Long paths and connectivity in 1‐independent random graphs

A probability measure μ on the subsets of the edge set of a graph G is a 1‐independent probability measure (1‐ipm) on G if events determined by edge sets that are at graph distance at least 1 apart in G are independent. Given a 1‐ipm μ, denote by Gμ the associated random graph model. Let ℳ1,⩾p(G) denote the collection of 1‐ipms μ on G for which each edge is included in Gμ with probability at least p. For G=Z2, Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all μ∈ℳ1,⩾p(G), Gμ almost surely contains an infinite component. In this paper, we significantly improve previous lower bounds on p ⋆. We also determine the 1‐independent critical probability for the emergence of long paths on the line and ladder lattices. Finally, for finite graphs G we study f 1, G(p), the infimum over all μ∈ℳ1,⩾p(G) of the probability that Gμ is connected. We determine f 1, G(p) exactly when G is a path, a complete graph and a cycle of length at most 5.

Remark 3. Given a measure ∈  k,⩾p (G), we may readily produce a measurẽ∈  k,⩾p (G) such that̃({e is open}) = p for all e ∈ E(G) via random sparsification: independently delete each edge e of G with probability p∕ ({e is open}) ∈ [0, 1]. The resulting bond percolation model on G is clearly k-independent and has the property that each edge is open with probability exactly p; the corresponding bond percolation measurẽthus has the required properties.

Critical probabilities for percolation and motivation for this paper
Percolation theory is the study of random subgraphs of infinite graphs. Since its inception in Oxford in the 1950s, it has blossomed into a rich theory and has been the subject of several monographs [11,17,27]. The central problem in percolation theory is to determine the relationship between edge-probabilities and the existence of infinite connected components in bond percolation models.
In the most fundamental instance of this problem, consider an infinite, locally finite connected graph G, and let be a 0-independent bond percolation model on G. We say that percolation occurs in a configuration H on G if H contains an infinite connected component of open edges. By Kolmogorov's zero-one law, for G and as above, percolation is a tail event whose -probability is either zero or one. This allows one to thus define the Harris critical probability p 0,c (G) for 0-independent percolation:

Problem 4.
Given an infinite, locally finite connected graph G, determine p 0,c (G).
One of the cornerstones of percolation theory-and indeed one of the triumphs of twentieth century probability theory-is the Harris-Kesten theorem, which established the value of p 0,c (Z 2 ) to be 1∕2.
Theorem 23]). Let be the p-random measure on Z 2 . Then In this paper, we focus on the question of what happens to the Harris critical probability in Z 2 if the assumption of 0-independence is weakened to k-independence. In particular, how much can local dependencies between the edges postpone the global phenomenon of percolation? Definition 5. Let G be an infinite, locally finite connected graph and let k ∈ N 0 . The Harris critical probability for k-independent percolation 1 in G is defined to be: Problem 6 was proposed by Balister and Bollobás [4] in a 2012 paper in which they began a systematic investigation of 1-independent percolation models. Study of 1-independent percolation far predates their work (see e.g., [1,6,7,13,22,24]), however, due to important applications of 1-independent percolation models.
A standard technique in percolation is renormalisation, which entails reducing a 0-independent model to a 1-independent one (possibly on a different host graph), trading in some dependency for a boost in edge-probabilities. Renormalisation arguments feature in many proofs in percolation theory; a powerful and particularly effective version of such arguments was developed by Balister, Bollobás, and Walters [7].
Their method, which relies on comparisons with 1-independent models on Z 2 (in almost all cases) and Monte-Carlo simulations to estimate the probabilities of bounded events, has been applied to give rigorous confidence intervals for critical probabilities/intensities in a wide variety of settings: various models of continuum percolation [3,7,8], hexagonal circle packings [10], coverage problems [5,19], stable Poisson matchings [14,15], the Divide-and-Color model [2], site and bond percolation on the eleven Archimedean lattices [29] and for site and bond percolation in the cubic lattice Z 3 [9]. The usefulness of comparison with 1-independent models and the plethora of applications give strong theoretical motivation for the study of 1-independent percolation.
From a more practical standpoint, many of the real-world structures motivating the study of percolation theory exhibit short-range interactions and local dependencies. For example a subunit within a polymer will interact and affect the state of nearby subunits, but perhaps not of distant ones. Similarly, the position or state of an atom within a crystalline network may have a significant influence on nearby atoms, while long-range interactions may be weaker. Within a social network, we would again expect individuals to exert some influence in esthetic tastes or political opinions, say, on their circle of acquaintance, and also expect that influence to fade once we move outside that circle. This suggests that k-independent bond percolation models for k ⩾ 1 are as natural an object of study as the more widely studied 0-independent ones.
Despite the motivation outlined above, 1-independent models remain poorly understood. To quote Balister and Bollobás from their 2012 paper: "1-independent percolation models have become a key tool in establishing bounds on critical probabilities […]. Given this, it is perhaps surprising that some of the most basic questions about 1-independent models are open." There are in fact some natural explanations for this state of affairs. As remarked on in the previous subsection, there are many very different 1-independent models with edge-probability p, and they tend to be harder to study than 0-independent ones due to the extra dependencies between edges. In particular simulations are often of no avail to formulate conjectures or to get an intuition for 1-independent models in general. Moreover, while the theoretical motivation outlined above is probabilistic in nature, the problem of determining a critical constant like p 1,c (Z 2 ) is extremal in nature-one has to determine what the worst possible 1-independent model is with respect to percolation-and calls for tools from the separate area of extremal combinatorics.
In this paper, we continue Balister and Bollobás's investigation into the many open problems and questions about and on these measures. Before we present our contributions to the topic, we first recall below previous work on 1-independent percolation.

Previous work on 1-independent models
Some general bounds for stochastic domination of k-independent models by 0-independent ones were given by Liggett, Schonmann and Stacey [24]. Among other things, their results implied p 1,c (Z 2 ) < 1.
Balister, Bollobás and Walters [7] improved this upper bound via an elegant renormalisation argument and some computations. They showed that in any 1-independent bond percolation model on Z 2 with edge-probability at least 0.8639, the origin has a strictly positive chance of belonging to an infinite open component. This remains to this day the best upper bound on p 1,c (Z 2 ). In a different direction, Balister and Bollobás [4] observed that trivially p 1,c (G) ⩾ 1 2 for any infinite, locally finite connected graph G. In the special case of the square integer lattice Z 2 , they recalled a simple construction due to Newman which gives where site is the critical value of the -parameter for site percolation, that is, the infimum of ∈ [0, 1] such that switching vertices of Z 2 on independently at random with probability almost surely yields an infinite connected component of on vertices. Plugging in the known rigorous bounds for 0.556 ⩽ site ⩽ 0.679492 [33,34] yields p 1,c (Z 2 ) ⩾ 0.5062, while using the nonrigorous estimate site ≈ 0.592746 (see e.g., [32]) yields the nonrigorous lower-bound p 1,c (Z 2 ) ⩾ 0.5172. With regards to other lattices, Balister and Bollobás completed a rigorous study of 1-independent percolation models on infinite trees [4], giving 1-independent analogs of classical results of Lyons [25] for the 0-independent case. Balister and Bollobás's results were later generalized to the k-independent setting by Mathieu and Temmel [26], who also showed interesting links between this problem and theoretical questions concerning the Lovász local lemma, in particular the work of Scott and Sokal [30,31] on hard-core lattice gases, independence polynomials and the local lemma.

Our contributions
In this paper, we make a three-fold contribution to the study of Problem 6. First of all, we improve previous lower bounds on p 1,c (Z 2 ) with the following theorems.

