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 G μ the associated random graph model. Let ℳ 1 , ⩾ p ( G ) denote the collection of 1-ipms μ on G for which each edge is included in G μ with probability at least p. For G = Z 2 , Balister and Bollobás asked for the value of the least p ⋆ such that for all p > p ⋆ and all μ ∈ ℳ 1 , ⩾ p ( G ) , G μ 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 ∈ ℳ 1 , ⩾ p ( G ) of the probability that G μ 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.

INTRODUCTION

Bond percolation models, 1‐independence and edge‐probability

Let G be a (possibly infinite) connected graph. Spanning subgraphs of G are called configurations. In a configuration H, an edge is said to be open if it belongs to H, and closed otherwise. A bond percolation model on the host graph G is a probability measure on the spanning subgraphs of G, that is, on the space of configurations. Given such a measure, we denote the corresponding random graph model by , and refer to it as the ‐random graph or ‐random configuration.

In this paper, we study bond percolation models where the states (open or closed) of edges in subsets F 1, F 2 of E in a ‐random configuration are independent provided that the edges in F 1 and F 2 are “sufficiently far apart.” To make this more precise, we make use of the following definition.

Two edge sets F 1, F 2 ⊆ E are k‐distant if F 1 ∩ F 2 = ∅ and the shortest path of G from an edge in F 1 to an edge in F 2 contains at least k edges. A bond percolation model on G is k‐independent if for any pair (F 1, F 2) of k‐distant edge sets, the intersections and are independent random variables.

So for example is 0‐independent if each edge of is open at random independently of all the others, that is, can be viewed as a product of Bernoulli measures on the edges of G. A well‐studied 0‐independent model is the Erdős‐Rényi random graph G , where the host graph is G = K , the complete graph on n vertices, and where , known as the p‐random measure, sets each edge to be open with probability p, independently of all the others.

In this paper, we focus on the next strongest notion of independence, namely 1‐independence. Measures that are 1‐independent have the property that events determined by vertex‐disjoint edge sets are independent. For many 1‐independent models, the randomness can be thought to “reside in the vertices.” An important example of a 1‐independent model is that of site percolation on the square integer lattice. In this case the host graph is the square integer lattice (where two vertices are joined by an edge if they lie at distance 1 apart), and the measure is obtained by switching each vertex of on at random with probability , independently of all the others, and by setting an edge to be open if and only if both of its endpoints are switched on. Site percolation measures may be defined more generally on any host graph in the natural way.

Site percolation is an example of a broader class of 1‐independent measures where we independently associate to each vertex v ∈ V(G) a state S at random, and set an edge uv to be open if and only if f(S , S ) = 1, for some deterministic function f (which may depend on u and v). We refer to such measures as vertex‐based measures (see Section 11 for a formal definition). Vertex‐based measures on are a generalization of the well‐studied notion of two‐block factors, which are vertex‐based measures on in which the vertex states S are i.i.d. random variables and the function f is independent of u and v, see 24 for further details.

An important point to note is that while all 0‐independent bond percolation models are a product of Bernoulli measures on the edges of G (with varying parameters), it is well‐known that a graph G may support many 1‐independent measures which cannot be realized as vertex‐based measures or as general “block factors,” see for instance 1, 13, 21. In particular for most graphs G, it is not feasible to generate or simulate the collection of 1‐independent measures of G.

Given a bond percolation model on a host graph G, the (lower)‐edge‐probability of is

So for instance a p‐random measure has edge‐probability p, while a site percolation measure with parameter has edge‐probability . The collection of k‐independent bond percolation models on a graph G with edge‐probability is denoted by .

Given a measure , we may readily produce a measure such that for all e ∈ E(G) via random sparsification: independently delete each edge e of with probability . 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 measure thus 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, (G) for 0‐independent percolation:

Given an infinite, locally finite connected graph G, determine p 0, (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 to be 1/2.

(Harris‐Kesten Theorem [20,23]). Let be the p‐random measure on . Then

In this paper, we focus on the question of what happens to the Harris critical probability in 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?

Let G be an infinite, locally finite connected graph and let . The Harris critical probability for k‐independent percolation1 in G is defined to be:

Determine .

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 (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  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 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 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 . 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 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 . In a different direction, Balister and Bollobás 4 observed that trivially for any infinite, locally finite connected graph G. In the special case of the square integer lattice , they recalled a simple construction due to Newman which gives where is the critical value of the ‐parameter for site percolation, that is, the infimum of such that switching vertices of on independently at random with probability almost surely yields an infinite connected component of on vertices. Plugging in the known rigorous bounds for  33, 34 yields , while using the nonrigorous estimate (see e.g., 32) yields the nonrigorous lower‐bound .

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 with the following theorems.

For all , we have that

Theorem 7 strictly improves on the previous best lower bound for d = 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 again denote the critical threshold for site percolation. Then the following holds.

Substituting the rigorous bound into Theorem 8 yields the lower bound , which does slightly worse than Theorem 7. However substituting in the widely believed but nonrigorous estimate yields a significantly stronger lower bound of .

Second, motivated by efforts to improve the upper bounds on , 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 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  × G the Cartesian product of P with G. A left‐right crossing of P  × 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  × G to be that is, the least edge‐probability guaranteeing that in any 1‐independent model on P  × G, there is a strictly positive probability of being able to cross P  × G from left to right.

Given and a finite, connected graph G, determine p 1, ×(P  × 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.

Given an infinite, locally finite, connected graph G, determine p 1, (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 and the integer ladder lattice .

We have that

, and

.

