1‐independent percolation on ℤ2×Kn

A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$ , the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$ . For an infinite connected graph H$$ H $$ , the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\in \left[0,1\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\left({\mathbb{Z}}^d\right) $$ tends to a limit in [12,1]$$ \left[\frac{1}{2},1\right] $$ as d→∞$$ d\to \infty $$ , and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \underset{n\to \infty }{\lim }{p}_{1,c}\left({\mathbb{Z}}^2\times {K}_n\right)=4-2\sqrt{3}=0.5358\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \omega \left(\log n\right) $$ . We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\sqrt{3} $$ .

] as → ∞, and they asked for the value of this limit. We make progress on a related problem by showing that lim n→∞ p 1,c (Z 2 × K n ) = 4 − 2 In fact, we show that the equality above remains true if the sequence of complete graphs K n is replaced by a sequence of weakly pseudorandom graphs on n vertices with average degree (log n). We conjecture the answer to Balister and Bollobás's question is also 4 − 2

Background
Percolation theory lies at the interface of probability theory, statistical physics and combinatorics. Its object of study is, roughly speaking, the connectivity properties of random subgraphs of infinite connected graphs, and in particular the points at which these undergo drastic transitions such as the emergence of infinite components. Since its inception in Oxford in the late 1950s, percolation theory has become a rich field of study (see e.g., the monographs [9,15,22]). One of the cornerstones of the discipline is the Harris-Kesten theorem [17,18], which states that if each edge of the integer square lattice In general, given an infinite connected graph H, determining the critical probability for independent bond percolation on H is a hard problem, with the answer known exactly only in a handful of cases. There is thus great interest in methods for rigorously estimating such critical probabilities. One of the most powerful and effective techniques for doing just that was developed by Balister et al. [5], and relies on comparing percolation processes with locally dependent bond percolation on Z 2 (to be more precise: 1-independent bond percolation; see below for a definition). The method of Balister, Bollobás and Walters has proved influential, and has been widely applied to obtain the best rigorous confidence interval estimates for the value of the critical parameter in a wide range of models, see, for example, [1, 3-8, 12, 13, 16, 24].
However, as noted by the authors of [5] and again by Balister and Bollobás [2] in 2012, locally dependent bond percolation is poorly understood. To quote from the latter work, "[given that] 1-independent percolation models have become a key tool in establishing bounds on critical probabilities [ … ], it is perhaps surprising that some of the most basic questions about 1-independent models are open." In particular, there is no known locally dependent analog of the Harris-Kesten theorem, nor even until now much of a sense of what the corresponding 1-independent critical probability ought to be. In this article, we contribute to the broader project initiated by Balister and Bollobás of addressing the gap in our knowledge about 1-independent bond percolation by making some first steps toward a 1-independent Harris-Kesten theorem. To state our results and place them in their proper context, we first need to give some definitions.
Let H = (V, E) be a graph. Given a probability measure on subsets of E, a -random graph H is a random spanning subgraph of H whose edge-set is chosen randomly from subsets of E according to the law given by . Each probability measure on subsets of E thus gives rise to a random graph model on the host graph H, and we use the two terms (probability measure on subsets of E/random graph model H on H) interchangeably. In this article we will be interested in random graph models where the state (present/absent) of edges is dependent only on the states of nearby edges. Recall that the graph distance between two subsets A, B ⊆ E is the length of the shortest path in H from an endpoint of an edge in A to an endpoint of an edge in B. So in particular if an edge in A shares a vertex with an edge in B, then the graph distance from A to B is zero, while if A and B are supported on disjoint vertex-sets, then the graph distance from A to B is at least one. Definition 1.1 (k-independence). A random graph model H on a host graph H is k-independent if whenever A, B are disjoint subsets of E(H) such that the graph distance between A and B is at least k, the random variables E(H ) ∩ A and E(H ) ∩ B are mutually independent. If H is k-independent, we say that the associated probability measure is a k-independent measure, or k-ipm, on H.
Let  k,≥p (H) denote the collection of all k-independent measures on E(H) in which each edge of H is included in H with probability at least p. We define  k,≤p (H) mutatis mutandis, and let  1,p (H) denote  k,≥p ∩  k,≤p -in other words  k,p is the collection of all k-ipm on H in which each edge of H is included in H with probability exactly p.
