The Metastability Threshold for Modified Bootstrap Percolation in d Dimensions

In the modified bootstrap percolation model, sites in the cube {1,...,L}^d are initially declared active independently with probability p. At subsequent steps, an inactive site becomes active if it has at least one active nearest neighbour in each of the d dimensions, while an active site remains active forever. We study the probability that the entire cube is eventually active. For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to 0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the n-th iterate of the exponential function, and the threshold lambda equals pi^2/6 for all d.

where β t denotes the t-th iterate of the function β. W is the final active set for the modified bootstrap percolation model starting with W active. Now fix p ∈ (0, 1) and let X be a random subset of Z d in which each site is independently included with probability p. More formally, denote by P p = P the product probability measure with parameter p on the product σ-algebra of {0, 1} Z d , and define the random variable X by X(ω) We say that a set W ⊆ Z d is internally spanned, or i.s., if X ∩W = W (that is, if the model restricted to W fills W up). For a positive integer L we define the d-dimensional cube of side L to be where λ = π 2 6 .

Remarks
The case d = 2 of Theorem 1 was proved in [8]. The modified bootstrap percolation model considered here is a minor variant of the standard bootstrap percolation model, which is defined in the same way except replacing the function β with (so a site becomes active if it has at least d active neighbours). In the case d = 2, the analogue of Theorem 1 was proved for the standard bootstrap percolation model in [8]; in this case the threshold λ becomes π 2 /18. Similar results were obtained for a further family of two-dimensional models in [9]. The present work is the first proof of the existence of a sharp threshold λ for a bootstrap percolation model in 3 or more dimensions; in addition we determine the value π 2 /6. The analogue of Theorem 1 but with two different constants c 1 , c 2 in place of λ + ǫ, λ − ǫ was proved earlier in [2] (d = 2), [4] (d = 3) and [5] (d ≥ 4). These works apply to the standard model (among others), but can easily be adapted to the modified model considered here. It is a fascinating open problem to prove the existence of a sharp threshold for the standard model in 3 or more dimensions. In [3] a different kind of "sharpness" is proved, by a general method, for various models including standard and modified bootstrap percolation: writing p α = p α (L) for the value such that I(L, p α ) = α, then p 1−ǫ − p ǫ = o(p 1/2 ) as L → ∞, with a certain explicit bound. (However this result says nothing about the behaviour of p 1/2 as a function of L). Similar results with the roles of p and L exchanged may be obtained using the methods of [2].
There have been numerous other beautiful rigorous contributions to the study of bootstrap percolation models, initiated by [11]. For example see the references in [5], [8].
Bootstrap percolation models have important applications, both directly and as tools in the study of more complicated systems (see for example the references in [5], [8]). The models have been extensively studied via simulation, and it is a remarkable fact that the resulting asymptotic predictions often differ greatly from rigorous asymptotic results, apparently because the convergence as (L, p) → (∞, 0) is extremely slow. See [10], [8] for examples. In the case of the modified bootstrap percolation model in d = 2, the value 0.47 ± 0.02 for λ was predicted numerically in [1], whereas the rigorous result from [8] is λ = π 2 /6 = 1.644934 · · · . It would be worthwhile to compare simulations with our rigorous result that λ = π 2 /6 for d ≥ 3. It is of interest to understand this slow convergence phenomenon in more detail, and it is relevant to applications: a typical physical system might have L d ≈ 10 20 particles, which is much larger than current computer simulations allow, but potentially not large enough to exhibit a threshold close to the limiting value. See [6] for an interesting partly non-rigorous investigation of some these issues.