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  × G) and for a variety of graphs G. We summarize the latter, less technical, set of results below. Let C and K denote the cycle and the complete graph on n vertices respectively.

We have that

for all ;

;

;

;

.

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  × 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.

Let G be a finite connected graph. For any p ∈ [0, 1], we define the k‐independent connectivity function of G to be

Given a finite connected graph G, determine f 1, (p).

We resolve Problem 14 exactly when G is a path, a complete graph or a cycle on at most 5 vertices.

Given and p ∈ [0, 1], let and . We have that

Given and p ∈ [0, 1], let and . We have that In particular,

For p ∈ [0, 1] we have that

For p ∈ [0, 1] we have that

We also consider the opposite problem to Problem 14, namely maximizing connectivity in 1‐independent random graph models. Let denote the collection of 1‐independent measures on G such that . Set

Given a finite connected graph G, determine F 1, (p).

We resolve Problem 19 exactly when G is a path, a complete graph or a cycle on at most 5 vertices.

For all with , .

For all with , .

Together, Theorems 15‐18 and 21‐23 determine the complete connectivity “profile” for 1‐independent measures on K , P , C 4 and C 5—i.e., the range of values 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, (p), F 1, (p) and , where G is the 0‐independent model on G obtained by setting each edge of G to be open with probability exactly p, independently at random.

FIGURE 1

The 1‐independent connectivity profile of G for G = K 3 K 4, C 4 and C 5. The green curve represents f 1, (p), the dashed black curve f 0, (p), and the union of the red, blue and purple segments represent the piecewise smooth function F 1, (p)

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 . 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, (p) and F 1, (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 for the set of natural numbers {1, 2, …}, for the set , and 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 ( 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 pair G = (V, E) where V = V(G) and E = E(G) ⊆ V(G)(2) denote the vertex set and edge set of G 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 . We refer to a random configuration as a ‐random subgraph of G. Finally we write for the expectation taken with respect to the probability measure . For any event X, we write X for the complement event.

LOWER BOUNDS ON

Let . For , let . Let . For each vertex in , 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 the vertex is in, and we assign these states to each vertex independently of all other vertices.

If v is a vertex in T , where k ≡ 0[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 6, then we color v Blue.

If v is a vertex in T , where k ≡ 1[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 6, then we color v Red with probability q/2 and color it Blue otherwise.

If v is a vertex in T , where k ≡ 2[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 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 , where k ≡ 3[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 6, then we color v Red.

If v is a vertex in T , where k ≡ 4[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 6, then we color v Blue with probability q/2 and color it Red otherwise.

If v is a vertex in T , where k ≡ 5[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 6, then we color v Blue with probability q and put it in the Inwards state I otherwise.

Note that the rules for T , T , T are the same as those for T , T , T 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 d = 2. Now suppose that e = {v 1, v 2} is an edge in . 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 , v 2 ∈ T , 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 as it is vertex‐based, and it is also easy to check that every edge is present with probability at least .

FIGURE 2

The possible states of the vertices in T 0, T 1, T 2 and T 3 when d = 2. The letter B stands for Blue, the letter R stands for Red, and the letter I stands for the Inwards state

Call this measure , and let . We claim that in , for all k ≡ 0[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 3, there is no path of open edges from T to T . Suppose this is not the case, and P is some path of open edges from a vertex in T to T . We first note that P cannot include a vertex in state I, as such a vertex would be in T 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 is sandwiched between some T and T , where k ≡ 0[[spimspace="0.3em"]][[spimspace="0.3em"]] mod[[spimspace="0.3em"]] 3, and so is of finite size. Thus we have that .

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 A ⊆ V(G), let be the closure of A under 2‐neighbor bootstrap percolation on G. That is, let , where A 0: = A and for We say that G has the finite 2‐percolation property if, for every finite set A ⊆ V(G), we have that is finite.

If G has the finite 2‐percolation property, then .

Partition V(G) in the following way: pick any vertex v and set T 0: = {v}. For let We have that if w ∈ T , then w is only adjacent to vertices in T , T and T . Moreover, w is adjacent to at most one vertex in T —this is the crucial property needed for our construction. Since G has the finite 2‐percolation property, each T is finite. Thus we can use the T to construct a nonpercolating 1‐ipm on G in the exact same fashion as done for 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 .

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 . As this is less than , we have that our construction gives the (rigorous) improvement of .

Fix ε > 0 sufficiently small so that is strictly larger than 1/4. For each vertex , 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 , and else we assign it to the D state with probability .

We now describe which edges are open based on the states of the vertices. We first say that the edge e is open if both of its vertices are in the On state. If a vertex is in state L, then the edge adjacent and to the left of it is open. Similarly, if a vertex is in state D, then the edge adjacent and down from it is open. All other edges are closed. See Figure 3 for an example of this construction.

FIGURE 3

This figure shows the open edges of the construction on a small subset of . The unlabeled vertices correspond to those in the On state. The black edges are the open edges that are adjacent to two On vertices, while the dashed red edges are the open edges that are either to the left of an L vertex or below a D vertex

It is easy to see that this is a 1‐independent measure on as it is vertex‐based, and every edge is present with probability . Call this measure and let . We will show that every component of has finite size. We begin by first proving an auxiliary lemma. Let , and let us define another 1‐independent measure on , which we call the left‐down measure with parameter t. In the left‐down measure, each vertex of is assigned to one of three states: Off, L or D, and we do this independently for every vertex. For each vertex , 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, if a vertex is in state L, then the edge adjacent and to the left of it is open, while if a vertex is in state D, then the edge adjacent and down from it is open. All other edges are closed. We use to denote the left‐down measure with parameter t.

