V = U + z. Clearly, U and V are independent and have identical probabilities under the product measure, so Pp (U ∩ V ) = Pp (U )Pp (V ) = Pp (U )2 . Substituting into (3.29), we have |Pp (A)2 − Pp (U ∩ V )| < 2ε. (3.30) Note that, by several applications of Definition 3.25.3 and by the translationinvariance of A, we have A 4 V = {ω1 | ω1 ∈ A and ω1 6∈ V } ∪ {ω1 | ω1 ∈ V and ω1 6∈ A} = {ω1 | ω1 ∈ A and ω1 6∈ U + z} ∪ {ω1 | ω1 ∈ U + z and ω1 6∈ A} = {ω1 | ω1 ∈ A and ω1 − z 6∈ U } ∪ {ω1 | ω1 − z ∈ U and ω1 6∈ A} = {ω2 − z | ω2 + z ∈ A and ω2 6∈ U } ∪ {ω2 − z | ω2 ∈ U and ω2 + z 6∈ A} = {ω2 − z | ω2 ∈ A and ω2 6∈ U } ∪ {ω2 − z | ω2 ∈ U and ω2 6∈ A} = (A 4 U ) − z, and thus Pp (A 4 V ) = Pp (A 4 U ) by the translation-invariance of Pp . Note also that A 4 (U ∩ V ) = (A ∩ (U ∩ V )c ) ∪ (Ac ∩ U ∩ V ) = (A ∩ (U c ∪ V c )) ∪ (Ac ∩ U ∩ V ) = (A ∩ U c ) ∪ (A ∩ V c ) ∪ (Ac ∩ U ∩ V ) ⊆ (A ∩ U c ) ∪ (Ac ∩ U ) ∪ (A ∩ V c ) ∪ (Ac ∩ V ) = (A 4 U ) ∪ (A 4 V ). Therefore, Pp (A 4 (U ∩ V )) ≤ Pp (A 4 U ) + Pp (A 4 V ) = 2Pp (A 4 U ) < 2ε, so (3.31) |Pp (A) − P(U ∩ V )| < 2ε. Combining (3.30) and (3.31) by the triangle inequality, we have that |Pp (A) − Pp (A)2 | < 4ε. But this holds for all ε > 0, so Pp (A) = Pp (A)2 , so Pp (A) must be equal to 0 or to 1. Remark 3.32. The key mechanism in the above proof is the application of Theorem 2.17. The translation-invariant event A certainly cannot depend on any specific region of the lattice, but Theorem 2.17 says that it can be approximated in measure by an event that depends only on some specific region of the lattice. This can happen only if A happens almost always or almost never. 3.4. Planar Duality. An important peculiarity of the two-dimensional square lattice L2 is that the lattice is “dual” to a copy of itself translated by (1/2, 1/2). This allows us to use arguments about the connectedness of the dual lattice to draw conclusions about the connectedness of the original lattice. Definition 3.33. We define the dual two-dimensional square lattice by L2d = L2 + (1/2, 1/2) (Figure 1a). Remark 3.34. The dual lattice L2d has a vertex at the center of every square in L2 . Every edge e of L2 intersects exactly one edge ed of L2 and vice versa, so the map e 7→ ed is a bijection. Definition 3.35. A configuration ω of L2 gives rise to a configuration ωd of L2d given by ωd (ed ) = ω(e), with ed defined as in Remark 3.34. We will sometimes omit the subscript d and speak of the configuration ω of L2d . TOPICS IN PERCOLATION 11 (a) Nodes and edges of L2 (dark) (b) Two open clusters (dark) in a and L2d (light). configuration of L2 are “separated” by a closed path (light) in L2d . Figure 1. The lattice and dual lattice. Remark 3.36. Informally, if x and y are members of two different clusters of L2 or of a subset of L2 , then there must be a closed path in L2d “separating” x from y (Figure 1b). We omit the formalization and proof of this statement, but will use this type of argument frequently in analyzing percolation on L2 . 