Proof outline
The proof of Theorem 1 is by induction on the dimension. The base case d = 2 is provided by the results in [8]. (The proof in [8] is quite involved, and very specific to the 2-dimensional model). It is interesting that the constant λ = π 2 /6 enters only here. The inductive step follows closely the pioneering work of [4], [5], although since our result is more precise we need to be more careful with the estimates. As in [5], the case d = 3 is the most delicate.
The proof of the lower bound in Theorem 1(i) is relatively straightforward, and many of the ideas were already present in [10]. The fundamental observation is that if a cube is already entirely active, then the sites lying on its faces evolve according to the modified bootstrap percolation model in d − 1 dimensions. Hence, by the inductive hypothesis, a cube of size L is likely to be internally spanned if it contains some internally spanned cube of size m = exp d−2 λ+ǫ p (sometimes called a "critical droplet" or "nucleation centre"), because such a cube will grow forever from its faces. For d ≥ 3, straightforward arguments show that such a cube is internally spanned with probability at least (roughly) e −m , and so in order to internally span the larger cube we should take approximately L > 1/(e −m ) = exp d−1 λ+ǫ p , completing the induction.
The proof of the upper bound in Theorem 1(ii) is more challenging, and is based on the more subtle construction originating in [4]. The idea is to find an upper bound on the probability that a cube of size m = exp d−2 λ−ǫ p has a left-right crossing in its final configuration. (Such a crossing plays the role of a nucleation centre in this bound). The proof proceeds by dividing this cube into "slices", and running the (d − 1)-dimensional model in each, to produce a configuration which dominates the d-dimensional model. By the inductive hypothesis, the probability that a slice becomes fully active is small, and, where a slice does not become fully active, its final configuration resembles subcritical percolation. Hence a connection of length m has probability at most (roughly) e −m , and again we can complete the induction. A key point in the present proof is that for the modified model, we can use slices of thickness 1, whereas in [4], [5] (for the standard model) it was necessary to use slices of thickness 2, and to replace the parameter p with 2p. Changing p in this way makes it impossible to obtain matching upper and lower bounds, and it is for this reason that our method cannot be adapted directly to prove an analogous result for the standard bootstrap percolation model. Another difference in the proof here as compared with [4], [5] is that (in the case d = 3) we need to carefully balance the probabilities of fully active slices with those of percolation connections. Equation (24) is the heart of this calculation.

Notation and conventions
It will be convenient to consider lower-dimensional versions of the model running on subsets Z d . Let δ ∈ {1, . . . , d}. We define the δ-dimensional cube By a copy of Q δ (L) we mean an image of Q δ (L) under any isometry of Z d . For a set W ⊆ Z d , which will always be a subset of some copy of a δ-dimensional cube, we define , and note that this is consistent with the earlier definition.
Theorem 1 involves an asymptotic statement as p → 0 with d and ǫ fixed. Many of the inequalities used in the proof will be valid "for p sufficiently small", be which we mean for all p less than some q = q(d, ǫ) > 0, whose value may vary from one instance to another. In some of following proofs we use C 1 , C 2 , . . . to denote constants in (0, ∞) which may depend on d and ǫ, but not on p.

Lower Bound
Lemma 2 For any d ≥ 3 and for a, ǫ > 0, if p is sufficiently small (depending on d, a, ǫ) then Proof. If d ≥ 4, note that a cube is internally spanned if all of its sites are occupied. Therefore for p sufficiently small. (To check the second inequality, take three successive logarithms of the reciprocal of both sides). The case d = 3 is a little more delicate. Write L = ⌊e a/p ⌋ and k = ⌊p −3 ⌋ (so k ≪ L for p sufficiently small). Let A be the event that every site having two of its coordinates in {1, . . . , k} and one coordinate in {1, . . . , L} is occupied (see Figure 1). Let B be the event that every copy of Q 1 (k) in Q 3 (L) contains at least one occupied site. It is straightforward to check that if A and B both occur then Q 3 (L) is internally spanned. Since A and B are increasing events, the Harris-FKG inequality (see e.g. [7]) yields We now estimate P(A) ≥ p 3Lk 2 ≥ 1/ exp 2 a + ǫ p for p sufficiently small (to check the second inequality, take two logarithms of the reciprocals), while for p sufficiently small. Therefore we have for p sufficiently small, and since ǫ was arbitrary this proves the result for d = 3. The following result from [2] states that I d (L) increases rapidly with L once it is large enough. [10] or [5]. The idea is to divide Q d (L) into disjoint or nearly-disjoint copies of Q d (ℓ). If Q d (L) is not i.s. then it is crossed by a path of non-i.s. copies of Q d (ℓ). The probability of this event can be estimated using standard percolation methods.
Proof of Theorem 1(i). The proof is by induction on d. The statement of the theorem holds in the case d = 2 by Theorems 4 and 1(i) of [8]. Now let d ≥ 3, and suppose that for all δ = 2, . . . , d − 1 we have We shall deduce that (4) holds for δ = d also. Fix ǫ > 0. We first claim that for p sufficiently small. To prove this, write By Lemma 2 we have We will deduce the claimed lower bound on I d (b) using the fact that an internally spanned cube will grow if each of its faces (of all possible dimensions) is internally spanned in the model of the appropriate lower dimension. More precisely, for L ≥ 1 and a proper subset S {1, . . . , d}, define the face Thus F S (L) is a copy of Q |S| (L), and we have the disjoint union where In order to bound P(G b a ), first note that we may take p sufficiently small that where c(δ) is as in Lemma 3. (The case δ = d − 1 follows directly from (4); the cases δ ∈ [2, d − 2] follow from (4) by an additional application of Lemma 3; the case δ = 1 is trivial.) Therefore writing C ′ = max δ∈[1,d−1] C(δ), Lemma 3 yields for p sufficiently small. Combining this with (6),(7) proves the claim (5).
Let E be the event that Q d (L) contains some d-i.s. copy of Q d (b), and let F be the event that for each δ ∈ [1, d − 1], every copy of It is straightforward to check that if E and F both occur then Q d (L) is d-i.s. Hence by the Harris-FKG inequality, By tiling Q d (L) with disjoint copies of Q d (b), we have But by (5), for p sufficiently small we have On the other hand, again taking for p sufficiently small to satisfy (8), we have by Lemma 3, for p sufficiently small. Hence P(F ) → 1 as p → 0. Thus by (9) we have proved that I d (exp d−1 λ+5ǫ p ) → 1 as p → 1. Since ǫ was arbitrary this is (4) with δ = d, and the induction is complete.