If , then all components in are finite almost surely.

Let . As we have that . We start by taking a random subgraph of 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 we close the edge e 1, and otherwise we close the edge e 2. We do this independently for every vertex v of .

It is easy to see that this is an equivalent formulation of , 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 , they are independent of one another just as in the measure.

Thus we have coupled to the 0‐independent bond percolation measure on with edge‐probability z. In this coupling we have that if an edge e is open in , then it is also open in . As we have that all components in are finite by the Harris‐Kesten theorem, and so we also have that all of the components in are finite too.

By considering an appropriate branching process it is possible to prove the stronger result that if , then almost surely all components in are finite. We make no use of this stronger result in this paper, so we omit its proof. It is also clear that when , every vertex in is part of an infinite path consisting solely of steps to the left or steps downwards, and so percolation occurs in 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 . 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 , 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 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 , 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 consists of at most one On‐section, and a finite number of finite LD‐sections attached to it. Thus almost surely every component in is finite.

A GENERAL UPPER BOUND FOR

Let G be a finite connected graph. Set v(G): = |V(G)|. Recall that for any 1‐independent bond percolation measure , we have .

If p satisfies for some , then for every

Consider an arbitrary measure . For any , the restriction of to [n] × V(G) is a measure from , and clearly all such measures can be obtained in this way. Furthermore, for every , the restriction of to {n} × V(G) is a measure from , and in particular the subgraph of induced by {n} × V(G) is connected with probability at least f 1, (p).

We consider the ‐random graph . For let Y be the event that [n] × V(G) induces a connected subgraph. For , let X be the event that [n − 1] × V(G) induces a connected subgraph and at least one vertex in {n} × V(G) is connected to a vertex in {n − 1} × V(G). For n = 1, set X 1 to be the trivially satisfied event occurring with probability 1. For , let V be the event that {n} × V(G) induces a connected subgraph, and for let H be the event that at least one of the edges from {n − 1} × V(G) to {n} × V(G) is present.

It easily follows that X  = Y  ∩ H and X  ∩ V  ⊆ Y . From here, we obtain the following inclusions:

(X ) ∩ Y  = (H ) ∩ Y ,

Y  ∩ Y  ⊇ (V  ∩ X ) ∩ Y , and

(Y ) ∩ X  ⊆ (V ) ∩ X .

Now set

We begin by establishing two inductive relations for the sequences x and y . First of all, using (a) and (b) we have,

Second, using (c), Now if (3.1) is satisfied, we claim that for all n. Indeed x 1 = 0, and if , then by (3.3) 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, (p) is a nondecreasing function of p with f 1, (1) = 1. Thus the function (f 1, (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) 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.

Let G be a finite connected graph. Let p ⋆ = p ⋆(G) be as above. Then

Apply Theorem 26 with .

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”).

Let G be a finite graph, and let be a set of polynomials with real coefficients, indexed by subgraphs of G. Given , let be the following function from subgraphs of G to : Suppose there exists a nontrivial interval such that, for all , the function defines a 1‐ipm on G. Suppose further that there exists a set such that, for all and all H ⊆ G, is a nonnegative real number. Then is a 1‐ipm on G for all .

We start by proving that is a measure on G for all . As is a nonnegative real number for all 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 . To do this we must show that the following holds true for all , for all A, B ⊆ V(G) such that A and B are disjoint, and all G 1 and G 2 such that G 1 is a subgraph of G[A] while G 2 is a subgraph of G[B]: 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 , 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

In this section we prove Theorem 15 on the connectivity function of paths. Recall that, given and p ∈ [0, 1], we let and . Let .

We begin by constructing a measure as follows. Let us start with the case . For each vertex of P , 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 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 , and closed otherwise. Thus, as , we have that each edge is open with probability p. Moreover will be connected if and only if there exists some j ∈ [n + 1] such that S  = 0 for all k < j, while S  = 1 for all . Therefore is connected with probability . As this construction is vertex‐based, it is clear that it is 1‐independent.

When 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 , 1]. For each subgraph G of P , set to be the polynomial for all . The following claim, together with Lemma 28, shows that in fact is a 1‐ipm on P for all p ∈ [p , 1].

For all and all G ⊆ P we have that is nonnegative real number.

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 . We have that and , 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 . We have that . For we can write

When we have that and are complex conjugates, and also that is a pure imaginary number. Thus both the numerator and denominator of the above fraction are pure imaginary, and so is a real number for all . By writing , where and , we can rewrite (5.1) as

Now implies , which in turn gives . Thus by (5.2) above, is a nonnegative real number for all p in the interval , as required.

We now deal with the case that G ≠ P . Let us consider the vertex‐based construction from which 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  = 1 while S  = 0. Note that if , then the edge {j − 1, j} is present in G regardless of the state of vertex j − 1. Similarly, if , then the edge {j + 1, j + 2} is present in G regardless of the state of vertex j + 2. If we write and , then we have that Now, by induction, we have that and are positive real numbers for all ; to make this inductive step work we are using the fact that forms an increasing sequence, and so implies that for all . As , we have that (5.3) is a positive real for all , and so we have proven the claim.

Note that as this proof shows that , we have that the probability is connected is equal to 0. As for all , we have that for all .

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 and its analysis are due to Balister and Bollobás 4 (they considered slightly different probabilities for vertex states, setting S  = 0 with probability q , where q is defined for k ∈ [n] by q 1 = 0 and by the recurrence relation for k ≥ 2, which corresponds exactly to the equality case in inequality (5.5) below).

