High-dimensional asymptotics for percolation of Gaussian free field level sets

We consider the Gaussian free field on $\mathbb{Z}^d$, $d$ greater or equal to $3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1 + o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d) \geq h_*(d)$ introduced by Rodriguez and Sznitman in arXiv:1202.5172 is in fact asymptotically equivalent to $h_*(d)$.


Introduction
When studying the statistical mechanics of random interfaces, which typically arise between coexisting phases of a (d + 1-dimensional) physical system in equilibrium, one often considers so-called effective models, which aim at describing the d-dimensional surface itself, free from its surroundings. Arguably the most notorious example in this class is the massless harmonic crystal, or Gaussian free field (precise definitions will follow, see (0.1) below). A natural approach in trying to gain some insights into the geometry of this field is to inquire about its level sets, say, above a given height h ∈ R. In case the underlying space is the cubic lattice Z d , with d ≥ 3, and due to the presence of strong correlations (the susceptibility is infinite), this gives rise to an interesting percolation model, which was originally introduced by Lebowitz and Saleur in [16], and has since then been investigated in [6], [8], [9], [18], [21] and [23], see also [2], [20] for related results.
Only recently has it been shown in [23] that the associated phase transition is non-trivial in all dimensions d ≥ 3 (partial results were already obtained in [6] and [9]). Our main focus in the present work is to examine the limiting behavior of certain critical parameters associated to this transition in high dimension. Of fundamental importance in this context is the heuristic principle by which this percolation model ought to fall in the "domain of attraction" of a corresponding model on the (2d)-regular tree. Our results indicate that this is indeed the case, and in fact, this paradigm permeates more or less explicitly many of the proofs below. Ideally, we would also like to compare these results with corresponding ones in the tree-case directly. We hope to return to this point elsewhere.
A first step in the direction of high-dimensional asymptotics is given by Theorem 3.3 of [23], which asserts that the critical height h * (d) for percolation of Gaussian free field level sets, defined in (0.3) below, is strictly positive when d is sufficiently large. In fact, a careful inspection of the proof of this theorem yields that h * (d) → ∞ as d → ∞, which amounts to saying that the critical density for this percolation model converges to 0 as d tends to infinity. We will considerably refine this result by providing the leading asymptotic behavior of the critical density, as well as principal asymptotics for h * (d) and a related critical parameter h * * (d) (see (0.4) below) as d becomes large. Our proof follows in its broad lines the general strategy underlying similar results in Bernoulli (see [12] and [3], Section 4) and interlacement (see [29] and [30]) percolation. However, the implementation of this program highly depends on the specific nature of the model, as it crucially relies on a precise understanding of its dependence structure in high dimension. In particular, in the present context, the random walk representation of the Gaussian free field, which will lead us to its "perturbative" (orthogonal) expansion around a suitable independent field, will play a pivotal role in allowing for a precise understanding of the local connectivity of the level set (we will explain this in greater detail below, see the discussion around (0.12)). Moreover, a severe technical obstruction is the absence of a BK-type inequality (companion to the long-range dependence). In the independent case, this inequality underlies the successful deployment of such elaborate tools as the lace expansion, see [13] and [24], Chapter 9. In our set-up, as a partial substitute, we develop suitable decoupling inequalities, much in the spirit of [22], [23], with the notable difference that they will need to work "uniformly well" for all sufficiently large d. These inequalities will typically allow for a certain kind of (static) renormalization procedure, to which the dimension d will be inextricably tied.