Observe that a 0-independent measure is what is known as a Bernoulli or product measure on E: each edge in E is included in H at random independently of all the others. We refer to such measures as independent measures. The collection  0,p (H) thus consists of a single measure, the p-random measure, in which each edge of H is included in the associated random graph with probability p, independently of all the other edges. When the host graph H is K n , the complete graph on n vertices, this gives rise to the celebrated Erdős-Rényi random graph model, while when H = Z 2 this is exactly the independent bond percolation model considered in the Harris-Kesten theorem.
In this article, we will focus instead on  1,≥p (H) and  1,p (H), whose probability measures allow for some local dependence between the edges. A simple and well-studied example of a model from  1,p (H) is given by site percolation: build a random spanning subgraph H site of H by assigning each vertex v ∈ V(H) a state S v independently at random, with S v = 1 with probability and S v = 0 otherwise, and including an edge uv ∈ E(H) in H site if and only if S u = S v = 1. Each edge in this random graph is open with probability p = 2 , and the model is clearly 1-independent since "randomness resides in the vertices," and so what happens inside two disjoint vertex sets is independent. More generally, any state-based model obtained by first assigning independent random states S v to vertices v ∈ V(H) and then adding an edge uv according to some deterministic or probabilistic rule depending only on the ordered pair (S u , S v ) will give rise to a 1-ipm on H. State-based models are a generalization of the probabilistic notion of a two-block factor, see [20] for details.
Given a 1-ipm on an infinite connected graph H, we say that percolates if H almost surely (i.e., with probability 1) contains an infinite connected component. 1 Definition 1.2. Given an infinite connected graph H, we define the 1-independent critical percolation probability for H to be Remark 1.3. Given ∈  1,≥p (H) we can obtain a random graph H from H by deleting each edge uv of H independently at random with probability 1−p∕ . Clearly H stochastically dominates (i.e., is a supergraph of) H and ∈  1,p (H). Thus the definition of p 1,c (H) above is unchanged if we replace  1,≥p (H) by  1,p (H). Balister et al. [5] devised a highly effective method for giving rigorous confidence interval results for critical parameters in percolation theory via comparison with 1-independent models on the square integer lattice Z 2 . Their method relies on estimating the probability of certain finite, bounded events (usually via Monte Carlo methods, whence the confidence intervals) and on bounds on the 1-independent critical probability p 1,c (Z 2 ). Work of Liggett et al. [20] on stochastic domination of independent models by 1-independent models implied p 1,c (Z 2 ) < 1. Balister et al. [5,Theorem 2] obtained the effective upper bound p 1,c (Z 2 ) < 0.8639 via a renormalization argument; this upper bound has not been improved since, and the authors of [5] noted "it would be of interest to give significantly better bounds for p 1,c (Z 2 ); unfortunately, we cannot even hazard a guess as to [its] value." The question of determining p 1,c (Z 2 ) was raised again by Balister and Bollobás [2,Question 2], who noted the difficulty of the problem: Balister and Bollobás [2] observed that a simple modification of site percolation due to to Newman shows that is the critical probability for site percolation in Z 2 . Since it is known that s ∈ [0.556, 0.679492] (see [26,27]), this shows that p 1,c (Z 2 ) ≥ 0.5062. Non-rigorous simulation-based estimates s ≈ 0.597246 [28] improve this to a non-rigorous lower bound of 0.5172. Recently, Day, Hancock and the first author gave significant improvements on these lower bounds. In [10,Theorem 7], they constructed measures based on an idea from the first author's PhD thesis [14,Theorem 62] showing that for any ∈ N, 3 for any host graph H satisfying what they call the finite 2-percolation property (see Section 3 for a formal definition), a family which includes the graphs Z 2 × K n for any n ∈ N. (Recall that the Cartesian product H × K n of a graph H with K n is the graph whose vertices are the pairs (v, i) ∈ V(H) × {1, 2, … n} and in which two distinct vertices (v, i) and (v ′ , i ′ ) are joined by an edge if either v = v ′ or vv ′ is an edge of H and i = i ′ ; see Section 1.4 for an illustration and a more general definition of the Cartesian product of two graphs.) Further, the same authors gave a different construction [10,Theorem 8] showing that where s = s (Z 2 ) is the critical probability for site percolation in Z 2 . Using the aforementioned simulation-based estimates for s , this gives a non-rigorous lower bound of 0.5549 on p 1,c (Z 2 ). All these lower bounds remain far apart from the upper bound of 0.8639 from [5], and, as noted in [5], part of the difficulty of Problem 1.5 has been the absence of a clear candidate conjecture to aim for. In view of the difficulty of Problem 1.5, there has been interest in increasing our understanding of 1-independent models on other host graphs than Z 2 . Balister and Bollobás noted p 1,c (Z ) is non-increasing in and must therefore converge to a limit as → ∞. They showed this limit is at least 1∕2 and posed the following problem [2, Question 2]: Problem 1.6 (Balister and Bollobás problem). Determine lim →∞ p 1,c (Z ).
By the construction of Day, Falgas-Ravry and Hancock mentioned above, this limit is in fact at least 4 − 2