The above construction discussed shows that It is clear that for all p, and so all that remains to show is that for all p ∈ [p , 1].

Let . For k ∈ [n], let X be the event that the subgraph of induced by the vertex set [k] is connected, and let H be the event that the edge {k − 1, k} is not present in . Applying random sparsification as in Remark 3 if necessary, we may assume without loss of generality that for every k, the event H occurs with probability exactly 1 − p.

Let , and for k > 2 let . We have that

Note that . 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 . In the measure , we have that every edge is present with probability exactly p, thus 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 , and so it is sufficient to show that equality holds for , as that will show they are the same polynomial (and so equality holds for all p ∈ [p , 1]). Suppose that the event (H  ∩ X ) occurs. As H has occurred we have that S  = 1 while S  = 0. As S  = 1, we have that edge {k − 2, k − 1} is open, regardless of S . Thus, as X has occurred we also have that X has occurred. Therefore (H  ∩ X ) has also occurred, and so we are done.

We remark in similar fashion to the above proof that the following holds for any : 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 . This leads us to another way to define : let , , and for all we have that We conclude this section with a proof of Theorem 11(i).

(i). For the upper bound we plug into equation (3.4), solve that equation to get and apply Corollary 27 to obtain .

For the lower bound, let be fixed. As the sequence is monotone increasing and tends to 3/4 as n → ∞, there exists such that p < p . We showed in Theorem 15 that there exists a measure such that the probability is connected is equal to zero.

We use this measure to create a measure . For each , we let the subgraphs on horizontal shifts of P by i(N − 1) be independent identically distributed random variables with distribution given by . This gives rise to a 1‐independent model on with edge‐probability at least p (in fact at least p ). Furthermore, all connected components of have size at most 2(N − 1) − 1. In particular, . Since was chosen arbitrarily, this gives the required lower bound .

THE LADDER LATTICE

In this section we construct a family of 1‐ipms on segments of the ladder 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  × 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 (. If n + y is even, then we let while if instead n + y is odd, then we let Here are suitably chosen sequences of real numbers, ensuring that the S ( are well‐defined random variables. We then define a random spanning subgraph of G: = P  × P 2 from the random vertex states S (: (n, y) ∈ [N] × [2] as follows:

for each n ∈ [N − 1] and y ∈ [2], the horizontal edge {(n, y), (n + 1, y)} is open in if and only if ,

for each n ∈ [N], the vertical edge {(n, 1), (n, 2)} is open in if and only if .

Note the condition for a vertical edge {(n, 1), (n, 2)} to be open can be rephrased as if and only if either S ( = S ( or one of S (, S ( is equal to 1. So intuitively, the value of the S ( must increase from left to right along open horizontal edges, and it must stay constant along open vertical edges unless one of the endpoints is in the special state 1, which allows free passage up or down.

Clearly the bond percolation measure associated to our random graph model is a 1‐ipm on the ladder G = P  × P 2 as it is vertex‐based. By making a judicious choice of the sequences and taking N sufficiently large, one can ensure that in addition satisfies and . In particular, with this construction we prove the following result.

Fix . Then there exists such that for all ,

Fix , with . We start by defining the sequences , and iteratively as follows. We set p 1 = r 1 = 1 and s 1 = 0. Then for , we let

The following hold for all :

p , r  ∈ [0, 1],

s  ∈ [0, 1 − r ],

,

,

.

We prove the lemma by induction on n. By definition of our sequences, , s 1 = 0, and , and thus (i)‐(v) all hold in the base case n = 1.

Suppose now (i)‐(v) hold for all , for some . Since p and r  + s both lie in [0, 1], the definition of p and r implies these also both lie in [0, 1]. This establishes (i) for n = N + 1. By construction, , and by the inductive hypotheses (ii) and (v), we have whence (ii) holds for n = N + 1.

If p  = 0, then trivially holds (since by (i)). On the other hand, suppose . Then we have r  + s  > 1 − p, which by our inductive hypothesis (v) implies . The definition of p then implies as desired, establishing that (iii) holds for n = N + 1. Arguing in exactly the same way (using the inductive hypothesis (iii) instead of (v)), we obtain that . Hence (iv) 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.

If p  = 0 or r  = 0, then s  = 0.

If p  = 0, then by construction s  = 0 and so we are done. If r  = 0, then by construction p  ≤ 1 − p, which by our inductive hypothesis (iii) implies p  ≤ 1 − p and hence .

If p and r are both strictly positive, then for all i ∈ {2, … , N + 1}, we have s  > 0.

Fix i ∈ {2, … , N + 1}. By our inductive hypotheses (iii) and (iv) (which we have already established up to n = N + 1) and since , we have and . Since r  > 0, we in fact have p  > 1 − p. We also have that Now for fixed y ∈ [1/2, 1], the function x ↦ f(x, y) is a nonincreasing function of x. Thus if , we have On the other hand for fixed y ∈ (0, 1/2), the function x ↦ f(x, y) is strictly increasing in x. Therefore if r  < 1/2, we have In either case, f(p , r ) > 0, and thus .

With these results in hand, we return to the proof of (v). If s  = 0, then (v) follows immediately from (iv). Thus we may assume that s  > 0, whence by Claim 32 p  > 0 and r  > 0. By Claim 33 and our inductive hypotheses (iii) and (iv), this implies that p , r and s 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:

Under our assumption that s  > 0, for all integers i ∈ [N + 1] we have

Since p 1 = 1 and p 2 = p, our claim holds for i = 1. Suppose it holds for some . Then by rearranging terms, we have Substituting this into the formula for p 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  + s as a function of p :

For p  > (1 − p) (which we recall holds since r  > 0), the expression above is an increasing function of p . By our inductive hypothesis (iii) that it follows that and we have verified that (v) holds for n = N + 1.