We now describe our results and refer to Section 1 for details. We consider the lattice Z d , d ≥ 3, endowed with the usual nearest-neighbor graph structure, and investigate the Gaussian free field on Z d , with canonical law P on Ω = R Z d (equipped with the product σ-algebra) such that, under P, the canonical field ϕ = (ϕ x ) x∈Z d is a centered Gaussian field with covariance E[ϕ x ϕ y ] = g(x, y), for all x, y ∈ Z d , (0. 1) where g(·, ·) denotes the Green function of simple random walk on Z d , see (1.3). In particular, this model exhibits strong correlations, see (1.4) below. For any level h ∈ R, we introduce the (random) subset of Z d often referred to as the excursion set (or level set) of the field ϕ above height h. In order to study its percolative properties, we set η(h) = P[0 ≥h ←→ ∞], the probability that the origin lies in an infinite cluster (i.e. connected component) of E ≥h ϕ . The function η( · ) being decreasing, we define the critical parameter for level-set percolation as (0. 3) h * (d) = inf{h ∈ R ; η(h) = 0} (with the convention inf ∅ = ∞). Following (0.6) of [23], we also introduce a second critical point ≥h ←→ S(0, 2L)} refers to the existence of a (nearest-neighbor) path in E ≥h ϕ connecting B(0, L), the ball of radius L around 0 in the ℓ ∞ -norm, to S(0, 2L), the ℓ ∞ -sphere of radius 2L around 0 (in fact (0.4) does not exactly coincide with (0.6) in [23], which requires the relevant probability to decay at least polynomially in L; the two are in fact equivalent, and (0.4) can be further weakened, see [21], Theorem 2.1). The definitions (0.3) and (0.4) immediately yield that h * (d) ≤ h * * (d) for all d ≥ 3. It is presently known that (0. 5) 0 ≤ h * (d) and h * * (d) < ∞, for all d ≥ 3, which implies that percolation of E ≥h ϕ exhibits a non-trivial phase transition (see [6], Corollary 2 for the former and [23], Theorem 2.6 for the latter result in (0.5); see also Theorem 3 in [6] for a proof of h * (3) < ∞). In particular, for all h > h * , E ≥h ϕ only contains finite clusters, and a (unique, see [23], Remark 1.6) infinite cluster for all h < h * . The parameter h * * is an important quantity because it characterizes a strongly subcritical regime. For all h > h * * and d ≥ 3, the probability P[0 ≥h ←→ S(0, L)] decays exponentially in L as L → ∞ (with logarithmic corrections when d = 3), as follows from Theorem 2.1. in [21]. Moreover, as mentioned in the expository paragraph (see [23], Theorem 3.3), (0. 6) h * (d) is strictly positive when d is large enough.
Our main goal in this paper is to prove the following asymptotic result concerning the critical density of this percolation model.
In fact, Theorem 0.1 will be an easy consequence of the following two results regarding the principal asymptotics of the critical parameters h * and h * * in high dimension. For future reference, we let (0. 8) h as (d) = 2g(0) log d, where g(0) refers to the Green function at the origin, cf. (1.3) below. We will show the following. Before describing our methods, we make a few remarks concerning these results and the heuristics lurking behind them. As alluded to above, a recurrent theme in the proofs below will be that, in high dimensions, the free field at small scales (at which the geometry felt by the random walk is roughly tree-like) can be viewed as a perturbation of an independent Gaussian field, and a considerable effort will go into understanding the precise effect of this perturbation on the connectivity of the level sets around the asymptotic value h as (see the comments below, and the proofs of Theorems 2.5 and 3.4). This perturbative behavior was already hinted at in the proof of (0.6) in [23], which was based on a decomposition of the covariance g(x, y) for x, y belonging to a (lower-dimensional) subspace Z d ′ , with d ′ ≪ d, into the sum of a dominant diagonal part with entries close to 1 (thus inducing a field of independent Gaussians) and a "small noise" (cf. Lemma 3.1. of [23] for a precise statement). Admitting this (local) resemblance of ϕ to an independent field, it is reasonable to compare (0.7) to p site c (Z d ), the critical parameter for Bernoulli site percolation on Z d , which is known to be asymptotically equal to 1/2d as d → ∞, see [3], [7], [12], [14].
Regarding Theorems 0.2 and 0.3, using h * ≤ h * * , (0.9) and (0.10) imply that . It is at present an important unresolved question whether both critical parameters are actually equal (in all dimensions).