Theorem 7.
For all ∈ N ⩾2 , we have that Theorem 7 strictly improves on the previous best lower bound for = 2 given in (1.1) above; moreover, it is based on a very different idea, which first appeared in the second author's PhD thesis [16]. In addition we give a separate improvement of (1.1): let site again denote the critical threshold for site percolation. Then the following holds.
Substituting the rigorous bound site ⩾ 0.556 into Theorem 8 yields the lower bound p 1,c (Z 2 ) ⩾ 0.531136, which does slightly worse than Theorem 7. However substituting in the widely believed but nonrigorous estimate site ≈ 0.592746 yields a significantly stronger lower bound of p 1,c (Z 2 ) ⩾ 0.554974.
Second, motivated by efforts to improve the upper bounds on p 1,c (Z 2 ), and in particular to establish some 1-independent analogs of the Russo-Seymour-Welsh (RSW) lemmas on the probability of crossing rectangles, we investigate the following problems. Let P n denote the graph on the vertex set {1, 2, … n} with edges {12, 23, … , (n − 1)n}, that is, a path on n vertices. Given a connected graph G, denote by P n × G the Cartesian product of P n with G. A left-right crossing of P n × G is a path from a vertex in {1} × V(G) to a vertex in {n} × V(G). We define the crossing critical probability for 1-independent percolation on P n × G to be that is, the least edge-probability guaranteeing that in any 1-independent model on P n × G, there is a strictly positive probability of being able to cross P n × G from left to right.
Problem 9. Given n ∈ N and a finite, connected graph G, determine p 1,× (P n × G).
Problem 9 can be thought of as a first step toward the development of 1-independent analogs of the RSW lemmas; these lemmas play a key role in modern proofs of the Harris-Kesten theorem, and one would expect appropriate 1-independent analogs to constitute a similarly important ingredient in a solution to Problem 6. By taking the limit as n → ∞ in Problem 9, one is led to consider another 1-independent critical probability. Let G be an infinite, locally finite connected graph. The long paths critical probability for 1-independent percolation on G is that is, the least edge-probability at which arbitrarily long open paths will appear in all 1-independent models in G.
In this paper, we resolve Problem 9 in a strong form when G consists of a vertex or an edge (see Theorems 15 and 30). This allows us to solve Problem 10 when G is the integer line lattice Z and the integer ladder lattice Z × P 2 .
Theorem 11. We have that Note that part (i) of Theorem 11 above can be read out of earlier work of Liggett, Schonman, and Stacey [24] and Balister and Bollobás [4]. We prove further bounds on both p 1,× (P n ×G) and p 1, (Z×G) for a variety of graphs G. We summarize the latter, less technical, set of results below. Let C n and K n denote the cycle and the complete graph on n vertices respectively.

Theorem 12.
We have that A key ingredient in the proof of Theorems 11 and 12 is a local lemma-type result, Theorem 26, relating the probability in a 1-independent model of finding an open left-right crossing of P n × G to the probability of a given copy of G being connected in that model. This motivated our third contribution to the study of 1-independent models in this paper, namely an investigation into the connectivity of 1-independent random graphs. Definition 13. Let G be a finite connected graph. For any p ∈ [0, 1], we define the k-independent connectivity function of G to be Problem 19. Given a finite connected graph G, determine F 1,G (p).
We resolve Problem 19 exactly when G is a path, a complete graph or a cycle on at most 5 vertices.
Together, Theorems 15-18 and 21-23 determine the complete connectivity "profile" for 1-independent measures on K n , P n , C 4 and C 5 -i.e., the range of values ({connected}) can take if every edge is open with probability p. In Figure 1, we illustrate these for four of these graphs G with plots of f 1,G (p), , where G p is the 0-independent model on G obtained by setting each edge of G to be open with probability exactly p, independently at random.

Organization of the paper
Our first set of results, Theorems 7 and 8 are proved in Section 2.
In Section 3, we use arguments reminiscent of those used in inductive proofs of the Lovász local lemma to obtain Theorem 26, which gives a general upper bound for crossing and long paths critical probabilities in 1-independent percolation models on Cartesian products Z × G. This result is used in Sections 5 and 6 to prove Theorem 11 on the long paths critical probability for the line and ladder lattices.
In Sections 5, 7 and 8 and 9, we prove our results on f 1,G (p) and F 1,G (p) when G is a path, a complete graph or a short cycle. We apply these results in Section 10 to prove Theorem 12. Finally we end the paper in Section 11 with a discussion of the many open problems arising from our work.

Notation
We write N for the set of natural numbers {1, 2, …}, N 0 for the set N ∪ {0}, and N ⩾k for the set of natural numbers greater than or equal to k. We set [n] ∶= {1, 2, … n}. Given a set A, we write A (r) for the collection of all subsets of A of size r, hereafter referred to as r-sets from A. We use standard graph theoretic notation. A graph is a (2) denote the vertex set and edge set of G The 1-independent connectivity profile of G for G = K 3 K 4 , C 4 and C 5 . The green curve represents f 1,G (p), the dashed black curve f 0,G (p), and the union of the red, blue and purple segments represent the piecewise smooth function respectively. Given a subset A ⊆ G, we denote by G [A] the subgraph of G induced by A. We also write N(A) for the set of vertices in G adjacent to at least one vertex in A.
Given two graphs G and H, we write G × H for the Cartesian product of G with H, which is the graph on the vertex set V(G) × V(H) having an edge between (x, u) and (y, v) if and only if either u = v and x is adjacent to y in G, or x = y and u is adjacent to v in H.
Throughout this paper, we shall use k-ipm as a shorthand for "k-independent percolation model/measure." In a slight abuse of language, we say that a bond percolation model on an infinite connected graph G percolates if ({percolation}) = 1. We refer to a random configuration G as a -random subgraph of G. Finally we write E for the expectation taken with respect to the probability measure . For any event X, we write X c for the complement event.