Recall that , for some fixed .

We have that p  = r  = s  = 0 for all , where N : = ⌈2ε −1⌉.

Suppose first that there exists m ∈ [N  − 1] such that r  = 0. Then s  = 0, so p  = 0 and s  = 0 by construction and r  = 0 by Lemma 31(iv). Lemma 31(iii)‐(v) then implies p  = r  = s  = 0 for all , as required.

Suppose instead that r  > 0 for all n ∈ [N  − 1] and there exists some m ∈ [N  − 2] such that . Then r  = 0, and thus by the argument above, we have that p  = r  = s  = 0 for all , as required.

Finally, suppose p  > 1 − p and r  > 0 both hold for all n ∈ [N  − 2]. By Claim 33, we have s  > 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 for all n ∈ [N  − 2]. In particular, as N  = ⌈2ε −1⌉ and , 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 on the graph G = P  × P 2 from independent random assignments of states S ( 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 ( 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 :

for each n ∈ [N − 1] and y ∈ [2], the horizontal edge {(n, y), (n + 1, y)} is open if and only if ,

for each n ∈ [N], the vertical edge {(n, 1), (n, 2)} is open if and only if either S ( = S ( or one of S (, S ( is equal to 1.

So intuitively, the value of the S ( must increase from left to right along open horizontal edges of , and it must stay constant along open vertical edges of unless one of the endpoints is in the special state 1 which allows free passage up or down.

We have that .

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 , Similarly if n + y is odd, then by definition of p , Finally, for a vertical edge {(n, 1), (n, 2)}, n ∈ [N], we have Now, if p  = 0, then by definition of p , whence by Lemma 31(iv), and so the expression above equals . On the other hand if p  ≠ 0, then by definition of s the expression above is at least p. Thus each horizontal edge and each vertical edge is open in with probability at least p, and as claimed.

There is no open path in from {1} × [2] to {N} × [2].

By construction, p 1 = r 1 = 1, whence S (1, 1) = S (1, 2) = 2. Furthermore, by Lemma 35 and our choice of N, p  = r  = s  = 0, whence S ( = S ( = 0.

Let N′ be the largest n ∈ [N] for which there exists an open path in from {1} × [2] to {n} × [2]. Let be such a path, and let v 0 ∈ {1} × [2], v 1 ∈ {2} × [2], v 2, … , v  ∈ {n} × [2] be the vertices of traversed from left to right. Observe that in this ordering of the vertices of , every horizontal edge {(n, y), (n + 1, y)} of is traversed from left to right.

We claim that for all i ∈ [ℓ], we have S  ∈ {1, 2}. Indeed, by construction . Suppose there exists some such that for all j < i. If , then the edge v v can be open in only if . What is more, can be equal to 1 if and only if v v is a vertical edge. On the other hand, suppose . Then v v was a vertical edge (since there is no edge both of whose endpoints are in state 1 and since horizontal edges are traversed from left to right by ), and so v  = v  + (1, 0). But then v v open in implies . Thus for every vertex v of , we have that is indeed in state 1 or 2.

This implies in particular that v  ∉ {N} × [2] (since as we remarked above S ( = S ( = 0). Thus there is no open path in from {1} × [2] to [N] × [2].

Thus is an element of for which Given , we may extend to an element by letting every edge in P  × P 2 ∖ P  × P 2 be open independently at random with probability p. In this way we obtain a 1‐independent bond percolation measure on P  × P 2 with edge‐probability p for which there almost surely are no open left‐right crossings of P  × P 2, giving the required lower bound on p 1, ×(P  × P 2).

We conclude this section by proving Theorem 11(ii), with the aid of Theorem 30.

(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 . Thus the constant p ⋆(P 2) defined by equation (3.4) is the unique solution in [0, 1] to the equation x 2 = 4(1 − x)2, namely . By Corollary 27, this implies .

For the lower bound, fix . In the proof of Theorem 30, we showed there exist some integer and such that

;

.

We use this measure to create a measure . Let . For each , we let the subgraphs on horizontal shifts of the ladder P  × 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 with edge‐probability p. Furthermore, property (i) implies all connected components in have size at most 4(N − 1) − 2 = 4N − 6. In particular, . Since was chosen arbitrarily, this gives the required lower bound .

COMPLETE GRAPHS

In this section we will prove Theorem 16. Recall that, given and p ∈ [0, 1], we let and . Let .

An upper bound for

Before proving Theorem 16, let us give a simple vertex‐based construction of a measure that shows for . We call this measure the Red‐Blue construction. We think of K 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 , we have that each edge is present in with probability p. Note that will either be either a disjoint union of two cliques, in which case it is disconnected, or the complete graph K , 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 is connected is equal to . As this construction is vertex‐based, it is clear that it is 1‐independent.

If 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 , 1]. Given j ∈ {0, 1, … , n}, let When j = 0 or j = n we have that and are each equal to , and so we just write the latter instead. Given some A ⊆ [n], let H 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 is equal to K [, and more generally that H  = H [. For p ∈ [0, 1], let be the following function on subgraphs G of K : For this function matches the Red‐Blue construction given above, and so by defining for all subgraphs G ⊆ K , we obtain a measure which is a 1‐ipm defined without making reference to states of vertices. The following claim, together with Lemma 28, shows that in fact is a 1‐ipm on K for all p ∈ [p , 1].