√
3; the only known upper bound is again the 0.8639 upper bound on p 1,c (Z 2 ) from [5]. Balister and Bollobás have further studied 1-independent models on infinite trees, obtaining in this setting 1-independent analogs of classical results of Lyons [21] for independent bond percolation. Day, Hancock and the first author for their part gave a number of results on the connectivity of 1-independent random graphs on paths and complete graphs, and on the almost sure emergence of arbitrarily long paths in 1-independent models. More precisely, they introduced the Long Paths critical probability p 1,LP (H) of H, given by

Contributions of this article
Our main result in this article is determining the limit of the 1-independent critical probability for percolation in Z 2 × K n as n → ∞: Theorem 1.8. The following hold: In particular, we have lim n→∞ p 1,c ( As a corollary to the key result in our proof of Theorem 1.8, we also obtain a solution to the problem of Day, Falgas-Ravry and Hancock on long paths in 1-independent percolation, Problem 1.7 above: 3. In fact, we are able to show the conclusions of Theorems 1.8 and 1.9 still hold if we replace the complete graph K n by a suitable pseudorandom graph. Recall that the study of pseudorandom graphs originates in the ground-breaking work of Thomason [25]. In this article we shall use the following notion of weak pseudorandomness (see Condition (3) in the survey of Krivelevich and Sudakov [19]): We conjecture that the conclusion of Theorem 1.8 still holds if we replace the complete graph K n by an n-dimensional hypercube. 3.
Observe that, since Z 2 × Q n is a subgraph of Z n+2 and p 1,c (Z n+2 ) ≥ 4 − 2 √ 3 [10, Theorem 7], Conjecture 1.13 implies that the answer to the problem of Balister and Bollobás (Problem 1.6 above) 3. In fact, we make the following bolder conjecture: Conjecture 1.14 (1-independent percolation in high dimension). There exists ≥ 3 such that Finally we prove some modest results on component evolution in 1-independent models on K n and on pseudorandom graphs. The main point of these results is that "the two-state measure minimizes the size of the largest component," a heuristic which in turn guides our Conjecture 1.13. Here by the two-state measure, we mean the following variant of site percolation, due to Newman (see [23]): Definition 1.15 (Two-state measure). Let H be a graph, and let p ∈ . The two-state measure 2s,p ∈  1,p (H) is constructed as follows: assign to each vertex v ∈ V(H) a state S v independently and uniformly at random, with S v = 1 with probability = (p) = (1 + √ 2p − 1)∕2 and S v = 0 otherwise. Then let H 2s,p be the random subgraph of H obtained by including an edge if and only if its endpoints are in the same state.
Day, Hancock and the first author showed in [10, Theorem 16] that 2s,p minimizes the probability of connected subgraphs over all 1-ipm ∈  1,p (K 2n ). We show below that it also minimizes the probability of having a component of size greater than n. Explicitly, given a set of edges F ⊆ E(H) in a graph H, we let C i (F) denote the ith largest connected component in the associated subgraph (V(H), F) of H. Then: ) and H = K 2n . Then for all p ∈ [p 2n , 1], Further, we show that the two-state measure also asymptotically minimizes the likely size of a largest component in 1-independent models on pseudorandom graphs: be fixed. Let (H n ) n∈N be a sequence of weakly q-pseudorandom graphs on n vertices with q = q(n) ≫ log(n)∕n. Then the following hold for H = H n : This leads us to the natural conjecture that the two-state measure asymptotically minimizes the size of a largest component in 1-independent models on the hypercube Q n : Conjecture 1.18. Let p ∈ ( 1 2 , 1] be fixed, and let H = Q n . Then for all ∈  1,≥p (Q n ), with We suspect that a proof of this conjecture combined with the ideas in the present paper would yield a proof of Conjecture 1.13.
Overall, our results would lead us to speculate that the true value of p 1,c (Z 2 ) is probably a lot closer to the lower bound of 0.5549 from (1.1) than to the upper bound of 0.8639 obtained from renormalization arguments in [5]. However a rigorous proof of improved upper bounds on p 1,c (Z 2 ) remains elusive for the time being.