LOWER BOUNDS ON
For each vertex in Z , we color it either Blue or Red, or set it to state I, which stands for Inwards. The probability that a given vertex will be in each of these states will depend on which of the T k the vertex is in, and we assign these states to each vertex independently of all other vertices.
• If v is a vertex in T k , where k ≡ 0 mod 6, then we color v Blue. • If v is a vertex in T k , where k ≡ 1 mod 6, then we color v Red with probability q∕2 and color it Blue otherwise. • If v is a vertex in T k , where k ≡ 2 mod 6, then we color v Red with probability q and put it in the Inwards state I otherwise. • If v is a vertex in T k , where k ≡ 3 mod 6, then we color v Red. • If v is a vertex in T k , where k ≡ 4 mod 6, then we color v Blue with probability q∕2 and color it Red otherwise. • If v is a vertex in T k , where k ≡ 5 mod 6, then we color v Blue with probability q and put it in the Inwards state I otherwise.
Note that the rules for T k+3 , T k+4 , T k+5 are the same as those for T k , T k+1 , T k+2 respectively, except with red and blue interchanged. See Figure 2 for the possible states of the vertices in T 0 , T 1 , T 2 and T 3 when = 2. Now suppose that e = {v 1 , v 2 } is an edge in Z . First we say that the edge e is open if either both v 1 and v 2 are Blue or both v 1 and v 2 are Red. We also say the edge e is open if, for some k, we have that v 1 ∈ T k , v 2 ∈ T k+1 , and v 2 is in state I. In all other cases we say that the edge e is closed. It is clear that this gives a 1-independent measure on Z as it is vertex-based, and it is also easy to check that every edge is present with probability at least 4 − 2 3. Call this measure , and let G ∶= Z . We claim that in G , for all k ≡ 0 mod 3, there is no path of open edges from T k to T k+3 . Suppose this is not the case, and P is some path of open edges from a vertex in T k to T k+3 . We first note that P cannot include a vertex in state I, as such a vertex would be in T k+2 and would only be adjacent to a single edge. Thus every vertex of P is either Blue or Red. However, as one end vertex of P is Blue and the other end vertex is Red, and there are no open edges with different colored end vertices, we have that such a path P cannot exist. As a result, every component of G is sandwiched between some T k−3 and T k+3 , where k ≡ 0 mod 3, and so is of finite size. Thus we have that p 1,c The construction in Theorem 7 can in fact be generalized to certain other graphs and lattices. Given an infinite, connected, locally finite graph G, and a vertex set We say that G has the finite 2-percolation property if, for every finite set A ⊆ V(G), we have that A is finite.  3.
Proof. Partition V(G) in the following way: pick any vertex v and set T 0 ∶= {v}. For k ⩾ 1 let We have that if w ∈ T k , then w is only adjacent to vertices in T k−1 , T k and T k+1 . Moreover, w is adjacent to at most one vertex in T k−1 -this is the crucial property needed for our construction. Since G has the finite 2-percolation property, each T k is finite. Thus we can use the T k to construct a nonpercolating 1-ipm on G in the exact same fashion as done for Z in Theorem 7 (the key being that vertices in state I are still dead ends, being incident to a unique edge), which in turn shows that p 1,c (G) ⩾ 4 − 2 √ 3. ▪ An example of a lattice with the finite 2-percolation property is the lattice (3,4,6,4), where here we are using the lattice notation of Grünbaum and Shephard [18]. Riordan and Walters [29] showed that the site percolation threshold of this lattice is very likely to lie in the interval [0.6216, 0.6221]. Thus this estimate, together with Newman's construction (see equation (1.1)), shows (nonrigorously) that p 1,c ((3, 4, 6, 4)) ⩾ 0.52981682. As this is less than 4 − 2 √ 3, we have that our construction gives the (rigorous) improvement of p 1,c ((3, 4, 6, 4)) ⩾ 4 − 2 √ 3.
Proof of Theorem 8. Fix > 0 sufficiently small so that q ∶= site (Z 2 ) − is strictly larger than 1∕4. For each vertex v ∈ Z 2 , we assign to it one of three states: On, L or D, and we do this independently for every vertex. We assign v to the On state with probability q, we assign it to the L state with probability It is easy to see that this is a 1-independent measure on Z 2 as it is vertex-based, and every edge is present with probability q 2 + 1 2 (1 − q). Call this measure and let G ∶= Z 2 . We will show that every component of G has finite size. We begin by first proving an auxiliary lemma. Let t ∈ [0, 1 2 ], and let us define another 1-independent measure on Z 2 , which we call the left-down measure with parameter t. In the left-down measure, each vertex of Z 2 is assigned to one of three states: Off, L or D, and we do this independently for every vertex. For each vertex v ∈ Z 2 , we assign it to state L with probability t, we assign it to state D with probability t, and we assign it to state Off with probability 1 − 2t. As above, We start by taking a random subgraph of Z 2 where every edge is open with probability z, independently of all other edges. We then further modify it as follows. For each vertex v = (x, y) we look at the state of the edge e 1 from v to the vertex (x − 1, y), and the state of the edge e 2 from v to the vertex (x, y − 1). If at least one of e 1 or e 2 is closed, we do not change anything. However, if both e 1 and e 2 are open, with probability 1 2 we close the edge e 1 , and otherwise we close the edge e 2 . We do this independently for every vertex v of Z 2 .
It is easy to see that this is an equivalent formulation of t , the left-down measure with parameter t. Indeed, to each vertex v = (x, y) as above we may assign a state Off if both the edge e 1 (v) to the vertex to the left of v and the edge e 2 (v) to the vertex below v are closed, a state L if e 1 (v) is open and a state D if e 2 (v) is open. The probabilities of these three states are (1 − z) 2 = 1 − 2t, t and t respectively, and since the vertex states depend only on the pairwise disjoint edge sets {e 1 (v), e 2 (v)} v∈Z 2 , they are independent of one another just as in the t measure.
Thus we have coupled t to the 0-independent bond percolation measure on Z 2 with edge-probability z. In this coupling we have that if an edge e is open in G t , then it is also open in G . As z ⩽ 0.5 we have that all components in G are finite by the Harris-Kesten theorem, and so we also have that all of the components in G t are finite too. ▪ By considering an appropriate branching process it is possible to prove the stronger result that if 0 ⩽ t < 1 2 , then almost surely all components in G t are finite. We make no use of this stronger result in this paper, so we omit its proof. It is also clear that when t = 1 2 , every vertex in G t is part of an infinite path consisting solely of steps to the left or steps downwards, and so percolation occurs in G t at this point.
Let us return to our original 1-independent measure , where every vertex is in state On, L or D. Recall that our aim is to show that all components have finite size in G . Consider removing all vertices in state L or D, and also any edges adjacent to these vertices. What is left will be a collection of components consisting only of edges between vertices in the On state, which we call the On-sections. The black edges in Figure 3 are the edges in the On-sections. As a vertex is On with probability q < site (Z 2 ), we have that almost surely every On-section is finite. Similarly, consider removing all edges in the On-sections. What is left will be a collection of edges adjacent to vertices in the L or D states. We call these components the LD-sections; the dashed red edges in Figure 3 are the edges in the LD-sections. As each vertex is in state L with probability 1 2 (1 − q) ⩽ 3 8 and in state D with the same probability, Lemma 25 tells us that almost surely every LD-section is finite.
For each vertex v in state L orient the open edge to the left of it away from v, while for each vertex v in state D orient the open edge below it away from v. This gives a partial orientation of the open edges of G , in which every vertex in state L or D has exactly one edge oriented away from it, and vertices in state On have no outgoing edge. Furthermore, if v 1 is a vertex in the On state and v 2 is a vertex in the L or D state, then the edge between them is oriented from v 2 to v 1 . Since the LD-sections are almost surely finite, this implies the LD sections under this orientation consist of directed trees, each of which is oriented from the leaves to a unique root, which is in the On state. In particular, every LD-section attaches to at most one On-section. As such, almost surely every component in G consists of at most one On-section, and a finite number of finite LD-sections attached to it. Thus almost surely every component in G is finite. ▪

A GENERAL UPPER BOUND FOR P 1, P (Z × G)
Let G be a finite connected graph. Set v(G) ∶= |V(G)|. Recall that for any 1-independent bond percolation measure ∈  1,⩾p (G), we have (G is connected) ⩾ f 1,G (p).

Theorem 26. If p satisfies
Proof. Consider an arbitrary measure ∈  1,⩾p (Z × G). For any n ∈ N, the restriction of to [n] × V(G) is a measure from  1,⩾p (P n × G), and clearly all such measures can be obtained in this way. Furthermore, for every n ∈ N, the restriction of to {n} × V(G) is a measure from  1,⩾p (G), and in particular the subgraph of (Z × G) induced by {n} × V(G) is connected with probability at least f 1,G (p).
We consider the -random graph (Z × G) . For n ⩾ 1 let Y n be the event that For n = 1, set X 1 to be the trivially satisfied event occurring with probability 1. For n ⩾ 1, let V n be the event that {n} × V(G) induces a connected subgraph, and for n ⩾ 2 let H n be the event that at least one of the edges from It easily follows that X n = Y n−1 ∩ H n and X n ∩ V n ⊆ Y n . From here, we obtain the following inclusions: .
We begin by establishing two inductive relations for the sequences x n and y n . First of all, using (a) and (b) we have, Second, using (c), Furthermore, we have by (3.2) and (3.1) that so our claim follows by induction.
Finally, we have that ) .
▪ For any finite connected graph G, f 1,G (p) is a nondecreasing function of p with f 1,G (1) = 1. Thus the function (f 1,G (p)) 2 is also nondecreasing in p and attains a maximum value of 1 at p = 1. On the other hand, the function 4(1 − p) v(G) is strictly decreasing in p and is equal to 4 at p = 0. Thus there exists a unique solution p ⋆ = p ⋆ (G) in the interval [0, 1] to the equation Theorem 26 thus has the following immediate corollary.

IMAGINARY LIMITS OF REAL CONSTRUCTIONS: A PRELIMINARY LEMMA
In this section we prove a lemma that we shall use in Sections 5 and 7. The lemma will allow us to use certain vertex-based constructions to create other 1-ipms that cannot be represented as vertex-based constructions (or would correspond to vertex-based constructions with "complex weights").