For all and all j ∈ {0, … , n} we have that is nonnegative real number.

Let us begin with the case j = n. As , we have that and are complex conjugates, and so is a real number for all p in this range. By writing , where and , we can write Now implies , which in turn gives . By (7.1), it follows that is a nonnegative real number for all , which proves the claim when j = n. For general j ∈ {0, … , n}, we have that Therefore the previous case of the claim shows that for all ; at this stage we are using the fact that forms an increasing sequence, and so implies that for all .

Note that as this proof shows that , we have that the probability is connected is equal to 0. As for all , we have that for all . We now prove that this construction is optimal with respect to the connectivity function.

A lower bound on

The previous constructions discussed show that It is clear that for all p, and so all that remains to show is that for p ∈ [p , 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 for all j ∈ {0, 1, … , n}. Thus, if we multiply both sides of this equation by and sum over all j ∈ {0, 1, … , n}, we have that Let and let C be the event that is connected. Given A ⊆ [n], let X be the event that and are each connected, where A  = [n] ∖ A. Moreover, let Y be the event that and are each connected, and there are no edges between A and A in . For all A ⊆ [n], we have that Note that when A = ∅ or A = [n], the above equation is trivially true due to the fact that C, X , X [, Y and Y [ 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 forms an increasing sequence, and so implies that for all . We are also using the fact that for all . 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 [ are all the same event to (7.5) to get Note that for all A ⊆ [n], the events Y and are the same event, and so . Moreover, the set {Y  : 1 ∈ A ⊆ [n]} consists of pairwise disjoint events, and so . Thus

We apply (7.7) to (7.6) to obtain . As n > 2, we have that and so we are done.

A remark on for

Clearly we can define f (p) analogously to f 1, (p) for . For k = 0, is exactly the probability that an instance of the Erdős‐Rényi random graph G 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 exactly, which is the other interesting case, as for the connectivity problem is trivial.

For all , we have that

For the lower bound, consider . Since any subgraph of K with at least 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 . With probability , select a vertex i ∈ [n] = V(K ) uniformly at random, and let G be the subgraph of K obtained by removing all edges incident with i. Otherwise, let G be the complete graph K . 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 , an event which occurs with probability .

CYCLES

Linear programming for calculating f 1, (p)

In this subsection we describe how we can represent the problem of finding f 1, (p), for any graph G, as a (possibly nonlinear) program.

Given a graph G on vertex set [n], let 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 , we have if and only if the following three conditions all hold:

is a probability measure on labeled subgraphs of G,

Every edge of G is open in with probability at least p,

Given nonempty such that S and T are supported on disjoint subsets of [n], .

As we are interested in determining f 1, (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 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:

,

For all edges e ∈ E(G), we have that,

For all nonempty , 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:

We have a row for the empty set such that A : = 1.

We have a row for e ∈ G; the entry ;

We have a row for each pair supported on disjoint subsets of [n]; the entry .

Let q = q(G) be a vector with indexing the same as the rows of A; let q : = 1, q : = p for e ∈ G, and q {: = 0 for each pair supported on disjoint subsets of [n]. Then a vector w, whose entries are indexed by , which satisfies for all , and also A w = q corresponds precisely to a 1‐ipm on G with for all edges e ∈ E(G).

Let c be a vector indexed by defined by . Just to make it clear, we say that is connected if it contains a spanning tree of [n]. Then for a given value of p the vector w(p) satisfying A w(p) = q and c w minimal corresponds to a measure such that .

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 . Thus for any choice of p, we can turn the problem of finding f 1, (p) into the following linear program: (Note that for graphs with six or more vertices, one may find S and T such that (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 ), w = (w ), c = (c ), q = (q ) and x = (x ) any solutions w and x must satisfy , , x  ≥ 0 and . Thus we have In particular for optimal solutions we have and so the inequality must be an equality, that is Consequently for each j we either have w  = 0 or . 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 have been set to zero; assume for these indices that we always have w  = 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 (p) = 0}.

Solve the set of equations {(A w) = q , w  = 0  : i ∈ [|w|], j ∈ J} to obtain functions of p for all w , k ∈ [|w|], which we write as wk′(p).

Solve the set of equations {(A x) = c  : i ∈ [|w|] ∖ J} to obtain functions of p for all x , k ∈ [|w|], which we write as xk′(p).

Write w ∗(p): =  c w′(p) and x ∗(p): =  q x′(p).

For a certain interval P ⊆ [0, 1] of values of p, check that and , for all i ∈ [|w|].

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, (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.

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.

For C 4 and an extremal construction is given by the measure , defined by For C 4 and an extremal construction is given by the measure , defined by

We can in fact give a direct combinatorial proof of the lower bound in Theorem 17: for any , 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 , we do not have a combinatorial proof, and our result relies solely on linear optimisation.

For C 5 and an extremal construction is given by the measure , defined by For C 5 and an extremal construction is given by the measure , defined by

General bounds for cycles of length at least 6

We can use Markov's inequality to derive the following simple lower bound on for .

For , with , and p ∈ [0, 1], we have .

A small adjustment to this argument gives the following improvement for n = 6.

For p ∈ [0, 1] we have that .

Let . Note that is connected if and only if it has at most one closed edge. Thus by Markov's inequality, we have

Let X be the number of closed edges in . Cyclically label the edges of C 6 as e 1, … , e 6. Then by simple counting, 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 and hence prove Theorem 21.

