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Plaquettes, Spheres, and Entanglement

The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e>= \mu^-2 where \mu is the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.


Introduction and results
The plaquette percolation model is a natural dual to bond percolation in two and more dimensions. Let Z d be the integer lattice; elements of Z d are called sites. For any site z, let Q(z) := [− 1 2 , 1 2 ] d + z be the topologically closed unit d-cube centred at z. A plaquette is any topologically closed unit (d − 1)-cube in R d that is a face of some Q(z) for z ∈ Z d . Let Π d be the set of all plaquettes. For a set of plaquettes S ⊆ Π d , we write [S] := π∈S π for the associated subset of R d . In the plaquette percolation model with parameter p ∈ [0, 1], each plaquette of Π d is declared occupied with probability p, otherwise unoccupied, with different plaquettes receiving independent states; the associated probability measure is denoted P p .
The ℓ s -norm on R d is denoted · s . A sphere of R d is a simplicial complex, embedded in R d , that is homeomorphic to the unit sphere {x ∈ R d : x 2 = 1}. By the generalized Schönflies theorem, the complement in R d of a sphere has a bounded and an unbounded pathcomponent, which we call respectively its inside and outside. The (ℓ 1 -)radius of a set A ⊆ R d with respect to the origin 0 ∈ R d is rad A = rad 0 A := sup{ x 1 : x ∈ A}.
Let µ d be the connective constant of Z d , given as where σ(k) is the number of (ℓ 1 -)nearest-neighbour self-avoiding paths from the origin with length k in Z d ; it is a straightforward observation that µ d ∈ [d, 2d − 1], and stronger bounds may be found, for example, in [20]. Theorem 1. Let d ≥ 2, and consider the plaquette percolation model. If p < µ −2 d then almost surely there exists a finite set S of unoccupied plaquettes whose union [S] is a sphere with 0 in its inside. Moreover S may be chosen so that for any α ∈ (µ d p, 1), and some C = C(p, d, α) < ∞.
When d = 2, the first assertion of Theorem 1 amounts to the well known fact that there exists a suitable circuit of unoccupied bonds of the dual lattice (see, e.g., [8]). The result is more subtle in higher dimensions.
When d = 3, Theorem 1 has an application to entanglement percolation, which we explain next. Define a bond to be the topologically closed line segment in R d joining any two sites x, y ∈ Z d with x−y 1 = 1. Let L d be the set of all bonds. In the bond percolation model, each bond is declared occupied with probability p, otherwise unoccupied, with the states of different bonds being independent. For a set of bonds K, write [K] := e∈K e. We say that K contains a site We say that a sphere Z ⊂ R d separates a set A ⊂ R d if A intersects both the inside and the outside of Z, but not Z itself. (We write A ⊂ B if A ⊆ B and A = B.) Let d = 3. We say that a set of bonds K ⊆ L 3 is 1-entangled if no sphere of R 3 separates [K]. The idea of this definition is that a 1-entangled set of bonds, if made of string or elastic, cannot be continuously "pulled apart". Any connected set of bonds is evidently 1-entangled. The simplest disconnected set that is 1-entangled consists of two linked loops. The prefix "1" reflects the fact that other natural definitions of entanglement are possible; see [10] and the discussion in Section 2 for more details. Entanglement of sets of bonds is intrinsically a three-dimensional issue, and therefore we shall always take d = 3 when discussing it.
In the bond percolation model in d = 3, let η 1 (p) be the probability that there exists an infinite 1-entangled set of occupied bonds containing the origin 0, and define the 1-entanglement critical probability p 1 e := sup{p : η 1 (p) = 0}. The maximal 1-entangled set of occupied bonds containing a site x is called the 1-entanglement cluster at x.
The connective constant of Z 3 satisfies the rigorous bound µ 3 ≤ 4.7387 (see [20]). Therefore, Corollary 2 gives This is a significant improvement on the previous best lower bound of [4], namely p 1 e > 1/597, which in turn substantially improved the first non-zero lower bound, p 1 e ≥ 1/15616, proved in [14]. In each case, the improvement is by a factor of approximately 26.