4. Percolation We consider percolation on the d-dimensional square lattice Ld . We will consider the sample space Ω consisting of all configurations ω of Ld , equipped with the product σ-algebra F and product measure Pp . With respect to Pp , the individual edge-states ω(e) are independent, identically-distributed random variables satisfying the law Pp (ω(e) = 1) = p. 4.1. Basic theory and the critical probability. Sometimes it will be more convenient to think about the size of the cluster containing a node in terms of the lengths of paths originating at that node, a concept we formalize next. Definition 4.1. Let G = (V, E) be a graph. Given n ∈ N and x ∈ V , a selfavoiding walk of length n starting at x is a sequence W = (x = x1 , x2 , x3 , . . . , xn ) of length n such that hxi , xi+1 i ∈ E for all i = 1, . . . , N and also that xi = xj implies i = j. Given a configuration ω of G, we say that W is open in ω if ω(hxi , xi+1 i) = 1 for all i = 1, . . . , N . Proposition 4.2. Given a configuration ω ∈ Ω, there is an open self-avoiding walk of length n starting at 0 for all n ∈ N if and only if |C0 (ω)| = ∞. Proof. If there is an open self-avoiding walk Wn of S length n starting at 0 for all n ∈ N, then each of the infinite number elements of n∈N Wn is certainly connected to 0. Conversely, if |C0 (ω)| = ∞, then for any n ∈ N the open cluster containing the origin must contain a point with a graph-theoretic distance of n from the origin, and the self-avoiding walk (which can be attained by erasing the cycles from an arbitrary path) from that point to the origin must have length at least n. 12 ALEXANDER DUNLAP Definition 4.3. Given Ω with product measure Pp , the percolation probability θ(p) is defined by θ(p) = Pp (|C0 | = ∞). Proposition 4.4. The percolation probability function θ : [0, 1] → [0, 1] is nondecreasing. Proof. The function f : Ω → R given by f (ω) = 1{|C0 )|=∞} (ω) is certainly nondecreasing, since a configuration with an infinite cluster containing the origin will continue to have an infinite cluster containing the origin if more edges are opened. Therefore, by Theorem 3.7, θ(p) = Ep [f ] is nondecreasing in p. Definition 4.5. The critical probability pc is defined by pc = sup(θ−1 (0)). Remark 4.6. By Proposition 4.4, θ(p) = 0 for all p < pc and θ(p) > 0 for all p > pc . It has been shown that θ(pc ) = 0 for d = 2 and d ≥ 19, and it is believed that this result holds for 3 ≤ d ≤ 18, but the latter statement has not been proven. Much of the remainder of this paper will be devoted to proving results about the nature of percolation when p < pc and when p > pc . However, some of these results would be vacuous if it turned out that pc = 0 or pc = 1. The next proposition and two theorems show that while d = 1 is an uninteresting case, we have 0 < pc < 1 whenever d ≥ 2. This establishes the existence of criticality in percolation: the behavior of the system changes abruptly at p = pc . Proposition 4.7. For d = 1, we have pc = 1. Proof. Let p < 1. For any ω ∈ Ω, we have |C0 (ω)| = ∞ if and only if either ω(hn − 1, ni) = 1 for all n ∈ N or ω(h−n + 1, −ni) = 1 for all n ∈ N. Both of