Organization of the article
The key step in the proof of our main results, Theorem 2.1, is proved in Section 2; it establishes that 3 is the threshold for ensuring there is a high probability in any 1-independent model of finding a path between the largest components in two disjoint copies of K n joined by a matching. The argument in a sense captures "what makes the 4 − 2

Notation
Given n ∈ N we write [n] for the discrete interval {1, 2, … , n}. We write S (2) for the collection of all unordered pairs from a set S. We use standard graph-theoretic notation throughout the article. Given a graph H, we use V = V(H) and E = E(H) to refer to its vertex-set and edge-set respectively, and write e(H) for the size of E(H). Given X ⊆ V, we write H[X] for the subgraph of H induced by X, that is, the graph (X, E(H) ∩ X (2) ). For disjoint subsets X, Y of V we also write H[X, Y] for the bipartite subgraph of H induced by X ⊔ Y, that is the graph (X ∪ Y, {xy ∈ E(H) ∶ x ∈ X, y ∈ Y}). We denote by K n the complete graph on n vertices, K n = ([n], [n] (2) ).
The Cartesian product of two graphs G 1 and G 2 is the graph . In particular if G 1 = K 2 , that is, a single edge, then G 1 × G 2 is the bunkbed graph of G 2 consisting of two disjoint copies of G 2 , the left copy {1} × G 2 and the right copy {2} × G 2 , together with a perfect matching joining each vertex (1, v) in the left copy to its image (2, v) in the right copy. See Figure 1 for an example.
Finally we use the standard Landau notation for asymptotic behavior: We also sometimes use f ≪ g and f ≫ g as a shorthand for f = o(g) and f = (g), respectively. Given a sequence of events (E n ) n∈N in some probability space, we say that E n occurs with high probability

WHEN LEFT MEETS RIGHT: JOINING THE LARGEST COMPONENTS ON EITHER SIDE OF K × G n
Let (G n ) n∈N be a sequence of weakly q-pseudorandom n-vertex graphs where qn ≫ log n. Consider the Cartesian product H = K 2 × G n . Given ∈  1,p (H), let "Left meets Right" denote the event that the -random graph H contains a connected component containing both strictly more than half of the vertices in {1} × [n] and strictly more than half of the vertices in {2} × [n]. Our main result in this section is showing that the event "Left meets Right" undergoes a sharp transition at p = 4 − 2 3 it is possible to construct 1-independent measures ∈  1,p (H) such that whp the event "Left meets Right" does not occur, while for p > 4 − 2 √ 3 it occurs whp regardless of the choice of .
3 be fixed. Then for every ∈  1,p (H), , let = (p) be given by The quantity will play an important role in the proof of both parts of Theorem 2.1. Observe that ∈ [p, 1] and satisfies Using the latter of these relations, we see that for p ∈ [0, 1], Our proofs will also make extensive use of the following Chernoff bound: given a binomial random variable X ∼ Binom(N, p) and ∈ (0, 1), we have