In Section 2 we discuss some history and background to our results. In Section 3 we prove Theorem 1 and Corollary 2. If p satisfies the stronger bound p < (2d − 1) −2 , we shall see that our methods yield versions of these results with explicit formulae for the constants C and α. In Section 4 we consider the critical value of p associated with the event in Theorem 1, and its relationship to certain other critical values.
2. Remarks 2.1. Duality. To each bond e ∈ L d there corresponds a unique plaquette π(e) ∈ Π d that intersects e. It is therefore natural to couple the bond and plaquette percolation models with common parameter p in such a way that π(e) is occupied if and only if e is occupied. If p is less than the critical probability p c for standard bond percolation (see, e.g., [8]), the connected component of occupied bonds at the origin is almost surely finite, and it is a straightforward consequence that there exists a finite set of unoccupied plaquettes whose union encloses the origin (i.e., the origin lies in some bounded component of its complement). Indeed, such plaquettes may be chosen so as to form a 'surface' enclosing the origin (although precise definition of such an object requires care). However, such a surface might be homeomorphic to a torus, or some other topological space. It is a key point of Theorem 1 that the surface [S] is a sphere.

2.2.
Entanglement. Entanglement in three-dimensional percolation was first studied, in a partly non-rigorous way, in [17] (some interesting remarks on the subject appeared earlier in [2]). The rigorous theory was systematically developed in [10], and further rigorous results appear in [3,13,14,15,16]. A discussion of physical applications of entanglement percolation may be found in [4].
As mentioned in Section 1, there are several (non-equivalent) ways of defining the property of entanglement for infinite graphs. One of these, namely 1-entanglement, was presented in that section, and a second follows next. We say that a set of bonds K ⊆ L 3 is 0-entangled if every finite subset of K is contained in some finite 1-entangled subset of K. It was shown in [10] that the notions of 0-entanglement and 1-entanglement are extremal members of a certain class of natural candidate definitions, called entanglement systems, and furthermore that these two entanglement systems correspond (respectively) in a natural way to free and wired boundary conditions.
By combining inequalities of [3,10,14], we find that where p 0 e , p 1 e , and p E e are the critical probabilities for 0-entanglement, 1-entanglement, and for an arbitrary entanglement system E, respectively. The inequality p 0 e ≤ p c reflects the straightforward fact that every connected set of bonds is 0-entangled. It was strengthened to the strict inequality p 0 e < p c in [3,16]. In [17] it was argued on the basis of numerical evidence that p c − p e ≈ 1.8 × 10 −7 , for a certain notion of 'entanglement critical probability' p e . It is an open question to decide whether or not p 0 e = p 1 e .
2.3. Spheres, lower bounds, and exponential decay. The inequality p 1 e > 0 expresses the fact that, for a sufficiently small density p of occupied bonds, there is no infinite entangled set of bonds. Prior to the current paper, proofs of this seemingly obvious statement have been very involved.
The proof in [14] employs topological arguments to show that, for p < 1/15616, almost surely the origin is enclosed by a sphere that intersects no occupied bond. The argument is specific to three dimensions, and does not resolve the question of the possible existence of a sphere of unoccupied plaquettes enclosing the origin. (See [10] for more on the distinction between spheres intersecting no occupied bond, and spheres of unoccupied plaquettes.) In [10], related arguments are used to show that, for sufficiently small p, the radius R of the 1-entanglement cluster at the origin has 'near-exponential' tail decay in that P(R > r) < exp −cr/ log · · · log r for an arbitrary iterate of the logarithm, and for some c > 0 depending on p and the number of logarithms.