these events have probability limn→∞ pn = 0, so θ(p) = 0. Theorem 4.8. For d ≥ 2, we have pc > 0. Proof. We show that pc > 0 by finding a p > 0 such that θ(p) = 0. By Proposition 4.2, |C0 (ω)| = ∞ if and only if for each n ∈ N, there is an open self-avoiding walk of length n starting at 0. Let Wn be the set of all self-avoiding walks of length n starting at 0, and let σn = | Wn |. Let Nn (ω) be the number of open self-avoiding T walks of length n in ω starting at 0. Then, θ(p) = Pp ( n∈N {Nn ≥ 1}), so for all n ∈ N, θ(p) ≤ Pp (Nn ≥ 1) = Ep [1Nn ≥1 ] ≤ Ep [Nn ] = Ep X 1{W W ∈ Wn open} = X Pp (W open) = σn pn . W ∈ Wn All that remains is to bound σn . The first step of a self-avoiding walk starting at the origin can be in any of 2d directions, and all subsequent steps can go in at most 2d − 1 directions, since a self-avoiding walk certainly cannot backtrack its previous 2d (p(2d − 1))n step. Therefore, σn ≤ 2d(2d − 1)n−1 , so θ(p) ≤ 2d(2d − 1)n−1 pn = 2d−1 1 for all n ∈ N. If p ∈ (0, 2d−1 ), this implies that θ(p) = 0, so pc ≥ 1/(2d−1) > 0. Theorem 4.9. For d ≥ 2, we have pc < 1. Proof. We must find a p < 1 so that θ(p) > 0. If θ(p) > 0 for d = 2, then θ(p) > 0 for all d ≥ 2. This is because we can define an embedding ` : L2 → Ld by `(x1 , x2 ) = (x1 , x2 , 0, . . . , 0) and mapping edges in L2 to corresponding edges in Ld . If there is an open cluster in `(L2 ) ⊂ Ld containing the origin (which, by the TOPICS IN PERCOLATION 13 properties of product measure, happens with probability θ(p) for d = 2), then there is certainly an open cluster in Ld containing the origin. Therefore, we can restrict our argument to d = 2, which allows us to use a planar duality argument to show the desired result. Note that, as in Remark 3.36, we have |C(ω)| < ∞ if and only if ωd has a connected “circuit” of closed edges encircling the origin in L2d . Let Qn be the set of all circuits of length n encircling the origin in L2d . If Q ∈ Qn , then Q must contain a node in the set {(k + 1/2, 1/2) | k ∈ {0, . . . , n − 1}}, which we can consider the “starting point” of Q. Then, M must have n edges, and there can be at most 4 (in all but the first case, no more than 3, but the better estimate is unnecessary) possibilities for the direction of each edge. Therefore, |Qn | ≤ n4n . If we define the random variable Mn by Mn (ω) = | {Q ∈ Qn | Q is closed in ω} |, we have ! ! ∞ ∞ [ X Pp (|C| < ∞) = Pp {Mn ≥ 1} = Pp Mn ≥ 1 = Ep 1(P∞ n=4 Mn ≥1) ≤ Ep n=4 ∞ X n=4 ! Mn n=4 = ∞ X X = ∞ X X Ep [1(M closed) ] n=4 M ∈Mn Pp (M closed) = n=4 M ∈Mn ∞ X |Qn |(1 − p)n ≤ n=4 ∞ X n(4(1 − p))n . n=4 By making p ∈ (0, 1) sufficiently close to 1, we can make this last sum less than 1, which is what we want. 4.2. Subcritical percolation: exponential decay. Percolation is subcritical when p < pc . In this case, there will almost never be an infinite cluster, and thus it is natural to ask what we can say about the size of the cluster C0 containing the origin. The chief result is that the probability of C0 extending to a box of radius n decays exponentially in n. Definition 