Upper Bound
The main step in the proof of Theorem 1(ii) will Theorem 10 below, which states that within a cube of appropriate size, the final configuration of the model resembles (highly) subcritical percolation.
We call a set of sites W ⊆ Z d connected if it induces a connected graph in the nearest-neighbour hypercubic lattice. A component is a maximal connected subset. For sites x, y ∈ Z d and a (random) set W ⊆ Z d we write "x W ←→ y" for the event that W has a component containing x and y. (Note that x W ←→ x is equivalent to x ∈ W ). For sites x, y ∈ Q d (m), we define ←→ y .
Theorem 10 Let d ≥ 3 and ǫ > 0. Let where λ = π 2 /6. There exist γ = γ(d, ǫ) > 0 and q = q(d, ǫ) > 0 such that for all p < q and all x, y ∈ Q d (m), The "+1" term in the exponent is important, since for the induction we need a bound which is o(1) as p → 0 even in the case x = y.
The following result from [2] is very useful. The diameter of a set W ⊆ Z d is diam W := sup x,y∈W x − y ∞ . Proof. See [2], [8] or [5]. The idea is to realize the bootstrap percolation model by an iterative algorithm. We keep track of a collection of disjoint i.s. sets S j . At each step, if there is a site in ∪ j S j \ ∪ j S j then we unite it with at most d of the sets to form a new set. In the case of the modified model, max j (diam S j ) is at most doubled at each step.
Proof of Theorem 1(ii). The case d = 2 was proved in [8]. Therefore fix d ≥ 3 and ǫ > 0, and let By Lemma 11, if Q d (L) is internally spanned then it contains some connected internally spanned set T with diam T ∈ [m/2, m]. This implies that there ←→ y. Hence by Theorem 10 we obtain as required. Here L d m d m d is a bound on the number of choices for v, x, y, and the second inequality holds for p sufficiently small, by taking the logarithm thus: The proof of Theorem 10 is by induction on the dimension. The key estimate is Lemma 13 below, for which we need to define two more quantities. Let where z := (n + 1, . . . , n + 1) is the site at the centre of Q d (2n + 1). (Thus χ d n is the expected volume of the component at z in the final configuration of the model on Q d (2n + 1)). For n ≤ m define Lemma 13 For any n ≤ m and x, y ∈ Q d (m) with ℓ = x − y ∞ we have (Perhaps the easiest way to understand Lemma 13 is to read the proof as far as (14),(15), and look at Figures 2,3.) Proof. The following construction is based on that of [4], [5]. Without loss of generality suppose that x − y ∞ = (y) d − (x) d ; if not we reorder the coordinates and/or reverse the direction of the dth coordinate. Write (x) d = u and (y) d = u + ℓ. Divide the cube Q d (m) into the slices so that x ∈ T 0 and y ∈ T ℓ . Let Y j := X ∩ T j d−1 be the final configuration of the (d − 1)-dimensional model restricted to T j . Let In the former case we say that the slice T j is full. Now let The point of this construction is that Z ⊇ X ∩ Q d (m) d . To see this note that Z j ⊇ Y j ⊇ X ∩ Q d (m) d ∩ T j ; the latter inclusion holds because running the (d − 1)-dimensional model on T j is equivalent to running the ddimensional model with the boundary condition that every site in Q d (m) \ T j is occupied -hence it must result in a larger configuration in T j than running the d-dimensional model on X ∩ Q d (m). (Note that the argument would not work in this form for the standard bootstrap percolation model, since the boundary condition adds two extra neighbours to each site in the slice.) Therefore We shall bound the above probability by splitting the event up according to which slices are full. Let I 1 < · · · < I K be the random indices of those slices among T 1 , . . . , T ℓ that are full. Also let W be the event than every path in Z from x to y intersects some full slice among T −u , . . . , T 0 , and let −A be the index of the last full slice among T −u , . . . , T 0 (or A = ∞ if there (See Figure 2 for an illustration). Using independence of the slices, the probability (15) above is at most where we have written for convenience i k+1 := ℓ + 1, and where Similarly, the probability (14) is at most where Next we bound the factors in (16),(17). For any slice T j we have so it remains only to bound the probabilities of E(i, i ′ ) and G(x, i, i ′ ). Suppose that the event E(i, i ′ ) occurs. Then there is a nearest-neighbour path in i ′ j=i Z j from some site in T i to some site in T i ′ . Let α be such a path of minimum length. Define sites v 1 , w 1 , v 2 , w 2 , . . . , w s along the path as follows (see Figure 3 for an illustration). Let v 1 ∈ T i be the first site of α. Given v 1 , . . . , v t , let v t+1 be the first site after v t at which the path enters a slice different from that of v t . Iterate this until when we reach a site v s ∈ T i ′ . Let w t be the site preceding v t+1 in α for each t < s, and let w s ∈ T i ′ be the last site of α. Thus α consists of a sequence of s sub-paths (v 1 , . . . , w 1 ), . . . , (v s , . . . , w s ), each one lying entirely within one slice, and with w t , v t+1 being adjacent but in different slices. (Note however that two non-adjacent sub-paths may lie in the same slice). More precisely we have the following facts. For each t = 1, . . . , s: (iv) the component of Z j(t) at v t has diameter ≤ n; ←→ v t ′ does not occur for any t ′ = t. (The definition of this event is delicate, and corrects a small error in [5]). Then we have Here the second sum is over all possible choices of v 1 , w 1 , v 2 , . . . , w s satisfying properties (i)-(iii) above, the symbol • denotes disjoint occurrence (which holds because of property (vi)), and the second inequality follows from the Van den Berg-Kesten inequality (see e.g. [7]). In order to bound the above, consider choosing v 1 , w 1 , v 2 , . . . , w s in order. There are #T i = m d−1 possible choices for v 1 . Once v t is chosen, the possible choices for w t lie in the cube ←→ w t . Once w t is chosen, there are (at most) 2 possible choices for v t+1 , corresponding to the two neighbouring slices. Hence we obtain v 1 ,...,ws where H(r) is as in the statement of Lemma 13. We use an almost identical argument to bound the probability of G(x, i, i ′ ). In this case the path starts at x, so there is no need for the factor m d−1 . We obtain Finally, substituting (18),(20),(21) into (16),(17), and substituting these into (14),(15) we obtain the conclusion of Lemma 13.
In the following proofs we use C 1 , C 2 , . . . to denote constants in (0, ∞) which may depend on d and ǫ, but not on p.
Proof of Theorem 10 (case d = 3). Let d = 3 and fix ǫ > 0. Since ǫ is arbitrary we can take for convenience We shall bound f 3 m (x, y) using Lemma 13; for this we need to choose n and find upper bounds on F 2 m,n and χ 2 n .
We first consider F 2 Now we use (22),(23) to bound the terms in Lemma 13. We have for p sufficiently small Writing C = C 2 we now bound the following expression from Lemma 13 by considering two possible cases for the value of r: for p sufficiently small. Looking again at Lemma 13 we therefore have for p sufficiently small, where we can drop the initial multiplicative constant because i 1 ≥ 1. Finally, substituting (24),(25) into Lemma 13 we obtain for p sufficiently small f 3 m (x, y) ≤ ℓ k=0 0<i 1 <···<i k <i k+1 =ℓ+1 for some γ ′ = γ ′ (d, ǫ) > 0, as required. so that the inductive hypothesis gives for x, y ∈ Q d−1 (n) that f d−1 n (x, y) ≤ f d−1 2n+1 (x, y) ≤ p γ( x−y ∞ +1) .
We shall apply Lemma 13. If X ∩ Q d−1 (m) d−1 has a component of diameter ≥ n then by Lemma 11, Q d−1 (m) contains a connected, (d − 1)i.s. subset T with diameter in [n/2, n]. Hence by the inductive hypothesis together with the argument used to obtain (12) in the proof of Theorem 1(ii), we have Hence, as in the case d = 3, substituting into Lemma 13 gives for p sufficiently small f d m (x, y) ≤ 2 ℓ p C 9 (ℓ+1) ≤ p γ ′ (ℓ+1) , as required.

Open Problems
(i) Prove the analogue of Theorem 1 for the standard bootstrap percolation model in 3 or more dimensions. What is the value of the threshold λ in this case?
(ii) Currently all proofs of the existence of a sharp threshold (in the sense of Theorem 1 as opposed to [3]) for bootstrap percolation models involve calculating its value. (See [8], [9] and the present work). Is there a simpler method of proving existence without determining the value?
(iii) What is the "second order" asymptotic behaviour of the model? Specifically, for example, if p = p 1/2 (L) is such that I d (L, p) = 1/2, what is the asymptotic growth rate of p log d−1 L − λ as L → ∞?