In the recent paper [4], the above results are substantially improved in several respects. The lower bound on the critical point is improved to approximately p 1 e > 1/597, and it is proved also that the radius of the 1-entanglement cluster at the origin has exponential tail decay for p below the same value. The key innovation is a proof of an exponential upper bound on the number of possible 1-entangled sets of N bonds containing the origin, thereby answering a question posed in [10]. The method of proof is very different from that of [14].
In the current article, we improve the proofs mentioned above in several regards. The lower bound on the critical point is further improved to approximately p 1 e > 1/23, and we establish exponential decay of the 1-entanglement cluster-radius at the origin for p below this value. We prove the existence of a sphere of unoccupied plaquettes enclosing the origin (rather than just a sphere intersecting no occupied bond), and we do so for all dimensions. Finally, our proofs are very simple. Our methods do not appear to imply the key result of [4] mentioned above, namely the exponential bound for the number of entangled sets containing the origin.

Proofs
The geometric lemma below is the key to our construction of a sphere. For x, y ∈ R d , write y x if for each i = 1, 2, . . . , d we have |y i | ≤ |x i | and x i y i ≥ 0 (equivalently, y lies in the closed cuboid with opposite corners at 0 and x). For a bond e ∈ L d , recall that π(e) ∈ Π d is the unique plaquette that intersects it.
S := π(e) : e is a bond with exactly one endvertex in K .
Then [S] is a sphere with 0 in its inside.
Proof. Let U := x∈K Q(x) be the union of the unit cubes corresponding to K. Note that [S] is the topological boundary of U in R d . Let Σ := {z ∈ R d : z 2 = 1} be the unit sphere; we will give an explicit homeomorphism between [S] and Σ.
We claim first that U is strictly star-shaped, which is to say: if x ∈ U then the line segment {αx : α ∈ [0, 1)} is a subset of the topological interior of U (i.e., of U \ [S]). To check this, suppose without loss of generality that x is in the non-negative orthant [0, ∞) d . By the given properties of K, the open cuboid is a subset of U (here it is important that the origin is at the centre of a cube, rather than on a boundary); now, H clearly contains the aforementioned line segment, and the claim is proved. In the above, x ∨ y denotes the maximum of x and y. It follows that, for any point z ∈ Σ, the ray {αz : α ∈ [0, ∞)} has exactly one point of intersection with [S]. Denote this point of intersection f (z). Clearly f is a bijection from Σ to [S]; we must prove that it is a homeomorphism. Since Σ and [S] are compact metric spaces, it suffices to express them as finite unions Σ = r j=1 X j and [S] = r j=1 Y j , where the X j and Y j are compact, and such that f restricted to X j is a homeomorphism from X j to Y j , for each j. This is achieved by taking {Y 1 , . . . , Y r } equal to the set of plaquettes S. Any plaquette in Π d is a subset of some (d − 1)-dimensional affine subspace (hyperplane) of R d that does not pass through 0 (here the offset of 1 2 is again important) and it is elementary to check that the projection through 0 from such a subspace to Σ is a homeomorphism to its image in Σ.
Finally, we must check that 0 lies in the inside of the sphere [S]; this is clear because 0 ∈ U \ [S], and any unbounded path in R d starting from 0 must leave U at some point, and thus must intersect [S].
The next lemma is closely related to a recent result on random surfaces in [7]. Consider the bond percolation model with parameter p on L d . By a path we mean a self-avoiding path comprising sites in Z d and bonds in L d . Recall that σ(k) is the number of paths starting at the origin and having k edges, and Lemma 4. Let K be the set of sites u ∈ Z d for which there is a good path from 0 to u. If p < µ −2 d then K is a.s. finite, and moreover, for any α ∈ (µ d p, 1), and some Proof. Let N(r) be the number of good paths that start at 0 and end on the ℓ 1 -sphere {x ∈ Z d : x 1 = r}. Then, For any path π with vertices 0 = v 0 , v 1 , . . . , v k = u with u 1 = r, let be respectively the number of steps Away from, and Back towards, the origin 0. Note that k = A + B, and u 1 = A − B. Thus, the probability that π is good is p A , while the number of possible paths having given values of A and B is at most σ(A + B). Hence, For any ǫ > 0, we have σ(k) ≤ (µ + ǫ) k for k sufficiently large, where µ = µ d . Therefore, (3) is at most B≥0 (µ + ǫ) 2B+r p B+r = [(µ + ǫ)p] r 1 − (µ + ǫ) 2 p provided (µ + ǫ) 2 p < 1 and r is sufficiently large; thus we can choose C ′ so that the required bound (2) holds for all r ≥ 0.