4.10. Given p ∈ [0, 1], the mean cluster size of Ld is defined by χ(p) = Ep [|C0 |], the expectation of the size of the cluster containing the origin. Fact 4.11. If p > pc , then there is a nonzero probability of an infinite cluster, so χ(p) = ∞. An important result for establishing exponential decay, the proof[3] of which we omit, is the following theorem. Theorem 4.12. If p < pc , then χ(p) < ∞. We can think of Theorem 4.12 as a statement about the coincidence of the critical points of θ and of χ. Assuming this result, we prove the exponential decay of subcritical percolation. Theorem 4.13. If p < pc , then there is a κ(p) > 0 so that, for all n ≥ 1, we have Pp (0 ↔ ∂Λ(n)) ≤ e−nκ(p) . Proof. By Theorem 4.12, we have χ(p) = Ep [|C0 |] < ∞. 14 ALEXANDER DUNLAP Λ(n + m) Λ(n) x Figure 2. Illustration of the disjoint-paths argument that lets us use the BK inequality. We first show that we can bound Pp (0 ↔ ∂Λ(n + m)) by Kp (m)Pp (0 ↔ ∂Λ(n)) for a function Kp . Moreover, we can find an m so that Kp (m) < 1, which allows us to use the division algorithm to establish exponential decay. Let m, n ∈ N. In a configuration ω in which there is a path from 0 to ∂Λ(n + m), there are also disjoint paths from 0 to some x ∈ ∂Λ(n) and from x to the translated box boundary x + ∂Λ(m) (Figure 2). Because these paths are disjoint, we can find a subset F ⊂ E so that ωF ∈ {0 ↔ x} and ωE\F ∈ {x ↔ x + ∂Λ(m)}. Therefore, in the notation of Definition 3.13, we have [ {ω ↔ ∂Λ(n + m)} ⊆ [{0 ↔ x} ◦ {x ↔ x + ∂Λ(m)}], x∈∂Λ(n) so the BK inequality (Theorem 3.16) says that X Pp (0 ↔ ∂Λ(m + n)) ≤ Pp (0 ↔ x)Pp (x ↔ x + ∂Λ(m)) x∈∂Λ(n) X = Pp (0 ↔ x)Pp (0 ↔ ∂Λ(m)) x∈∂Λ(m) = Pp (0 ↔ ∂Λ(m)) X Pp (0 ↔ x) x∈∂Λ(m) = Pp (0 ↔ ∂Λ(m))Ep [| {x ∈ Λ(m) | 0 ↔ x} |]. We thus choose Kp (m) = Ep [| {x ∈ Λ(m) | 0 ↔ x} |], and we want to analyze Kp to find an M such that Kp (M ) < 1. In fact, summing over all m, we have ∞ ∞ X X X Kp (m) = Pp (0 ↔ x) m=0 m=0 = X x∈Zd x∈∂Λ(n) Pp (0 ↔ x) = Ep [|C0 |] = χ(p) < ∞ TOPICS IN PERCOLATION 15 P∞ by hypothesis. Thus, since n=0 Kp (m) converges, limm→∞ Kp (m) = 0, and in particular, there is an M ∈ N so that Kp (M ) < 1, which we fix for the remainder of the proof. Let n ∈ N. By the division algorithm, we have q, r ∈ N (with 0 ≤ r < M ) so that n = qM + r. Since a path from 0 to ∂Λ(n) = ∂Λ(qM + r) certainly intersects ∂Λ(qM ), we have Pp (0 ↔ ∂Λ(n)) ≤ Pp (0 ↔ ∂Λ(qM )) ≤ Kp (m)Pp (0 ↔ ∂Λ((q − 1)M ) .. . ≤ (Kp (m))q = eq log(Kp (m)) . Since Kp (m) < 1, we have log(Kp (m)) < 0. Also, since n < (q + 1)M , we have that 1 1 − n1 ) ≥ n( M − M1+1 ) for all n ≥ M + 1. Therefore, letting q > −1 + n/M = n( M 1 1 0 − log(Kp (M )), κ (p) = − M M +1 we have κ0 (p) > 0 and q log(Kp (m)) < −nκ0 (p) 0 for all n ≥ M + 1, which implies that Pp (0 ↔ ∂Λ(n)) < e−nκ (p) for all n ≥ M + 1. Since there are only a finite number of n ≤ M , we can choose an S so that Pp (0 ↔ ∂Λ(n)) ≤ e−nS for all n ≤ M , and then letting κ(p) = max{κ0 (p), S} gives the desired inequality. 