Lemma 28.
Let G be a finite graph, and let  ∶= {Q H ( ) ∶ H ⊆ G} be a set of polynomials with real coefficients, indexed by subgraphs of G. Given ∈ C, let be the following function from subgraphs of G to C: Suppose there exists a nontrivial interval I ⊆ R such that, for all ∈ I, the function defines a 1-ipm on G. Suppose further that there exists a set X ⊆ C such that, for all ∈ X and all H ⊆ G, (H) is a nonnegative real number. Then is a 1-ipm on G for all ∈ X.
Proof. We start by proving that is a measure on G for all ∈ X. As (H) is a nonnegative real number for all ∈ X and all H ⊆ G, all that is left to prove is that The left hand side of (4.1) is a polynomial in with real coefficients, and is equal to zero for all in the interval I. By the fact that a nonzero polynomial over any field has only finitely many roots, the polynomial is identically zero and so (4.1) holds for all .
We now show that is a 1-ipm on G for all ∈ X. To do this we must show that the following holds true for all ∈ X, for all A, B ⊆ V(G) such that A and B are disjoint, and all G 1 and G 2 such that Both sides of (4.2) are polynomials in with real coefficients-the left hand side, for example, can be written as As is a 1-ipm on G for all ∈ I, we have that these two polynomials agree on I, and so must be the same polynomial. Thus (4.2) holds as required. ▪

THE LINE LATTICE Z
In this section we prove Theorem 15 on the connectivity function of paths. Recall that, given n ∈ N ⩾2 and p ∈ [0, 1], we let = (p) ∶= We begin by constructing a measure p ∈  1,⩾p (P n ) as follows. Let us start with the case p ⩾ 3 4 . For each vertex of P n , we set it to state 0 with probability , and set it to state 1 otherwise, and we do this independently for every vertex. Recall that for each j ∈ [n] we write S j for the state of vertex j; in this construction, the states are independent and identically distributed random variables. We set the edge {j, j + 1} to be open if S j ⩽ S j+1 , and closed otherwise. Thus, as p = + (1 − ) 2 , we have that each edge is open with probability p. Moreover (P n ) p will be connected if and only if there exists some j ∈ [n + 1] such that S k = 0 for all k < j, while S k = 1 for all k ⩾ j. Therefore (P n ) p is connected with probability g n ( ). As this construction is vertex-based, it is clear that it is 1-independent.
When p < 3 4 we have that is a complex number, and so the above construction is no longer valid. However, as discussed in Section 4, we will show that it is possible to extend this construction to all p ∈ [p n , 1]. For each subgraph G of P n , set Q G ( ) to be the polynomial p (( ] . The following claim, together with Lemma 28, shows that in fact p is a 1-ipm on P n for all p ∈ [p n , 1]. Claim 29. For all p ∈ [p n , 3 4 ) and all G ⊆ P n we have that Q G ( (p)) is nonnegative real number.
Proof. We proceed by induction on n. When n = 2 we have that there are only two possible subgraphs of P 2 , which are P 2 itself and its complement P 2 . We have that Q P 2 ( (p)) = p and Q P 2 ( (p)) = 1 − p, so the claim holds as required for n = 2.
Let us now assume that n > 2 and that the claim is true for all cases from 2 up to n − 1. We split into two further subcases. We first deal with the case that G = P n . We have that Q P n ( (p)) = g n ( ). For p < 3 4 we can write When p < 3 4 we have that and 1 − are complex conjugates, and also that 2 − 1 is a pure imaginary number. Thus both the numerator and denominator of the above fraction are pure imaginary, and so g n ( (p)) is a real number for all p < 3 4 . By writing = re i , where r ∶= , we can rewrite (5.1) as , which in turn gives sin ((n + 1) ) ≥ 0. Thus by (5.2) above, g n ( (p)) is a nonnegative real number for all p in the interval [p n , 3 4 ), as required.
We now deal with the case that G ≠ P n . Let us consider the vertex-based construction from which Q G ( ) was defined. As not every edge is present in G we have that there exists some j ∈ [n − 1] such {j, j + 1} is not an edge, and so S j = 1 while S j+1 = 0. Note that if j ⩾ 2, then the edge , then we have that Now, by induction, we have that Q G 1 ( (p)) and Q G 2 ( (p)) are positive real numbers for all p ∈ [p n , 3 4 ); to make this inductive step work we are using the fact that (p n ) n⩾2 forms an increasing sequence, and so p ⩾ p n implies that p ⩾ p s for all s ⩽ n. As (p) (1 − (p)) = 1 − p, we have that (5.3) is a positive real for all p ∈ [p n , 3 4 ), and so we have proven the claim. ▪ Note that as this proof shows that g n ( (p n )) = 0, we have that the probability (P n ) p n is connected is equal to 0. As p n ∈  1,⩾p (P n ) for all p ⩽ p n , we have that f 1,P n (p) = 0 for all p ⩽ p n .
We now prove that this construction is optimal with respect to the connectivity function. Note that the following proof involves essentially following the proof of Theorem 26 when G consists of a single point and checking that the above construction is tight at every stage of this proof. Finally, we should emphasize that the main ideas in the construction of p and its analysis are due to Balister and Bollobás [4] (they considered slightly different probabilities for vertex states, setting S k = 0 with probability q k , where q k is defined for k ∈ [n] by q 1 = 0 and by the recurrence relation q k = min for k ≥ 2, which corresponds exactly to the equality case in inequality (5.5) below).
Proof of Theorem 15. The above construction discussed shows that It is clear that f 1,P n (p) ⩾ 0 for all p, and so all that remains to show is that f 1,P n (p) ⩾ g n ( (p)) for all p ∈ [p n , 1]. Let ∈  1,⩾p (P n ). For k ∈ [n], let X k be the event that the subgraph of (P n ) induced by the vertex set [k] is connected, and let H k be the event that the edge {k − 1, k} is not present in (P n ) . Applying random sparsification as in Remark 3 if necessary, we may assume without loss of generality that for every k, the event H k occurs with probability exactly 1 − p.
Let q 2 ∶= ((X 2 ) c ) = 1 − p, and for k > 2 let q k ∶= ((X k ) c |X k−1 ). We have that Note that ((P n ) = P n ) = ∏ n j=2 (1 − q n ). Thus to show that the previous construction is optimal with respect to the connectivity function it is enough to show that equality holds for inequalities (5.4) and (5.5) when = p . In the measure p , we have that every edge is present with probability exactly p, thus p (H k ) = 1 − p and so equality holds in (5.5). To prove that equality holds in (5.4), it is sufficient so show that Both the left and right hand sides of (5.6) can be expressed as polynomials in (p), and so it is sufficient to show that equality holds for p ⩾ 3 4 , as that will show they are the same polynomial (and so equality holds for all p ∈ [p n , 1]). Suppose that the event (H k ∩ X k−2 ) occurs. As H k has occurred we have that S k−1 = 1 while S k = 0. As S k−1 = 1, we have that edge {k − 2, k − 1} is open, regardless of S k−2 . Thus, as X k−2 has occurred we also have that X k−1 has occurred. Therefore (H k ∩ X k−1 ) has also occurred, and so we are done. ▪ We remark in similar fashion to the above proof that the following holds for any ∈  1,⩾p (P n ): Moreover, by once again considering what states of vertices can lead to the various events, we have that equality holds for all of the above inequalities when = p . This leads us to another way to define g n ( (p)): let g 1 ( (p)) ∶= 1, g 2 ( (p)) ∶= p, and for all n ⩾ 3 we have that g n ( (p)) = g n−1 ( (p)) − (1 − p)g n−2 ( (p)) .
We conclude this section with a proof of Theorem 11(i).