Proof of Theorem 1. Couple the bond and plaquette percolation models by considering e to be occupied if and only π(e) is occupied. Let p < µ −2 d and let K be the random set of sites u for which there exists a good path from 0 to u. By Lemma 4, K satisfies (2) with the given constants α, C ′ .
Since a good path may always be extended by a step towards the origin (provided the new site is not already in the path), K satisfies the condition that x ∈ K and y x imply y ∈ K. Therefore, Proposition 3 applies. If e is a bond with exactly one endvertex in K, then by the definition of a good path, it is the end closer to 0 that is in K, and e must be unoccupied. Therefore, all plaquettes in the set S in (1) are unoccupied. Finally, the tail bound in (2) implies the bound in Theorem 1 because rad [S] ≤ rad K + d/2.
Proof of Corollary 2. Couple the bond and plaquette models as usual. Let p < µ −2 3 , and let S be the set of plaquettes from Theorem 1. The sphere [S] intersects no occupied bond and has 0 in its inside, hence it has [E] in its inside.

Critical values
We consider next the critical value of p for the event of Theorem 1, namely the event that there exists a finite set S of unoccupied plaquettes whose union [S] is a sphere with 0 in its inside. In so doing, we shall make use of the definition of a good path from Section 3. We shall frequently regard Z d as a graph with bond-set L d . A directed path of Z d is called oriented if every step is in the direction of increasing coordinate-value.
Let d ≥ 2, and (as after Theorem 1) declare a bond of L d to be occupied with probability p. We write v → oo w (respectively, v → oo ∞) if there exists an oriented occupied path from v to w (respectively, an infinite oriented occupied path from v). Let p c (d) denote the critical probability of oriented percolation on Z d .
Let θ g (p) be the probability that there exists an infinite good path beginning at the origin. Since θ g is a non-decreasing function, we define a critical value p g := sup{p : θ g (p) = 0}. Note that (4) µ −2 d ≤ p g ≤ p c (d). That µ −2 d ≤ p g follows by Theorem 1; the second inequality p g ≤ p c (d) holds since every oriented occupied path from 0 is necessarily good.
For  By the definition of admissibility, and indeed the associated events are equal. If x → a y by an admissible path using only sites of H + except possibly for the first site x, we write x → H a y. We write x → H a ∞ if x is the endvertex of some infinite admissible path, all of whose vertices except possibly x lie in H + . Let θ H a (p) = P p (0 → H a ∞), with associated critical value p H a := sup{p : θ H a (p) = 0}. Also define the orthant Let θ K a (p) be the probability of an infinite admissible path in K starting at 0, and let p K a be the associated critical value.
Since every admissible path in the orthant K is good, we have that p g ≤ p K a , and therefore p g ≤ p H a by Theorem 5. We pose two questions. Open Question 1. For d ≥ 3, is it the case that p a = p H a ? Open Question 2. For d ≥ 3, is it the case that p g = p H a ? These matters are resolved as follows when d = 2.
Theorem 6. For d = 2 we have p g = p H a = p a = 1 − p c (2). In advance of the proofs, we present a brief discussion of Open Question 1 above. By Proposition 7(b) below, one has that If one could prove that x∈Hn P p (0 → H a x) → 0 as n → ∞, whenever p < p H a , it would follow by Theorem 5 that p a = p H a . This is similar to the percolation problem solved by Aizenman-Barsky and Menshikov, [1,18,19] (see also [8,Chap. 5]). It seems possible to adapt Menshikov's proof to prove an exponential-decay theorem for admissible paths (as in [11,Thm 2]), but perhaps not for admissible connections restricted to H.