4.3. Supercritical percolation: uniqueness of the infinite cluster. If p > pc , then there is a nonzero probability of an infinite cluster containing the origin. By the zero-one law (Theorem 3.28), for a given p there is either almost surely or almost never an infinite cluster. Thus, if p > pc , there is almost surely an infinite cluster. The goal of this section is to show that this infinite cluster is almost surely unique. We first must establish a technical result regarding the number of ways of partitioning a set, which we will use in the proof of the theorem to establish a relationship between special points in the interior of a box and points on the boundary of the box. Definition 1. (1)4.14. Let Y be a set. A partition of Y is a collection P = P , P (2) , P (3) of three nonempty, disjoint subsets of Y such that P (1) ∪ P (2) ∪ P (3) = Y . (Note that we consider two partitions of Y to be equal if they consist of the same three subsets of Y , regardless of the ordering of the three subsets.) 2. Two partitions P and Q of a set Y are compatible if there are orderings of P and Q so that P (1) ⊇ Q(2) ∪ Q(3) . 3. A collection of partitions is compatible if the partitions are pairwise compatible. Lemma 4.15. Suppose that S = {Si }i is a compatible collection of partitions of a set Y . Then |S| ≤ |Y | − 2. 16 ALEXANDER DUNLAP Proof. We proceed by induction on n = |Y |. Because elements of a partition must be nonempty, there can only be one partition of Y if n = 3, so the lemma holds in this case. For the rest of the proof, we will fix n ≥ 4 and assume that the lemma holds for all Z with |Z| < n. Let S ∈ S be an arbitrary partition. Fix an ordering S = S (1) , S (2) , S (3) of the elements of S. Because S is compatible, for each T ∈ S \ S there is an ordering of T and a unique i ∈ {1, 2, 3} so that S (i) ⊇ T (2) ∪ T (3) ; therefore, we can write S \ S as the disjoint union S = S(1) ∪ S(2) ∪ S(3) , where T ∈ S(i) whenever there is an ordering of T such that S (i) ⊇ T (2) ∪ T (3) . Fix i ∈ {1, 2, 3}. Let Zi = S (i) ∪ {∆}, where ∆ is an arbitrary object (not an element of Y ). Note that |Zi | = |S (i) | + 1 ≤ n − 1 < n, so there can be no more than |S (i )| − 1 partitions of Zi by the inductive hypothesis. If T ∈ S(i) , then we can define a partition T 0 of Zi by n o T 0 = (T (1) ∩ S (i) ) ∪ {∆}, T (2) , T (3) . It is clear that the map that takes T to T 0 is injective. Thus, since there can be at most |S (i )| − 1 partitions of Zi , we have that |S(i) | ≤ |S (i )| − 1. This holds for all i ∈ {1, 2, 3}, so |S| = |S(1) | + |S(2) | + |S(3) | + 1 ≤ (|S (1) | − 1) + (|S (2) | − 1) + (|S (3) | − 1) + 1 = n − 2. We can now prove the main theorem of this section. Theorem 4.16 (Uniqueness of the infinite cluster). If p > pc , then there is almost surely exactly one infinite cluster. In other words, if N (ω) is the number of infinite clusters in a configuration ω, then Pp (N (ω) = 1 : ω ∈ Ω) = 1. Proof. We will use the notation ω F and ωF from 3.3. be the diamond of radius n around the origin in Ld , so V (S(n)) = Let dS(n) v ∈ L kvk1 ≤ n and E(S(n)) is the set all edges between adjacent elements of V (S(n)). For