Given a 1‐independent model G on K with edge‐probability at least 1 − p, observe that the complement G of G in K is a 1‐independent model in which every edge is open with probability at most p. Furthermore, G is connected whenever G fails to be connected. This immediately implies Furthermore, observe that the Red‐Blue measure we constructed to obtain the upper bound on in the proof of Theorem 16 has the property that a ‐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 .

For any measure , we have by 1‐independence that which implies . For the lower bound, we construct a 1‐ipm as follows. For each integer i: , 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 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 in which every edge is open with probability p, and for which

Thus , as claimed.

The case of cycles C 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.

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 , we have by 1‐independence that and, by a simple union bound and 1‐independence, Combining these two inequalities and using 1 − (1 − p)2 = 2p − p 2, we obtain which gives the claimed upper bound on .

For the lower bound, we give two different constructions, depending on the value of p. For consider the measure defined by It is easily checked that and that , which is maximal for p in that range.

For , consider the measure defined by Again, it is easily checked that and that , which is maximal for p in that range.

We simply apply the linear optimisation method from Section 8—here again we do not have a combinatorial proof. In addition to establishing the theorem, this gives us constructions of extremal 1‐independent measures maximizing connectivity.

For an extremal construction is given by the measure , defined by For an extremal construction is given by the measure , defined by For an extremal construction is given by the measure , defined by

PROOF OF THEOREM 1.12

Combining Corollary 27 with our results on 1‐independent connectivity, much of Theorem 12 is immediate.

For the lower bound in part (i), we note that has the finite 2‐percolation property. Thus, as described after the proof of Theorem 7, we have that . 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, (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 is nonincreasing in [0, 1] and hence tends to a limit as n → ∞. For the lower bound in (v), observe that for any the graph has the finite 2‐percolation property—indeed for any finite k, the closure of a copy of P  × K under 2‐neighbor bootstrap percolation in is equal to itself. We construct a 1‐ipm on as in Corollary 24 but with starting set T 0 = {0} × V(K ) and hence T  = ({k} × V(K )) ∪ ({− k} × V(K )). 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 , this proves for all . For the upper bound, we perform some simple analysis. By solving a quadratic equation, we see that for all fixed . Then by Theorem 16, for any such fixed p and all n sufficiently large, we have that Thus p ⋆(K ) < p for all n sufficiently large, which by Corollary 27 implies .

OPEN PROBLEMS

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 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.

Let G be a graph. A G‐partition is a partitioned set , with nonempty parts indexed by the vertices of G. A G‐partite graph is a graph H on a G‐partition whose edges are a subset of the union of the complete bipartite graphs corresponding to the edges of G.

Given a G‐partite graph H on a G‐partition , we have a natural way of constructing 1‐independent bond percolation models: given a family of independent random variables with S taking values in , the (H,  of G, denoted by H[X], is the random configuration on E(G) obtained by setting uv to be open if and only if S S  ∈ E(H).

Let G be a graph. A measure is said to be vertex‐based if there exist

a G‐partition ,

an associated G‐partite graph H, and

a collection of independent random variables (S ) with S taking values in ,

such that the (H, X)‐random subgraph H[X] has the same distribution as the ‐random graph .

Let denote the collection of all vertex‐based measures on G with edge‐probability at least p.

Determine .

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 have bounded size.

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), . Furthermore, a vertex‐based measure on a graph G is uniformly bounded if it is N‐uniformly bounded for some .

Let and 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.

For , determine

Determine

Finally, let us note that the second most obvious problem arising from our work, besides that of improving the bounds on , is arguably that of giving bounds on 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 , it is natural to ask about bounds on p 1, (G) for some of the other commonly studied lattices in percolation theory.

Give good bounds on the value of p 1, (G) when G is one of the eleven Archimedean lattices in the plane or the d‐dimensional integer lattice .

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 (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 contains a copy of , whence the sequence is nonincreasing in [0, 1] and converges to a limit. Balister and Bollobás asked for its value:

[Balister, Bollobás  4 ] Determine .

Note that by Theorem 7 proved in this paper, this limit must be at least .

Other notions of 1‐independent critical probabilities

Let G be an infinite, locally finite connected graph, and v 0 a fixed vertex of G. Given a bond percolation model on G, we let denote the connected component of containing v 0.

If is 0‐independent, then if and only if . However this need not be true for a 1‐independent measure. Indeed, consider the 1‐ipm on obtained by taking the measure constructed in the proof of Theorem 7 to determine the state of the edges in the ℓ ball of radius 3 around the origin and setting every other edge to be open independently at random with probability . Then in this model percolation occurs almost surely, but the origin is contained inside a component of order at most 28.

Thus in principle there are different edge‐probability thresholds in 1‐independent percolation on a graph G for percolation to occur somewhere with probability 1 and for it to occur anywhere with strictly positive probability. Indeed, if were strictly less than 3/4, then one could obtain examples of such a graph G by attaching a long path to the origin in .

Another critical edge‐probability of interest is the Temperley critical probability, which in 0‐independent percolation is the threshold p at which for any vertex v (and every 0‐independent measure with edge‐probability >p ). In general this threshold is different from the Harris critical probability. Again for 1‐independent percolation we have that the threshold for some vertex v ∈ V(G) to satisfy and for the threshold for all vertices of G to satisfy this are different.