Lower bound construction: Proof of Theorem 2.1(ii)
For each 1∕2 < p ≤ 4 − 2 √ 3, we construct a state-based measure F ∈  ≥p (K 2 × G n ), based on the ideas behind constructions in [10,14]. Assume without loss of generality that V(G n ) = [n]. We randomly assign to each vertex (i, v) ∈ [2] × [n] a state S v ∈ {0, 1, ⋆}, independently of all the other vertices, with (a) S (1,v) = 1 with probability and S (1,v) We then include edges of H = K 2 × G n in our random subgraph H F according to the following rules: See Figure 2 for an illustration of the construction. Since F is state-based, it is clearly a 1-ipm. Our state distributions (a)-(b) imply that every edge in the left copy of G n is open (included in our random graph) with probability 2 + (1 − ) 2 = p (by the edge-rule (i) above), and that every edge in the right copy of G n is open with probability ( √ p) 2 = p (by the edge-rule (ii) above). On the other hand, (by the edge-rule (iii) above) an edge {(1, v), (2, v)} from the left copy to the right copy is closed if and only if S (1,v) = 1 and S (2,v) 3 is fixed. We shall show that for n sufficiently large this implies that for any ∈  1,p (H), whp "Left meets Right" occurs. Our strategy for doing this is as follows: first of all we show in Lemma 2.5 that, for each i ∈ [2], in any fixed tripartition are present in H , we can combine the probabilistic information from Lemma 2.5 to show that whp the relative sizes of the V c,c ′ almost satisfy a certain system  = (p) of inequalities (2.7)-(2.10) (or more precisely that we can extract from the |V c,c ′ |∕n a solution to (p ⋆ ) for some p ⋆ a little smaller than p). For p > 4 − 2 √ 3 and n sufficiently large, we are able to show this leads to a contradiction (Lemma 2.6). Having outlined our proof strategy, we now fill in the details. We shall use the following path-decomposition theorem due to Dean and Kouider. [11]). Let G be an n-vertex graph. Then there exists a set  of edge-disjoint paths in G such that || ≤ 2n 3 and

Theorem 2.2 (Dean and Kouider
Recall that a matching in a graph is a set of vertex-disjoint edges.

Corollary 2.3.
Let > 0 and let G be an n-vertex graph with e(G) ≥ 2n∕ . Then there exists a set  of edge-disjoint matchings in G such that 2n for every M ∈ .
Proof. By Theorem 2.2, there exists a set  of edge-disjoint paths in G such that || ≤ 2n 3 and E(G) = ⋃ P∈ E(P). Let  short = {P ∈  ∶ e(P) ≤ 2 e(G) n }. Let  be the set of matchings obtained by decomposing each path in  ⧵  short into two matchings. We have || ≤ 2 || ≤ 2n.
Matchings are useful in a 1-independent context since the states of their edges (present or absent) are independent. We can thus combine Corollary 2.3 with a Chernoff bound to show the number of edges in a 1-independent model is concentrated around its mean. > 0 and p ∈ (0, 1]. Let G be an n-vertex graph with e(G) ≥ 2n∕ and let ∈  1,p (G). Then
Proof. We apply Corollary 2.3 to obtain a set  of edge-disjoint matchings in G such that properties (M1) to (M3) hold. For every M ∈ , we have |M| ≥ e(G) 2n . Thus by (2.2) and 1-independence, ) .
By a union bound, we have ) .
Thus with probability at least 1 − 4n exp This completes the proof. , and let = (p) > 0 be fixed and sufficiently small. Let G be an n-vertex graph satisfying for all U ⊆ V(G), where q(n) ≫ log n n . Consider a fixed tripartition V(G) = V 1 ⊔ V 2 ⊔ V 3 . Then for every ∈  1,p (G), the following hold whp: where the inequality in the third line follows from Lemma 2.4. So (P1) holds whp. Next we show that (P2) holds whp. Fix 1 ≤ i < j ≤ 3. If |V i | ≤ n or | | V j | | ≤ n, then (P2) trivially holds. Hence we may assume that qn 2 , which for n sufficiently large is greater than 2n . We now apply Lemma 2.4 to show that (P2) holds whp. We have