Proof of Theorem 11(i).
For the upper bound we plug f 1,P 1 (p) = 1 into equation (3.4), solve that equation to get p ⋆ (P 1 ) = 3 4 and apply Corollary 27 to obtain p 1, (Z) ⩽ 3 4 . For the lower bound, let p < 3 4 be fixed. As the sequence (p n ) n∈N is monotone increasing and tends to 3∕4 as n → ∞, there exists N ∈ N such that p < p N . We showed in Theorem 15 that there exists a measure p N ∈  1,⩾p N (P N ) such that the probability (P N ) p N is connected is equal to zero. We use this measure to create a measure ∈  1,⩾p (Z). For each i ∈ Z, we let the subgraphs Z [(i (N − 1) + [N])] on horizontal shifts of P N by i(N − 1) be independent identically distributed random variables with distribution given by p N . This gives rise to a 1-independent model on Z with edge-probability at least p (in fact at least p N ). Furthermore, all connected components of Z have size at most 2(N − 1) − 1. In particular, p 1, (Z) ⩾ p. Since p < 3 4 was chosen arbitrarily, this gives the required lower bound p 1, (Z) ⩾ 3 4 . ▪

THE LADDER LATTICE Z × P 2
In this section we construct a family of 1-ipms on segments of the ladder Z × P 2 with edge-probability close to 2∕3 for which with probability 1 there are no open left-right crossings. The idea of this construction is due to Walters and the second author [16] (though the technical work involved in rigorously showing the construction works is new). Let us begin by giving an outline of our construction. We write the vertex set V(P N ×P 2 ) as [N]× [2]. As in the case of the line lattice, we independently assign to each vertex (n, y) a random state S (n,y) . If n + y is even, then we let S (n,y) ∶= Clearly the bond percolation measure associated to our random graph model G is a 1-ipm on the ladder G = P N × P 2 as it is vertex-based. By making a judicious choice of the sequences  . Then there exists N ∈ N such that for all n ⩾ N, . We start by defining the sequences (p n ) n∈N , (r n ) n∈N and (s n ) n∈N iteratively as follows. We set p 1 = r 1 = 1 and s 1 = 0. Then for n ∈ N, we let otherwise.

Lemma 31.
The following hold for all n ∈ N: Proof. We prove the lemma by induction on n. By definition of our sequences, whence (ii) holds for n = N + 1.
Finally we consider (v) for n = N + 1, which is the most delicate part of the induction. We begin by recording two useful facts, the second of which we shall reuse later.
Proof. If p N+2 = 0, then by construction s N+2 = 0 and so we are done. If r N+2 = 0, then by construction p N+1 ≤ 1 − p, which by our inductive hypothesis (iii) implies p N+2 ≤ 1 − p and hence Proof. Fix i ∈ {2, … , N + 1}. By our inductive hypotheses (iii) and (iv) (which we have already established up to n = N + 1) and since i ⩾ 2, we have 0 < p N+2 ⩽ p i ⩽ p 2 = p and 0 < r N+2 ⩽ r i ⩽ r 2 = p. Since r i+1 > 0, we in fact have p i > 1 − p. We also have that On the other hand for fixed y ∈ (0, 1∕2), the function x  → f (x, y) is strictly increasing in x. Therefore if r i < 1∕2, we have In either case, f (p i , r i ) > 0, and thus s i = max With these results in hand, we return to the proof of (v). If s N+2 = 0, then (v) follows immediately from (iv). Thus we may assume that s N+2 > 0, whence by Claim 32 p N+2 > 0 and r N+2 > 0. By Claim 33 and our inductive hypotheses (iii) and (iv), this implies that p i , r i and s i are all strictly positive for i ∈ {2, 3 … , N + 2}. By definition of our sequences we thus have for all i ∈ [N + 1] that Combining these equations we obtain for i ∈ {2, … , N + 1} that: Claim 34. Under our assumption that s N+2 > 0, for all integers i ∈ [N + 1] we have Proof. Since p 1 = 1 and p 2 = p, our claim holds for i = 1. Suppose it holds for some i ⩽ N. Then by rearranging terms, we have Substituting this into the formula for p i+2 given by (6.2), we see our claim holds for i + 1 as well. ▪ It follows from Claim 34 and (6.1) that for all i ∈ [N + 1], we can write r i+1 + s i+1 as a function of p i : For p i > (1 − p) (which we recall holds since r i+1 > 0), the expression above is an increasing function of p i . By our inductive hypothesis (iii) that p N+1 ⩽ p N it follows that r N+2 +s N+2 ⩽ r N+1 +s N+1 and we have verified that (v) holds for n = N + 1. ▪ Recall that p = 2 3 − , for some fixed ∈ Then r m+1 = 0, and thus by the argument above, we have that p n = r n = s n = 0 for all n ∈ N ⩾m+2 , as required.
Finally, suppose p n > 1 − p and r n > 0 both hold for all n ∈ [N − 2]. By Claim 33, we have s n > 0 for all n ∈ {2, … , N − 3}. This allows us in turn to apply Claim 34 to all n in this interval and to deduce that Recall that p 1 = 1. As such, it follows from inequality (6.4) that p n ⩽ 1 − (n − 1) 3 4 for all n ∈ [N − 2]. In particular, as N = ⌈2 −1 ⌉ and ∈ ( 0, 1 6 ) , we have which is a contradiction. ▪ Now let N = N be the integer constant whose existence is given by Lemma 35 and construct the 1-ipm G on the graph G = P N × P 2 from independent random assignments of states S (n,y) to vertices (n, y) in V(G) = [N] × [2], as described at the beginning of this section.
We observe here that by Lemma 31(i) and (ii), the states S (n,y) are well-defined random variables for every (n, y) ∈ [N] × [2], and so is a well-defined 1-ipm. We recall here for the reader's convenience the state-based rules governing which edges are open in G : Claim 36. We have that ( ) ⩾ p.
Proof. For (n, y) ∈ [N − 1] × [2], consider the horizontal edge {(n, y), (n + 1, y)}, . If n + y is even, then by definition of r n+1 , Similarly if n + y is odd, then by definition of p n+1 ,  Given n ⩾ N, we may extend to an element ′ ∈  1,⩾p (P n × P 2 ) by letting every edge in P n × P 2 ⧵ P N × P 2 be open independently at random with probability p. In this way we obtain a 1-independent bond percolation measure ′ on P n × P 2 with edge-probability p for which there almost surely are no open left-right crossings of P n × P 2 , giving the required lower bound on p 1,× (P n × P 2 ). ▪ We conclude this section by proving Theorem 11(ii), with the aid of Theorem 30.
Proof of Theorem 11(ii). Trivially, the 1-independent connectivity function of the path on 2 vertices P 2 (i.e., the graph consisting of a single edge) is f 1,P 2 (p) = p. Thus the constant p ⋆ (P 2 ) defined by equation (   . In the proof of Theorem 30, we showed there exist some integer N ∈ N and ∈  1,⩾p (P N × P 2 ) such that We use this measure to create a measure ∈  1,⩾p (Z × P 2 ). Let G ∶= Z × P 2 . For each i ∈ Z, we let the subgraphs G [(i(N − 1) + [N]) × [2]] on horizontal shifts of the ladder P N × P 2 by i(N − 1) be independent identically distributed random variables with distribution given by . Thanks to property (ii) recorded above, the random subgraphs agree on the vertical rungs {1 + i(N − 1)} × P 2 of the ladder, and this gives rise to a bona fide 1-independent model on Z × P 2 with edge-probability p. Furthermore, property (i) implies all connected components in G have size at most 4(N − 1) − 2 = 4N − 6. In particular, p 1, (Z × P 2 ) ⩾ p. Since p < 2 3 was chosen arbitrarily, this gives the required lower bound p 1, (Z × P 2 ) ⩾ 2 3 . ▪