Given an infinite, locally finite connected graph G, determine the following four critical probabilities:

It follows from their definition that these four critical probabilities satisfy

In general, these four critical probabilities are all different. Indeed, Balister and Bollobás showed in [4, Theorem 1.5] that there exists an infinite, locally finite connected graph G with . For any p: , we have shown in Theorem 15 that there exists N such that . Attach one end of a path of length N to an arbitrary vertex of G to form a graph G 1, and let v denote the other end of the path. Then there exist 1‐ipm such that with probability 1 the component of v in a ‐random graph has order at most N, which is finite. Thus we have On the other hand consider a graph G 2 obtained from the line lattice by attaching to each vertex a collection of 2| leaves. Clearly, . Now consider a 1‐ipm . By Theorem 26 applied to G = K 1 and , for any path P of length i in G 2, the ‐probability that all edges in P are open is at least 2−(. Thus for any v 0 ∈ V(G 2), the expected size of is Thus we have

None of the inequalities in ( 11.1 ) may be replaced by an equality.

Neither nor the reverse inequality are true in general. □

Observe that is the 1‐independent Harris critical probability p 1, (G) studied in this paper; given Corollary 50, we more precisely should call it the first Harris critical probability for 1‐independent percolation. Our construction for the proof of Theorem 7 and the argument of Balister, Bollobás and Walters from 7 give the following bounds when :

Are any of the four critical probabilities from Problem  49 equal when ?

Finally, note that Problem 49 asks, in essence, how much we can delay percolation phenomena relative to the 0‐independent case by exploiting the local dependencies between the edges allowed by 1‐independence. While perhaps less useful in applications, it is an equally natural and appealing extremal problem to ask how much we can use these local dependencies to instead hasten the emergence of an infinite connected component. Balister and Bollobás were the first to consider this problem in 4, which it would be remiss not to mention here.

Let G be an infinite, locally finite connected graph, and let be as before the collection of k‐ipms on G satisfying . The critical threshold for fast k‐independent percolation on G is

Balister and Bollobás determined p 1, (G) when G is an infinite, locally finite tree, and also gave the simple general bounds where is the connective constant of G and the critical value of the ‐parameter for site percolation on G. For the square integer lattice, this gives a lower bound on of 0.1393 from known upper bounds on . In the other direction, we get a rigorous upper bound of of 0.4618 and nonrigorous upper bound of 0.3515 from bounds and estimates for . This obviously leaves a big gap, which Balister and Bollobás asked to reduce.

[Balister and Bollobás  4 ] What is ?

Long paths critical probability

An obvious problem is to tighten the bounds in Theorem 12(v), which are not too far apart (compared to many of the other bounds on critical probabilities for 1‐independent model).

Determine .

In a similar vein, the sequence is a nonincreasing function of n (since contains as a subgraph). In this paper, we have given constructions showing that for all integers , Thus the sequence tends to a limit in the interval as n → ∞.

Determine

An in principle different but related problem is determining the value of the long paths critical probability in (which need not be equal to the quantity defined above).

Determine .

We can also ask for k‐independent versions of the long paths critical probability. Defining mutatis mutandis, it is straightforward to adapt our arguments and constructions from Section 5 to show the following result, which also follows directly from the work of Liggett, Schonmann and Stacey 24 on stochastic domination of 0‐independent measures on by k‐independent ones.

[Liggett, Schonman and Stacey  24 ] For any , we have with the convention that 00 = 1.

Given k fixed, it is easy to construct a 3k‐ipm on with and no open path of length more than (2k + 1)2. Indeed, build a random graph model as follows:

begin with all edges of open;

independently for each , choose H  ∈ [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  − 1, j′), (i(k + 1) + H , j′) to be closed;

independently for each , choose V  ∈ [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  − 1), (i′, j(k + 1) + V ) to be closed.

It is easy to check that this random graph model is 3k‐independent, has edge probability at least and that every connected component has order at most (2k + 1)2.

For any fixed ,

[[spimspace="1em"]]□

In particular we have (and in fact a similar construction shows this remains true in ).

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 every vertex of G has a strictly positive probability of being part of a path of length at least ℓ. Thus we may actually define a second long paths critical probability,

Determine .

Our construction in the proof of Theorem 7 shows that , and we know it is upper‐bounded by . 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 ‐partite graph H with partition . 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 for all . The G‐partite density of H is

Suppose H is an n‐uniformly bounded ‐partite graph in which in every transversal subgraph the connected component containing the origin is…

… of size at most C, for some constant .

… finite.

How large can d (H) be?

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 for cycles C of length at most 5. It is natural to ask what happens for longer cycles.

Determine for .

As mentioned in Section 8, the problem of finding 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 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 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.

Determine for all p ∈ [0, 1], and integers .

Determine for all p ∈ [0, 1], and integers .

Beyond paths, cycles and complete graphs, the 1‐independent connectivity problem is perhaps most natural to study in the hypercube graph Q and in the n × n toroidal grid C  × C . Progress on either of these would likely lead to progress on other problems in 1‐independent percolation as well.

Determine for all .

Determine for all .

In a different direction, we can ask whether the extremal measures attaining f 1, (p) can be required to have “nice” properties. For C 4 and p ∈ [0, 1/2] another extremal construction for is given by the measure , defined by Motivated by the above, we call a measure 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 . Note that the above measure is an example of a nonsymmetric extremal construction for , whereas the measure given at the end of Section 8.1 is symmetric. This leads to the following question.

For any G and any p ∈ [0, 1], does there always exist a symmetric measure which achieves f 1, (p)?

If the program for solving f 1, (p) attained via our method in Section 8.1 is linear, then the answer for G is yes(see the appendix for a proof of this fact). Another natural question is when the extremal connectivity can be attained by vertex‐based measures.

For which G and which p does there exist a vertex‐based measure which achieves f 1, (p)?