The proofs of the two theorems above will make use of the next proposition. Let d ≥ 2 and 0 ≤ a < b ≤ ∞. Define the cone K a,b to be the set of sites x = (x 1 , x 2 , . . . , x d ) ∈ Z d satisfying: (6) x 1 ≥ 0, and ax 1 ≤ x j ≤ bx 1 for j = 2, 3, . . . , d.
The subgraph of Z d induced by K a,b comprises a unique infinite component, denoted I(K a,b ), together with a finite number of finite components. This may be seen as follows. The set of x ∈ K a,b with x 1 = k is the set S k of all integer-vectors belonging to {k} × [ak, bk] d−1 . Each S k is connected, and, for sufficiently large k, S k is connected to S k+1 . and for all v ∈ I(K a,b ), (b) If p > p H a , then P p (K a,b contains some infinite admissible path) = 1, and for all v ∈ I(K a,b ), The remainder of this section is set out as follows. First, we deduce Theorems 5 and 6 from Proposition 7. The proof of Proposition 7 is not presented in this paper, since it would be long and would repeat many constructions found elsewhere. Instead, this section ends with some comments concerning that proof.
Proof of Theorem 5. As noted before the statement of Theorem 5, K ⊆ H + ∪ {0}, whence p H a ≤ p K a . Let p > p H a . By Proposition 7(b), we have θ K a (p) > 0, so that p ≥ p K a . Therefore, p H a = p K a . There is more than one way of showing p a ≤ p H a , of which the following is one. Let d ≥ 3; the proof is similar when d = 2. Let p > p H a and fix 0 < a ≤ 1 < b < ∞ arbitrarily. By Proposition 7(b), K a,b contains a.s. some infinite admissible path π. Any infinite path π in K a,b has the property that, for all x ∈ Z d , there exists z ∈ π with x ≤ z (in that x i ≤ z i for every coordinate i). By (5), θ a (p) > 0, so that p ≥ p a and p H a ≥ p a as claimed. Proof of Theorem 6. We shall make extensive use of two-dimensional duality. We call the graph Z 2 the primal lattice, and we call the shifted graph Z 2 + ( 1 2 , 1 2 ) the dual lattice. Thus, the dual bonds are precisely the plaquettes of Π 2 . Recall that a dual bond is declared occupied if and only if the primal bond that crosses it is occupied. For consistency with standard terminology, we now call a dual bond open if and only if it is unoccupied (so a dual bond is open with probability q := 1 − p). We assign directions to dual bonds as follows: a horizontal dual bond is directed from left to right, and a vertical bond from top to bottom.
Consider the sets  Figure 1). If 1 − p < p c (respectively, 1 − p > p c ) the latter occurs with strictly positive probability (respectively, probability 0). Therefore, p H a = 1 − p c .
We show next that p g = p H a . Since p g ≤ p H a , it suffices to show that p g ≥ 1 − p c . Let p < 1 − p c , so that 1 − p > p c . We shall prove the required inequality p ≤ p g . Define the set of dual sites n − x ≤ y < ∞, x ≤ n}. By Proposition 7(a), P p (V n ) ≥ δ for some δ = δ(p) > 0 not depending on n. By reversing the directions of all dual bonds, we see that P p (W n ) = P p (V n ). On the event V n ∩ W n , there exists a directed open dual path of B(n) \ B( 1 3 n) from v n to w n , and hence, by the Harris-FKG inequality, as required for (7).
By considering corresponding events in the other three quadrants of Z 2 (with appropriately chosen bond-orientations), we conclude that each annulus of R 2 with inner (respectively, outer) ℓ ∞ -radius 1 3 n (respectively, n + 1 2 ) contains, with probability at least (1 − p) 4 β 4 , a dual cycle blocking good paths from the origin. It follows that p ≤ p g as required.