any k ∈ N, the event Ek = {N (ω) = k} is translation-invariant, so by the zero-one law (Theorem 3.28), we have Pp (Ek ) ∈ {0, 1}. Thus, there is a k ∈ Z≥0 ∪ {∞} so that Pp (Ek ) = 1, and our goal is to show that k = 1. By the assumption that p > pc , we know that k ≥ 1. Suppose first that k ∈Z. For any natural number n, both of the two events On = ω ∈ Ω ω = ω S(n) and Cn = ω ∈ Ω ω = ωS(n) are cylinders and thus have nonzero measure under Pp . Let NS(n) (ω) be the number of distinct infinite open clusters in ω that intersect S(n). Note that closing a finite number of edges can only increase the number of infinite clusters intersecting S(n), so NS(n) (ω) ≤ NS(n) (ωS(n) ) for all n ∈ N, ω ∈ Ω. Suppose that Pp (N (ω) 6= k | ω ∈ On ) > 0. Then Pp (N (ω) 6= k) ≥ Pp (N (ω) 6= k and ω ∈ On ) = Pp (N (ω) 6= k | ω ∈ On )Pp (ω ∈ On ) > 0, contradicting the definition of k. Therefore (using an analogous argument for Cn ), Pp (N (ω) = k | ω ∈ On ) = Pp (N (ω) = k | ω ∈ Cn ) = 1, which implies that Pp (N (ω S(n) ) = k) = Pp (N (ωS(n) ) = k) = 1. The intersection of two almost-sure events is also almost-sure, so we have Pp (N (ω S(n) ) = k = N (ωS(n) )) = 1. TOPICS IN PERCOLATION 17 But if ω is such that N (ω S(n) ) = N (ωS(n) ), then ω can have no more than one open cluster intersecting S(n), since otherwise opening all of the vertices of ωS(n) (to form ω S(n) ) would decrease the total number of open clusters. Therefore, Pp (NS(n) (ω) ≥ 2) = 0. If ω ∈ Ω is such that N (ω) ≥2, then there is an n∈ N so that NS(n) (ω) ≥ 2, S so {ω ∈ Ω | N (ω) ≥ 2} = n∈N ω ∈ Ω NS(n) (ω) ≥ 2 . Therefore, by continuity of measures, Pp (N (ω) ≥ 2) = limn→∞ Pp (NS(n) (ω) ≥ 2) = 0, which implies that k = 1 if k ∈ Z. We now need only to dismiss the case k = ∞. Given a configuration ω ∈ Ω, we say that x ∈ Ld is a trifurcation of ω, and write Tx (ω), if the following three conditions hold: 1. x is a member of an infinite cluster of ω. 2. x is an endpoint of exactly three open edges e1 = hx, y1 i , e2 = hx, y2 i , e3 = hx, y3 i. 3. In the configuration ω{e1 ,e2 ,e3 } created by declaring the edges surrounding x to be closed, we have that y1 , y2 , y3 are members of three distinct infinite clusters. By translation-invariance of Pp , it is clear that Pp (Tx ) = Pp (T0 ) for all x ∈ Ld . Therefore, X X Pp (Tx ) = |S(n)|Pp (Tx ). Ep 1 Tx = x∈S(n) x∈S(n) We wish to show that, if k = ∞, the expectation Ep hP x∈S(n) i 1Tx grows in pro- portion to |S(n)| as n → ∞, which means we must prove that Pp (T 0 ) 6= 0. This hP i conclusion will lead to a contradiction, since we will show that Ep 1 x∈S(n) Tx in fact cannot grow faster than |∂S(n)|, which grows more slowly than |S(n)|. We assume that k = ∞, so Pp (N (ω) = ∞) = 1, which certainly implies that Pp (N (ω) ≥ 3) = 1. Given a configuration ω such that N (ω) ≥ 3, there is clearly an n ∈ N so that NS(n) (ω) ≥ 3, which implies that NS(n) (ωS(n) ) ≥ 3. Therefore, S {ω | N (ω) ≥ 3} ⊆ n∈N ω NS(n) (ωS(n) ) ≥ 3 , so 1 = Pp (N (ω) ≥ 3) ≤ lim Pp (NS(n) (ωS(n) ) ≥ 3) ≤ 1, n→∞ so limn→∞ Pp (NS(n) (ωS(n) ) ≥ 3) = 1, so there