So (P2) holds whp. Now we show that (P1) implies (P3). Assume that (P1) holds. Fix i ∈ [3] and assume that
If |C| ≤ |Vi| 2 , then there is a partition of V i into at most 4 sets, each of size at most |Vi| 2 , such that every connected component of G [V i ] is entirely contained in one of the sets of the partition. Indeed, such a partition can be obtained by starting with a partition of V i into the connected components of G [V i ] and then as long as the partition contains two parts of size at most |Vi| 4 choosing two such parts arbitrarily and merging them into a single part. Since for any quadruple (x 1 , x 2 , x 3 , x 4 ) with 1 2 ≥ x i ≥ 0 and Rearranging terms, this gives which is a contradiction for chosen sufficiently small. Thus we may assume |C| ≥ |Vi| 2 . Now by (P1) and (2.3) again, we have Dividing by q |Vi| 2 2 and using |V i | ≥ 1∕4 n, we deduce that 2 is an increasing function in the interval Next we show that (P2) and (P3) together imply (P4). Assume that (P2) and (P3) hold. Fix 1 ≤ i < j ≤ 3 and assume that be the sets of vertices which cannot be reached by a path in G [V i , V j ] from C j and C i , respectively. Since there is no path from C i to C j , we must have C i ⊆ A i and C j ⊆ A j . By (P2), by the definition of A i and A j , and by (2 where the last inequality follows since (x, y)  → x+y−2xy is non-increasing in both x and y for x, y ≥ 1 2 . Note that (2.6) gives a contradiction for sufficiently small since p > 1 2 . So (P4) holds whp. Finally, we observe that (P5) follows directly from (P3) and (P4). Indeed let k ∈ [3] denote the number of i ∈ [3] for which |V i | ≥ 1∕4 n (note we can guarantee k ≥ 1 provided < 3 −4 ). Then (P3) and (P4) together imply there is a unique connected component C in G of size at least , satisfying the following inequalities: (2.10) The key step in our proof of Theorem 2.1 will be, assuming that "Left meets Right" does not occur whp, to use Lemma 2.5 to exhibit a partition of [n] into 9 parts whose relative sizes can be used to find a solution to (p ⋆ ), for some p ⋆ satisfying 4 − 2 √ 3 < p ⋆ < p. We will then be able to use the following lemma to derive a contradiction.
Proof. Suppose not and let A ∈ (p). Note that the bound for ∑ i,j A ij in (2.7) implies By transpose-symmetry of (p) and (2.7), we may assume without loss of generality that and since A 31 ≥ 1 2 ∑ j A 33j by (2.8), Substituting this expression into our upper bound on w, we get For A 11 fixed, the continuous function f A 11 (y) = − (1− )y − A 11 + √ (A 11 ) 2 +y 2 √ p is convex in (0, +∞) as its is increasing in y in that interval. By (2.14), 0 ≤ A 21 ≤ 1− A 11 , which together with the convexity of f A 11 gives: where the last inequality follows from the upper bound (2.11) on A 11 . We now claim that this contradicts (2.12), that is, that − p 2 and p  → 1−p (1 − ) − p 2 are both strictly decreasing functions (as is increasing in p). Hence to prove the claim above, it suffices to show that for p = 4 − 2 . Now it is easy to check that which completes the proof. ▪ We are now ready to complete the proof of Theorem 2.1(i).