An upper bound for f 1,K n (p)
Before proving Theorem 16, let us give a simple vertex-based construction of a measure p ∈  1,⩾p (K n ) that shows f 1,K n (p) ⩽ g n ( ) for p ⩾ 1 2 . We call this measure the Red-Blue construction. We think of K n as the complete graph on vertex set [n], and we color each vertex Red with probability and color it Blue otherwise, and we do this independently for all vertices. The edge {i, j} ∈ [n] (2) is open if and only if i and j have the same color. As p = 2 +(1− ) 2 , we have that each edge is present in (K n ) p with probability p. Note that (K n ) p will either be either a disjoint union of two cliques, in which case it is disconnected, or the complete graph K n , in which case it is connected. This latter case occurs if and only if every vertex receives the same color, and so the probability that (K n ) p is connected is equal to g n ( ). As this construction is vertex-based, it is clear that it is 1-independent.
If p < 1 2 then is a complex number, and so the Red-Blue construction is no longer valid. However, as discussed in Section 4, we will show that it is possible to extend this construction to all p ∈ [p n , 1]. Given j ∈ {0, 1, … , n}, let When j = 0 or j = n we have that g n,0 ( ) and g n,n ( ) are each equal to g n ( ), and so we just write the latter instead. Given some A ⊆ [n], let H A be the disjoint union of a clique on A with a clique on [n] ⧵ A. Note that when A = ∅ or [n] we have that H A is equal to K [n] , and more generally that H A = H [n]⧵A . For p ∈ [0, 1], let p be the following function on subgraphs G of K n : , 0 e l s e .
For p ∈ this function matches the Red-Blue construction given above, and so by defining for all subgraphs G ⊆ K n , we obtain a measure p which is a 1-ipm defined without making reference to states of vertices. The following claim, together with Lemma 28, shows that in fact p is a 1-ipm on K n for all p ∈ [p n , 1].
and all j ∈ {0, … , n} we have that g n,j ( (p)) is nonnegative real number.
Proof. Let us begin with the case j = n. As p ⩽ 1 2 , we have that and 1 − are complex conjugates, and so g n ( (p) ) is a real number for all p in this range. By writing = re i , where r ∶= , we can write g n ( (p)) = 2r n cos (n ) .
implies 0 ≤ ≤ 2n , which in turn gives cos(n ) ≥ 0. By (7.1), it follows that g n ( (p)) is a nonnegative real number for all p ∈ [ p n , 1 2 ] , which proves the claim when j = n. For general j ∈ {0, … , n}, we have that Therefore the previous case of the claim shows that g n,j ( (p)) ∈ [0, 1] for all p ∈ ; at this stage we are using the fact that (p n ) n⩾2 forms an increasing sequence, and so p ⩾ p n implies that p ⩾ p s for all s ⩽ n. ▪ Note that as this proof shows that g n ( (p n )) = 0, we have that the probability (K n ) p n is connected is equal to 0. As p n ∈  1,⩾p (K n ) for all p ⩽ p n , we have that f 1,K n (p) = 0 for all p ⩽ p n . We now prove that this construction is optimal with respect to the connectivity function.

7.2
A lower bound on f 1,K n (p) Proof of Theorem 16. The previous constructions discussed show that It is clear that f 1,K n (p) ⩾ 0 for all p, and so all that remains to show is that f 1,K n (p) ⩾ g n ( ) for p ∈ [p n , 1]. We will prove this result by induction on n. The inequality is trivially true when n = 2, so let us assume that n > 2 and that the inequality is true for all cases from 2 up to n − 1. First, we note that g n ( ) = g j ( )g n−j ( ) − g n,j ( ) for all j ∈ {0, 1, … , n}. Thus, if we multiply both sides of this equation by ( n j ) and sum over all j ∈ {0, 1, … , n}, we have that Let ∈  1,⩾p (K n ) and let C be the event that (K n ) is connected.
Note that when A = ∅ or A = [n], the above equation is trivially true due to the fact that C, X ∅ , X [n] , Y ∅ and Y [n] are all the same event. As is 1-independent we have that if A is a nonempty proper subset of [n], then, by induction on n, we have Note that here we are using the fact that (p n ) n⩾2 forms an increasing sequence, and so p ⩾ p n implies that p ⩾ p s for all s ⩽ n. We are also using the fact that g 1 ( ) = 1 for all ∈ [0, 1]. We proceed by summing (7.3) over all nonempty proper subsets of [n], and then applying (7.4) to obtain We apply (7.2) and the fact that the events C, Y ∅ and Y [n] are all the same event to (7.5) to get We apply (7.7) to (7.6) to obtain (2 n − 4) (C) ⩾ (2 n − 4)g n ( ). As n > 2, we have that (C) ⩾ g n ( ) and so we are done. ▪

7.3
A remark on f k,K n (p) for k ⩾ 2 Clearly we can define f k,G (p) analogously to f 1,G (p) for k ∈ N 0 . For k = 0, f 0,K n (p) is exactly the probability that an instance of the Erdős-Rényi random graph G n,p contains a spanning tree. As far as we know, there is no nice closed form expression for this function.
In this section, we have computed f 1,K n (p) exactly, which is the other interesting case, as for k ⩾ 2 the connectivity problem is trivial.
Proposition 39. For all k, n ∈ N ⩾2 , we have that Proof. For the lower bound, consider ∈  k,⩾p (K n ). Since any subgraph of K n with at least ( n 2 ) − (n − 1) edges is connected, we can apply Markov's inequality to show that For the upper bound, consider the random graph G obtained as follows. Let x ∶= 1−p 2 . With probability min(nx, 1), select a vertex i ∈ [n] = V(K n ) uniformly at random, and let G be the subgraph of K n obtained by removing all edges incident with i. Otherwise, let G be the complete graph K n . It is easy to check that G is a 2-independent model with edge-probability p and that G is connected if and only if G = K n , an event which occurs with probability 1 − min(1, nx) = max (0, 1 − n(1 − p)∕2). ▪

8.1
Linear programming for calculating f 1,G (p) In this subsection we describe how we can represent the problem of finding f 1,G (p), for any graph G, as a (possibly nonlinear) program. Given a graph G on vertex set [n], let  = (G) be the set of all labeled subgraphs of G. Throughout this section we treat these subgraphs as subsets of E(G), and always imagine them to be on the full vertex set [n]. For each labeled subgraph of G we write Recall that for a function ∶  → R ⩾0 , we have ∈  1,⩾p (G) if and only if the following three conditions all hold: 1. is a probability measure on labeled subgraphs of G, 2. Every edge of G is open in G with probability at least p, 3. Given nonempty S, T ∈  such that S and T are supported on disjoint subsets of [n], (S) ⋅ (T) = (S ∪ T).
As we are interested in determining f 1,G (p), and as randomly deleting edges cannot increase the probability of being connected, we may assume that in fact every edge of G is open in G with probability exactly p (by applying random sparsification as in Remark 3 if necessary).We can thus rewrite the conditions above in the following way: 1. ∑ H∈ (Ĥ) = 1, 2. For all edges e ∈ E(G), we have that ∑ H∈ 1(e ∈ H) (Ĥ) = p, 3. For all nonempty S, T ∈ , such that S and T are supported on disjoint subsets of [n], we have that Let A = A(G) be a matrix which has columns indexed by , and a row for each piece of information given by one of the above conditions. That is: 1. We have a row for the empty set such that A ∅,H ∶= 1. Let q = q(G) be a vector with indexing the same as the rows of A; let q ∅ ∶= 1, q e ∶= p for e ∈ G, and q {S,T} ∶= 0 for each pair S, T ∈  ⧵ {∅} supported on disjoint subsets of [n]. Then a vector w, whose entries are indexed by , which satisfies w H ⩾ 0 for all H ∈ , and also Aw = q corresponds precisely to a 1-ipm on G with {e open} = p for all edges e ∈ E(G). Let c be a vector indexed by  defined by c H ∶= 1(H is connected). Just to make it clear, we say that H ∈  is connected if it contains a spanning tree of [n]. Then for a given value of p the vector w(p) satisfying Aw(p) = q and c T w minimal corresponds to a measure ∈  1,⩾p (G) such that (H is connected) = f 1,G (p).
Observe that for any graph with five vertices or fewer, any partition of the graph into two parts has that one part must have at most two vertices in it. In particular, if G is a graph on [5], and S and T are nonempty subgraphs of G supported on disjoint subsets of [5], then one of S and T must consist of precisely one edge of G. By choosing T to be this subgraph, we can always choose S and T for (8.1) so that (T) = p. Thus for any choice of p, we can turn the problem of finding f 1,G (p) into the following linear program: a * = min w c T w subject to Aw = q, w ⩾ 0. (8.2) (Note that for graphs with six or more vertices, one may find S and T such that (T) (in (8.1)) is an unknown function of p, and thus the program is not linear; for example, this indeed is the case for C 6 .) The duality theorem states that the asymmetric dual problem has the same optimal solution a * : One can easily solve the linear programs above for a specific value of p, for example using the software Maple, and the LPSolve function it contains. However we of course wish to find solutions for all values of p ∈ [0, 1]. By writing A = (a ij ), w = (w j ), c = (c j ), q = (q i ) and x = (x i ) any solutions w and x must satisfy ∑ j a ij w j = q i , ∑ i a ij x i ⩽ c j , x i ≥ 0 and w i ⩾ 0. Thus we have In particular for optimal solutions we have ∑ i q i x i = ∑ j c j w j and so the inequality must be an equality, that is ( ∑ i a ij x i ) w j = c j w j , for all j.
Consequently for each j we either have w j = 0 or ∑ i a ij x i = c j . Thus in our attempt to obtain a function for all p, it seems reasonable to look at an optimal solution for one value of p and see which w j have been set to zero; assume for these indices that we always have w j = 0 and attempt to directly solve the equations that result from this. This motivates the following method: • Solve (8.2) with a specific value of p to obtain a solution w(p) and a set J ∶= {j ∈ [|w|] ∶ w j (p) = 0}.
For the given interval P which works above, the conditions above ensure that the w ′ (p) and x ′ (p) obtained in this way are feasible solutions to (8.2) and (8.3) respectively. Thus if w * (p) = x * (p), then by the duality theorem we have f 1,G (p) = w * (p). Furthermore, a measure on the subgraphs of G which is extremal is given directly by w ′ (p). In the following subsection we give, as examples, two results which are proved using the above method.