is an m ∈ N such that Pp (NS(m) (ωS(m) ) ≥ 3) ≥ 1/2, with 1/2 chosen as an arbitrary constant in (0, 1). Suppose ω is a configuration such that NS(m) (ωS(m) ) ≥ 3. Then there are three points x(ω), y(ω), z(ω) ∈ ∂S(m) so that x(ω), y(ω), z(ω) are in different infinite clusters of ω. It can be shown geometrically that there is a set of edges Eω ⊆ E(S(m)) connecting x(ω), y(ω), z(ω) through disjoint paths to the origin so that 0 Eω is a trifurcation of the configuration ωE(S(m))\E (which is given by taking all edges ω 18 ALEXANDER DUNLAP of Eω to be open and all other edges of S(m) to be closed). Now Eω Pp (T0 (ω)) ≥ Pp NS(m) (ωS(m) ) ≥ 3 and ω = ωE(S(m))\E ω Eω = Pp NS(m) (ωS(m) ) ≥ 3 Pp ω = ωE(S(m))\E N (ω ) ≥ 3 S(m) S(m) ω 1 Eω ≥ Pp ω = ωE(S(m))\E NS(m) (ωS(m) ) ≥ 3 ω 2 > 0, o n Eω is a cylinder event. since ω ω = ωE(S(m))\E ω hP i Thus, we can write Ep = K|S(n)|, where K = Pp (Tx ) > 0. x∈S(n) 1Tx However, we will useP Lemma 4.15 to show that this is absurd, because for any ω ∈ Ω, we must have x∈S(n) 1Tx (ω) < |∂S(n)|. Let ω ∈ Ω and n ∈ N. Fix an infinite cluster C that intersects S(n). Let UC = ∂S(n) ∩ C. If x ∈ C ∩ S(n) is a trifurcation of ω, then closing the edges (3) (2) (1) around x separates C into Cx , Cx , Cx and thus gives o n three infinite clusters (i) (i) of UC . Given two trifurcations rise to a partition Px = Px = Cx ∩ UC i=1,2,3 (1) (1) x, x0 ∈ C∩S(n) of ω, ordering Px and Px0 so that Px 3 x0 and Px0 3 x ensures that (3) (2) (1) Px ⊇ Px0 ∪ Px0 . Thus, in the language of Definition 4.14, the class of partitions P = {Px | x ∈ C ∩ S(n) and x is a trifurcation of ω} is compatible, so |P| < |UC | − 2 by Lemma 4.15. Moreover, P by the definition of trifurcation, the map that takes x to Px is a bijection, so x∈S(n)∩C 1Tx (ω) ≤ |UC | − 2 < |U |. Summing over all P clusters C, we have that x∈S(n) 1Tx (ω) < |∂S(n)| since the UC s are disjoint. But hP i this means that for large n, we cannot possibly have Ep x∈S(n) 1Tx = K|S(n)|, since |S(n)| grows in proportion to nd while |∂S(n)| grows in proportion to nd−1 . This dismisses the case k = ∞, so we have proved that k = 1. 4.4. The critical value in two dimensions. In general, it is very difficult to compute exactly the critical probability pc for a given lattice. For the two-dimensional square lattice L2 , however, we can use planar duality and the results of the previous two sections to prove that pc = 1/2. We use separate arguments to show that pc ≥ 1/2 and that pc ≤ 1/2. Theorem 4.17. On L2 , we have pc ≥ 1/2. Proof. Let p = 1/2. Suppose that pc < 1/2, so θ(1/2) > 0. For each n ∈ N, define the event An = {∂Λ(n) ↔ ∞}, and note that the An s form an increasing sequence. Since p > pc , there is almost surely an infinite cluster in any configuration, S and the box Λ(n) will intersect this infinite cluster if n is sufficiently large, so n An = Ω. Therefore, limn→∞ P1/2 (An ) = 1, so there is an N ∈ N so that (4.18) P1/2 (∂Λ(n) ↔ ∞) = P1/2 (An ) ≥ 1 − (1/8)4 for all n ≥ N . Fix n = N + 1. Let AN , AS , AE , and AW be the events that the north, south, east, and