Proof.
Let p > 4 − 2 √ 3 be fixed. Let = (p) > 0 be fixed and chosen sufficiently small. Let . Finally, let n be sufficiently large so that for G = G n the pseudorandomness assumption (2.3) holds, and let ∈  1,p (H), where H = K 2 × G n . For with i ≠ j, let  ij be the event that for any partition Suppose for a contradiction that  good holds but "Left meets Right" does not. For i ∈ [2], let C i be the unique largest connected component in G i (this exist by (P5)). Let U 1 ⊔ U 2 ⊔ U 3 = [n] and W 1 ⊔ W 2 ⊔ W 3 = [n] be such that the following hold. The key properties of this coloring are that the large components C 1 and C 2 in G 1 and G 2 are colored red and blue respectively, that there are no edges from red vertices to blue vertices, and that the green vertices span few edges in G i , i ∈ [2]. Our 3-coloring of V(H) gives rise to a partition of [n] into 9 sets in a natural way, by considering the possible color pairs for ((1, v), (2, v)), v ∈ [n]. This partition is illustrated in Figure 3.
We now investigate the relative sizes of this 9-partition.
where the penultimate inequality uses (2.16). Next, consider j ∈ [3].  [3]. If |V i2 | ≥ 1∕9 n, then by (2.8) which we have just established and the definition of A i1 , we have |V i1 | ≥ 1∕9 n also. In this case (2.19) and an appropriately small choice of ensure that holds trivially, since the right hand-side is zero.
So suppose that |V i1 | ≥ 1∕9 n > |V i2 |. Then (2.19), and p > 1∕2 imply that Together with an appropriately small choice of , this ensures Thus in every case (2.9) is satisfied by A (with p ⋆ instead of p). A symmetric argument shows A satisfies (2.10) for p ⋆ as well.
Thus A ∈ (p ⋆ ) as claimed. However, since p ⋆ > 4 − 2 √ 3, Lemma 2.6 implies that (p ⋆ ) = ∅, a contradiction. Thus the event  good , which holds whp, does imply the event "Left meets Right," proving the theorem. 3 and a suitable choice of G, we can use Theorem 2.1(i) to ensure that each edge in the 1-independent random graph H is present with probability 1 − o (1). With such a high edge probability, we can then establish the almost sure existence of infinite components or long paths in H in a straightforward way-either by using results in the literature, or by a direct argument.
On the other hand if p ≤ 4 − 2 √ 3, we can use ideas from the lower bound construction in the proof of Theorem 2.1(ii), which date back to [10,14], in order to construct a 1-independent random subgraph G of H × K n that fails to percolate (or, if H = Z, that only contain paths of length O(n)). For the convenience of the reader, we sketch below how this works in the special case H = Z 2 .
We now use these random states to build a 1-independent random graph G as follows. Given an edge{(x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 )} of H × K n , include it in G if one of the following holds: Then the choice of probabilities for our random states ensure each edge is open with probability at least p = 4 − 2

√
3, and our edge rules further imply that every connected component C in G meets at most four consecutive cylinders  r ∶= {(x, y, z) ∶ ||(x, y)|| ∞ = r}, r ∈ Z ≥0 since, as is easily checked, a connected component in G cannot both contain a vertex assigned state 0 and a vertex assigned state 1-we leave this as an exercise to the reader, and refer them to [10,Corollary 24] for a proof of this fact in a more general setting. In particular, we have that G does not percolate.
Having thus outlined our proof ideas, we now fill in the details. First we formalize our renormalization argument with the following lemma. 3 and > 0 be fixed. By Theorem 2.1(i), there exists n 0 ∈ N such that for all n ≥ n 0 and all ∈  1,≥p (K 2 × G n ), the -probability of the event "Left meets Right" is at least 1 − . For n ≥ n 0 , G = G n and ∈  1,≥p (H × G), define a random graph model H from (H × G) as follows: for each edge uv ∈ E(H), we add uv to H if and only if there is a connected component in containing strictly more than half of the vertices in {u} × V(G n ) and strictly more than half of the vertices {v}×V(G n ). The model H is clearly 1-independent, has edge-probability at least 1 − , and has the property that any path in H can be lifted up to a path in (H × G) . This proves the lemma. ▪ Recall that 2-neighbor bootstrap percolation on a graph G is a discrete-time process defined as follows. At time t = 0, an initial set of infected vertices A = A 0 is given. At every time t ≥ 1, every vertex of G which has at least 2 neighbors in A t−1 becomes infected and is added to A t−1 to form A t . We denote by A the set of all vertices of G which are eventually infected, A = ⋃ t≥0 A t . Following Day et al. [10], we say that a graph G has the finite 2-percolation property if for every finite set of initially the vertices in V(G). Moreover, a graph G is a complete multipartite graph if there exists a partition of V(G) such that two vertices in V(G) are joined by an edge in G if and only if they are contained in different parts of the partition. Finally, the complement G c of a graph G is the graph on V(G) whose edges are the non-edges of G, G c ∶= (V(G), V(G) (2) ⧵ E(G)).