Finally we show that p a = 1 − p c . Since p a ≤ p H a by Theorem 5, we have only to show that p a ≥ 1 − p c . This follows by (5) and the fact that, when q = 1−p > p c , there exists P p -a.s. a doubly-infinite directed open dual path intersecting the positive y-axis. Here is a proof of the latter assertion. Let ψ(q) be the probability that there exists an infinite oriented path from the origin in oriented percolation with density q. By reversing the arrows in the fourth quadrant, the probability that 0 lies in a doubly-infinite directed open path of Z 2 is ψ 2 . The event J := {there exists a doubly-infinite open dual path} is a zero-one event and P p (J) ≥ ψ 2 , so that P p (J) = 1. Let J + (respectively, J − ) be the event that such a path exists and intersects the positive (respectively, the non-positive) y-axis. By reversing the directions of bonds, we have that P p (J + ) = P p (J − ). By the Harris-FKG inequality, so that P p (J + ) = 1.
Proposition 7 may be proved by the dynamic-renormalization arguments developed for percolation in [5,12], for the contact model in [6], and elaborated for directed percolation in [9]. An account of dynamic renormalization for percolation may be found in [8]. The proof of Proposition 7 is omitted, since it requires no novelty beyond the above works, but extensive duplication of material therein. The reader is directed mainly to [9], since the present proposition involves a model in which the edge-orientations are important. The method yields substantially more than the statement of the proposition, but this is not developed here.
We highlight several specific aspects of the proof of Proposition 7, since they involve minor variations on the method of [9]. First, since Proposition 7 is concerned with admissible paths in sub-cones of the orthant K, we require a straightforward fact about oriented percolation on sub-cones of Z 2 , namely that the associated critical probability is strictly less than 1. This weak statement leads via renormalization to the stronger Proposition 7. The following lemma is slightly stronger than the minimum needed for the proof of Proposition 7, and the proof is given at the end of this section. Lemma 8. Let 0 ≤ a < b ≤ ∞, and let K a,b be the cone of Z 2 containing all sites (x, y) with ax ≤ y ≤ bx and x ≥ 0. There exists ǫ = ǫ a,b > 0 such that: if p > 1 − ǫ, P p (K a,b contains some infinite oriented occupied path) = 1, and for all v ∈ I(K a,b ), Our second remark is concerned with part (a) of Proposition 7. It is proved in [9] that, when p > p c (d), there exists a.s. an infinite oriented occupied path within some two-dimensional slab of Z d of the form S × [−A, A] d−2 for some A < ∞ and some sub-cone S of the orthant [0, ∞) 2 . In order to build such a path within K a,b , one adapts the construction of [9] using the so-called 'steering' arguments of [5,6,8,12]. . Finally, the current proof uses the technique known as 'sprinkling', in a manner similar to that of the proofs presented in [12] and [8,Sect. 7.2].
Proof of Lemma 8. Let r = r 2 /r 1 , s = s 2 /s 1 be rationals (in their minimal representation with positive integers r i , s i ) satisfying a < r < s < b. Since r < s, we may choose a rational β/α such that s 1 r 1 < β α < s 2 r 2 .
We may consider I(K a,b ) as the subgraph of Z 2 induced by the corresponding site-set. Since π is finite, we may choose v ∈ I(K a,b ) such that v+π is a subgraph of I(K a,b ), and we consider the set v i,j := v+iR+jS, i, j ≥ 0, of sites of K a,b . Let α = p N where N = |π|. By the definition of K a,b , each v i,j + π is a subgraph of I(K a,b ), and we declare v i,j black if every bond in v i,j + π is open. Note by the construction of the set π that the states of different sites v i,j are independent.
If 1−α < (1− p c ) 2 , the set of black vertices dominates (stochastically) the set of sites w of a supercritical oriented percolation model with the property that both bonds directed away from w are open. The claims of the lemma follow by standard properties of oriented percolation.