8.2
The connectivity function of small cycles In this subsection we prove Theorems 17 and 18 using the above method. Furthermore, the method gives us an extremal example in each case.
Proof of Theorem 17. For C 4 and p ∈ an extremal construction is given by the measure , defined by if H is contains precisely two edges, which are adjacent; ( We can in fact give a direct combinatorial proof of the lower bound in Theorem 17: for any ∈  1,⩾p (C 4 ), we have by 1-independence that Together with the first of the constructions of measures above (which can be found by analyzing how the bound in the inequality above can be tight), this gives a second and perhaps more insightful proof of Theorem 17 than the one obtained from applying the linear optimisation method. However for the next result, on f 1,C 5 (p), we do not have a combinatorial proof, and our result relies solely on linear optimisation.
Proof of Theorem 18. For C 5 and p ∈ an extremal construction is given by the measure , defined by if H is missing precisely two edges, which are adjacent; if H is missing precisely two edges, which are not adjacent; if H is the empty graph; 0 otherwise.
an extremal construction is given by the measure , defined by if H is the empty graph; if H consists of precisely two edges, which are adjacent; if H is missing precisely two edges, which are adjacent; if H is missing precisely two edges, which are not adjacent; 0 otherwise. ▪

General bounds for cycles of length at least 6
We can use Markov's inequality to derive the following simple lower bound on f 1,C n (p) for n ⩾ 6. A small adjustment to this argument gives the following improvement for n = 6.
Proof of Proposition 40. Let ∈  1,⩾p (C n ). Note that G is connected if and only if it has at most one closed edge. Thus by Markov's inequality, we have Now by simple counting again, linearity of expectation and the inequality above, we get:

MAXIMIZING CONNECTIVITY
In this section, we derive our results for maximizing connectivity in 1-independent modes. First of all Theorem 16 allow us to easily determine the value of F 1,K n (p) and hence prove Theorem 21.
Proof of Theorem 21. Given a 1-independent model G on K n with edge-probability at least 1 − p, observe that the complement G c of G in K n is a 1-independent model in which every edge is open with probability at most p. Furthermore, G c is connected whenever G fails to be connected. This immediately implies Furthermore, observe that the Red-Blue measure p we constructed to obtain the upper bound on f 1,K n (p) in the proof of Theorem 16 has the property that a p -random graph is connected if and only if its complement fails to be connected. This immediately implies that we have equality in (9.1). ▪ For paths, a simple construction achieves the obvious upper bound for F 1,P n (p).
Proof of Theorem 20. For any measure ∈  1,⩽p (P n ), we have by 1-independence that which implies F 1,P n (p) ⩽ p ⌊ n 2 ⌋ . For the lower bound, we construct a 1-ipm as follows. For each integer i: 1 ⩽ i ⩽ n∕2, we assign a state On to the vertex 2i with probability p, and a state Off otherwise, independently at random. Then set an edge of P n to be open if one of its endpoints is in state On, and closed otherwise. This is easily seen to yield a 1-ipm on P n in which every edge is open with probability p, and for which Thus F 1,P n (p) ⩾ p ⌊ n 2 ⌋ , as claimed. ▪ The case of cycles C n appears to be slightly more subtle. For the 4-cycle, as in the previous section, we can give two proofs, one combinatorial and the other via linear optimisation.
Proof of Theorem 17. The theorem immediately follows from an application of the linear optimisation techniques from Section 8. Alternatively, we can obtain the upper bound by a direct argument. For any measure ∈  1,⩽p (C 4 ), we have by 1-independence that Combining these two inequalities and using 1 − (1 − p) 2 = 2p − p 2 , we obtain which gives the claimed upper bound on F 1,C 4 (p).
For the lower bound, we give two different constructions, depending on the value of p. For p ∈ [ consider the measure defined by if H contains precisely three edges; ] an extremal construction is given by the measure , defined by if H contains precisely four edges; if H contains precisely two edges, which are adjacent; if H contains precisely one edge; 0 otherwise.
an extremal construction is given by the measure , defined by if H contains precisely four edges; if H contains precisely two edges, which are adjacent; if H contains precisely one edge; if H is the empty graph with no edges; 0 otherwise.
an extremal construction is given by the measure , defined by if H contains precisely four edges; if H is missing precisely two edges, which are adjacent; if H contains precisely two edges, which are adjacent; if H is the empty graph with no edges; 0 otherwise. ▪

PROOF OF THEOREM 1.12
Combining Corollary 27 with our results on 1-independent connectivity, much of Theorem 12 is immediate.
Proof of Theorem 12. For the lower bound in part (i), we note that Z×C n has the finite 2-percolation property. Thus, as described after the proof of Theorem 7, we have that p 1, (Z × C n ) ⩾ 4 − 2 √ 3. For the upper bound in part (i), since the long paths critical probability is nondecreasing under the addition of edges, we have which is at most 2∕3 by Theorem 11(ii).
For the upper bounds (ii)-(iv) Theorem 12 follow directly from our results on 1-independent connectivity functions. For G = K 3 , C 4 , C 5 , we plug in the value of f 1,G (p) in equation (3.4), solve for p ⋆ (G) and apply Corollary 27.
In part (v), we begin by noting that as we are considering an increasing nested sequence of graphs, the sequence ( p 1, (Z × K n ) ) n∈N is nonincreasing in [0, 1] and hence tends to a limit as n → ∞. For the lower bound in (v), observe that for any n ∈ N the graph Z × K n has the finite 2-percolation property-indeed for any finite k, the closure of a copy of P k × K n under 2-neighbor bootstrap percolation in Z × K n is equal to itself. We construct a 1-ipm on Z × K n as in Corollary 24 but with starting set T 0 = {0} × V(K n ) and hence T k = ({k} × V(K n )) ∪ ({−k} × V(K n )). It is easily checked that -almost surely, all components (and hence all paths) in a -random graph have length at most 5n. Since by construction ( ) = 4 − 2 √ 3, this proves for all n ∈ N. For the upper bound, we perform some simple analysis. By solving a quadratic equation, we see that for all fixed p ∈  . Then by Theorem 16, for any such fixed p and all n sufficiently large, we have that Thus p ⋆ (K n ) < p for all n sufficiently large, which by Corollary 27 implies p 1, (Z × K n ) < p. ▪

More tractable subclasses of 1-independent measures
The most obvious open problem about 1-independent percolation is of course whether the known lower and upper bounds on p 1,c (Z 2 ) can be improved. This problem is, we suspect, very hard in general. However, it may prove more tractable if we restrict our attention to a smaller family of measures.
Definition 42. Let G be a graph. A G-partition is a partitioned set ⊔ v∈V(G) Ω v , with nonempty parts indexed by the vertices of G. A G-partite graph is a graph H on a G-partition V(H) = ⊔ v∈V(G) Ω v whose edges are a subset of the union of the complete bipartite graphs Given a G-partite graph H on a G-partition ⊔ v∈V(G) Ω v , we have a natural way of constructing 1-independent bond percolation models: given a family X = (S v ) v∈V(G) of independent random variables with S v taking values in Ω v , the (H, X) Let  vb,⩾p (G) denote the collection of all vertex-based measures on G with edge-probability at least p.
Vertex-based measures arise naturally in renormalising arguments, and are thus a natural class of examples to consider. A special case of Problems 6 and 44 is obtained by further restricting our attention to the case where the Ω v have bounded size.