west sides of Λ(n), respectively, are members of infinite open clusters. Note that these events have equal probability by symmetry. We have that P1/2 (Λ(n) 6↔ ∞) = P1/2 ((AN )c ∩ (AS )c ∩ (AE )c ∩ (AW )c ) ≥ [P1/2 ((AN )c )]4 TOPICS IN PERCOLATION 19 by Corollary 3.11 (since (AN )c , (AS )c , (AE )c , (AW )c are decreasing events), so P1/2 ((AN )c ) ≤ [1 − P1/2 (Λ(n) ↔ ∞)]1/4 ≤ 1/8 by (4.18). Consider next the dual box Λ(n)d in L2d with vertex set [−n, n − 1]2 + (1/2, 1/2). S E W Let AN d , Ad , Ad , Ad be the events that the north, south, east, and west sides of Λ(n)d are members of infinite closed clusters in L2d . The dual box Λ(n)d has side c length n − 1 = N , so (4.18) applies to Λ(n)d as well, and thus P1/2 ((AN d ) ) ≤ 1/8. N S E W Let A = A ∩ A ∩ Ad ∩ Ad . If A occurs, then by Theorem 4.16, there must be an open path in L2 joining the north and south sides of A and a closed path in L2d joining the east and west sides of A. But this is impossible since it would require the paths to cross. Therefore, P1/2 (A) = 0. On the other hand, we have c W c P1/2 (Ac ) = P1/2 ((AN )c ∪ (AS )c ∪ (AE d ) ∪ (Ad ) ) c W c ≤ P1/2 ((AN )c ) + P1/2 ((AS )c ) + P1/2 ((AE d ) ) + P1/2 ((Ad ) ≤ 1/2, so P1/2 (A) ≥ 1/2, contradicting the fact proven above that P1/2 (A) = 0. Therefore, pc ≥ 1/2, as desired. Theorem 4.19. On L2 , we have pc ≤ 1/2. Proof. Let p = 1/2. Suppose for the sake of contradiction that pc > 1/2. By Theorem 4.13, there is a κ(1/2) > 0 so that P1/2 (0 ↔ ∂Λ(n)) ≤ exp(−nκ(1/2)) for all n ∈ N. Let B(n) be the modified box given by removing the edges on the east and west sides of the box given by [0, n + 1] × [0, n]. Let An be the event that there is an open path from the west side of B(n) to the east side of B(n). We consider the dual box B(n)d in L2d generated by [0, n] × [0, n + 1] + (1/2, −1/2), with the edges on the north and south sides removed. Let A0n be the event that there is a closed path from the north side of B(n)d to the south side of B(n)d . It is clear that P1/2 (An ) = P1/2 (A0n ) and that exactly one of An and A0n must occur (see Remark 3.36), so (4.20) P1/2 (An ) = 1/2. However, we also have An ⊆ n+1 [ {(0, i) ↔ i + Λ(n)} , i=1 which implies P1/2 (An ) ≤ P1/2 n+1 [ ! {(0, i) ↔ i + Λ(n)} i=1 ≤ n+1 X Pp ((0, i) ↔ i + Λ(n)) i=1 = (n + 1)Pp (0 ↔ Λ(n)) ≤ (n + 1) exp(−κ(1/2)n), which will eventually fall below 1/2 as n grows large, contradicting (4.20). 20 ALEXANDER DUNLAP Acknowledgments. I would like to thank my mentor, Mohammad Abbas Rezaei, for his valuable guidance throughout this project, and Professor Peter May for his work organizing and directing this REU. References [1] Patrick Billingsley. Probability and Measure. Second edition. John Wiley and Sons. 1986. [2] R. M. Burton and M. Keane. Density and Uniqueness in Percolation. Communications in Mathematical Physics 121, 501-505. 1989. [3] Geoffrey Grimmett. Percolation. Second edition. Springer. 1999. [4] Geoffrey Grimmett. Probability on Graphs: Random Processes on Graphs and Lattices. Statistical Laboratory, University of Cambridge. 2010.