Lemma 4.1.
If G is a complete multipartite graph on 2n vertices with independence number (G) ≤ n, then G contains at least n! perfect matchings.
Proof. Let G be a complete multipartite graph on 2n vertices with the minimum number of perfect matchings subject to (G) ≤ n. Let V 1 , V 2 , … , V r denote the parts of G with |V 1 | ≥ |V 2 | ≥ … ≥ |V r |. If |V r−1 | + |V r | ≤ n, then the graph G ′ obtained from G by deleting all edges in G[V r−1 , V r ] satisfies (G ′ ) ≤ n and has at most as many perfect matchings as G. We may therefore assume that |V r−1 | + |V r | ≥ n, and thus in particular that r ≤ 3. Consider a perfect matching M in G and let i be the number of edges in Hence the number PM(G) of perfect matchings in G is: i! counts the number of different ways of selecting i-sets of vertices from each of V 1 and V 2 and joining them by a perfect matching, while ! counts the number of ways of joining the vertices of V 3 by a perfect matching to the remaining vertices of V 1 ∪V 2 .) If |V 3 | > 0, then let G ′ be the complete tripartite graph with parts of size |V 1 | , |V 2 | + 1, |V 3 | − 1. Note that (G ′ ) ≤ n. By the formula above, we have PM(G) PM(G ′ ) = |V 3 | (n − |V 3 | + 1) (|V 2 | + 1)(n − |V 2 |) ≥ 1, since |V 3 | (n − |V 3 | + 1) − (|V 2 | + 1)(n − |V 2 |) = (|V 2 | − |V 3 | + 1)(|V 2 | + |V 3 | − n) ≥ 0 (as |V 2 | ≥ |V 3 | and |V 2 | + |V 3 | ≥ n). It follows that PM(G) ≥ PM(K n,n ) = n! as claimed. ▪ Proof of Proposition 1.16. Let H = K 2n . For all p ∈ [ 1 2 , 1], we may construct the two-state measure 2s,p ∈  1,p (H) which satisfies: proving the upper bound in that range. For p 2n ≤ p ≤ 1 2 , we note that = (p) is no longer a real number. However, as shown in [10, Section 7.1], we may take a "complex limit" of the 2-state measure 2s,p , and the conclusion above still holds.
For the lower bound, let C 1 , C 2 , … , C r be the connected components of a -random subgraph H of K 2n . Let G denote the complete multipartite graph associated with the partition ⊔ i C i of V(K 2n ) = [2n]. Observe that G is a subgraph of the complement H c of H . If |C i | ≤ n for all i, then (G) ≤ n, whence by Lemma 4.1 G contains at least n! perfect matchings. In particular, H c must contain at least n! perfect matchings. By Markov's inequality, we thus have )) counts the number of perfect matchings in K 2n by selecting n vertex-disjoint edges sequentially one after the other, and dividing through by n!.) The lower bound follows. which is strictly greater than qn 2 2(r+1) for = (p) chosen sufficiently small. Assume (4.1). We show this implies the claimed lower bound on the size of a largest component.
Dividing through by qn 2 ∕2, rearranging terms and using the fact is chosen sufficiently small, we get Solving for , we get that For part (ii), consider the r + 1-state measure in which each vertex is assigned state r + 1 with probability 1− √ r((r+1)p−1) r+1 and a uniform random state from the set {1, 2, … , r} otherwise, and in which an edge is open if and only if its vertices are in the same state. This is easily seen to be a 1-ipm with the requisite properties. ▪