Definition 45.
A vertex-based measure on a graph G is N-uniformly bounded if it as in Definition 43 above and in addition for each v ∈ V(G), |Ω v | ⩽ N. Furthermore, a vertex-based measure on a graph G is uniformly bounded if it is N-uniformly bounded for some N ∈ N.
Let  N−ubvb,⩾p (G) and  ubvb,⩾p (G) denote the collection of all vertex-based measures on G with edge-probability at least p that are N-uniformly bounded and uniformly bounded respectively.
Finally, let us note that the second most obvious problem arising from our work, besides that of improving the bounds on p 1,c (Z 2 ), is arguably that of giving bounds on p 1, (Z 2 ) and closely related variants. Such problems, which correspond to new questions in extremal graph theory, are discussed in the subsections below. For these problems too we believe restrictions to the class of uniformly bounded vertex-based 1-ipms could be both fruitful and interesting in their own right.

Harris critical probability for other lattices
Beyond Z 2 , it is natural to ask about bounds on p 1,c (G) for some of the other commonly studied lattices in percolation theory.
Problem 47. Give good bounds on the value of p 1,c (G) when G is one of the eleven Archimedean lattices in the plane or the -dimensional integer lattice Z .
This problem is particularly interesting when G is the triangular lattice or the honeycomb lattice (two lattices for which the 0-independent Harris critical probability is known exactly), or the cubic integer lattice Z 3 (which is important in applications). A challenge in all cases is finding constructions of nonpercolating 1-independent measures with high edge-probability-indeed, our arsenal of constructions for 1-independent percolation problems is so sparse that any new construction could be of independent interest. In a different direction, we can observe that Z +1 contains a copy of Z , whence the sequence ( p 1,c (Z ) ) ∈N is nonincreasing in [0, 1] and converges to a limit. Balister and Bollobás asked for its value: Given k fixed, it is easy to construct a 3k-ipm on Z 2 with ( ) = 1 − 2 k and no open path of length more than (2k + 1) 2 . Indeed, build a random graph model as follows: • begin with all edges of Z 2 open; • independently for each (i, j) ∈ Z 2 , choose H ij ∈ [k + 1] uniformly at random and then for all j ′ : j(k + 1) − k ≤ j ′ ≤ j(k + 1) + k, set the horizontal edge {(i(k + 1) + H ij − 1, j ′ ), (i(k + 1) + H ij , j ′ ) to be closed; • independently for each (i, j) ∈ Z 2 , choose V ij ∈ [k + 1] uniformly at random and then for all i ′ : i(k+1)−k ≤ i ′ ≤ i(k+1)+k, set the vertical edge {(i ′ , j(k+1)+V ij −1), (i ′ , j(k+1)+V ij ) to be closed.
It is easy to check that this random graph model is 3k-independent, has edge probability at least k 2 (k+1) 2 ≥ 1 − 2 k and that every connected component has order at most (2k + 1) 2 .
Corollary 58. For any fixed k ∈ N, In particular we have lim k→∞ p k, (Z 2 ) = 1 (and in fact a similar construction shows this remains true in Z ). Finally, as in Section 11.3, we should observe that the almost sure existence of arbitrarily long open paths in a 1-independent model on G does not imply that for every ∈ N every vertex of G has a strictly positive probability of being part of a path of length at least . Thus we may actually define a second long paths critical probability, Our construction in the proof of Theorem 7 shows that p 1, 2 (Z 2 ) ⩾ 4 − 2 √ 3, and we know it is upper-bounded by p 1,H 2 (Z 2 ) ⩽ 0.8639. As in Section 11.1, it may be fruitful to study the long paths critical constant when one restricts one's attention to a smaller class of 1-ipms. In particular, by considering the class of uniformly bounded vertex-based measures, one is led to the following intriguing problem in graph theory.
Given an n-uniformly bounded Z 2 -partite graph H with partition ⊔ v∈Z 2 Ω v . A transversal subgraph of H is a subgraph of H induced by a set of distinct representatives S for the parts of H, that is, a set of vertices of H such that |S ∩ Ω v | = 1 for all v ∈ Z 2 . The G-partite density of H is } . This question can be viewed as a problem from extremal multipartite graph theory. Plausibly some tools from that area, in particular the work of Bondy, Shen, Thomassé and Thomassen [12] and Pfender [28], could be brought to bear on it.

Connectivity function
We determined in Sections 8 and 9 the connectivity function f 1,C n (p) for cycles C n of length at most 5.
It is natural to ask what happens for longer cycles.
As mentioned in Section 8, the problem of finding f 1,C 6 (p) is nonlinear. Nevertheless, one can use software, such as Maple and its contained NLPSolve function, to try to estimate the answer. This suggests the following: • The threshold at which f 1,C 6 (p) becomes nonzero is approximately p = 0.59733; • For p just above this threshold, the best "asymmetric" (see the next subsection for a definition) measure is better than the best "symmetric" measure; e.g. at p = 0.62 we have f 1,C 6 (0.62) is approximately 0.007, but is as high as 0.11 when restricted to "symmetric" measures.
More generally, one can ask what happens in cycles if we have higher dependency or if we try to maximize connectivity rather than minimize.
Beyond paths, cycles and complete graphs, the 1-independent connectivity problem is perhaps most natural to study in the hypercube graph Q n and in the n × n toroidal grid C n × C n . Progress on either of these would likely lead to progress on other problems in 1-independent percolation as well.
Problem 65. Determine f 1,C n ×C n (p) for all n ⩾ 3.
In a different direction, we can ask whether the extremal measures attaining f 1,G (p) can be required to have "nice" properties. For C 4 and p ∈ [0, 1∕2] another extremal construction for f 1,C 4 (p) is given by the measure , defined by if H is the empty graph; if H is contains precisely two edges, which are not adjacent; 0 otherwise.
Motivated by the above, we call a measure ∈  1,⩾p (G) symmetric if for any pair of labeled subgraphs S and T of G such that there exists an automorphism of G mapping S to T, then (Ŝ) = (T). Note that the above measure is an example of a nonsymmetric extremal construction for f 1,C 4 (p), whereas the measure given at the end of Section 8.1 is symmetric. This leads to the following question.
Proof. Let be the nonsymmetric measure which achieves f 1,G (p). For all H ⊆ G define First note that we have the following: (A1) The first and final equalities follow by definition. The second equality follows by summing through each automorphism of H and the fact that ′ (Ĥ i ) = ′ (Ĥ j ) for all i, j. The third equality follows by swapping automorphisms of H to automorphisms of S, which again works since ′ (Ĥ i ) = ′ (Ĥ j ). Now note that if S is the empty graph or a single edge, then (S i ) = (S j ) for all i, j and thus we obtain ′ (S i ) = (S i ) for all i. It easily follows that ′ is a measure with edge-probability p. We must show ′ (S) ⋅ ′ (T) = ′ (S ∪ T) for all S, T which are labeled nonempty subgraphs of G supported on disjoint subsets of vertices. If the optimisation problem is linear, then without loss of generality we have ′ (T) = p, and so this follows by linearity and (A1). It remains to show that ′ also achieves f G (p). Again this follows easily since ′ (a ′ -random graph is connected) = ∑