We now comment on the proofs. The proof of the upper bound follows a strategy inspired by that used in [23] to prove finiteness of h * * (d), for arbitrary, but fixed dimension d (see also [22], [26], [27], [28], [30] for similar ideas in the context of random interlacements). In particular, we use a variant of the renormalization scheme developed therein. However, the present task requires a scheme which works "uniformly in d" as d becomes large. Using careful estimates on the behavior of the Green function of simple random walk on a high-dimensional lattice developed in [29,30], we obtain a "decoupling inequality," which enables us to propagate bounds on the relevant crossing events in E ≥h ϕ (cf. the definition (0.4) of h * * ), where h = h as (1 + ε) for some ε > 0, at small scale (the so-called seed estimates) to controls of such crossing probabilities at arbitrarily large scale in E ≥h+ε ϕ (this is in fact much more than we need). A significant part of the problem is to produce sufficiently sharp seed estimates, in order to initiate the renormalization, see Remark 2.3, 2) below (note that, in contrast to the proof of the finiteness of h * * (d) in [23], which allowed one to look for a corresponding regime at arbitrary large h, we are now constrained to remain in the vicinity of h as ).
Obtaining the desired bounds at small scales involves controlling the probability to see a crossing in E ≥h ϕ from a given point x to the boundary of the ℓ 1 -ball centered at x of radius R = c(ε)d, see (2.4) and Theorem 2.5 below (we emphasize that the use of the ℓ 1 -norm is essential here, as this distance controls the short-range behavior of g(·) in high dimensions, where the tree-like nature of the lattice manifests itself, cf. (1.14)). This is the first instance where the aforementioned (local) perturbative expansion comes into play. In rough terms, the domain Markov property for the free field ϕ allows us to "discover" it along any given path (in the present case, one joining x to S 1 (x, R)) "dynamically," starting from a suitable independent Gaussian field ψ, and introducing the required dependence at each step. Specifically, if K = {x 1 , . . . , x n } ⊂ Z d , n ≥ 1, denotes the trace of the path in question, we represent where the error term is a suitable linear combination of ψ x 1 , . . . , ψ x k−1 , cf. Lemma 1.3 below for a precise statement. This procedure enables us to "pass" from ϕ to the independent field ψ, provided we carefully keep track of the error we make in each step. For the lower bound, Theorem 0.3, our method mimics in its broad lines the approach of [12] and [3], Section 4, to the corresponding problem in Bernoulli (bond and site) percolation on Z d , involving ideas of [1] from hypercube percolation (see also [14] for a completely different solution, and [29] for a corresponding result in interlacement percolation, following a similar spirit). The proof essentially comprises two parts, which we briefly detail. In the first part, we show the following finite-size criterion. Suppose we partition the set H + Z 2 into translates of the hypercube. Roughly speaking, we show in i) contains with sufficiently high probability a "giant" connected component in each of the sets H and its four neighboring translates, and ii) all these components are connected in E ≥h ϕ within the union of H and its four neighboring translates (in Z 2 ), then E ≥h−ε ϕ percolates (the "additive" sprinkling in h is more than enough for the sake of proving (0.10)). Due to the long-range dependence, this reduction step does not follow from standard stochastic domination arguments (see for example [17]), and Theorem 3.1 is established by means of a (two-dimensional) renormalization argument.
In the second (and more difficult) part of the proof of Theorem 0.3, we show that this criterion actually holds. In the independent setting of [12] and [3], the validity of such a criterion is essentially guaranteed by the analysis in [1] of (independent) percolation in the hypercube. Our procedure essentially comprises two steps. First, drawing inspiration from [1], cf. in particular Lemma 1 therein, we grow substantial connected components in E ≥h ϕ ∩ H (with h = h as (d)(1 − ε)), which have cardinality growing polynomially in d. To achieve this, we embed into H a deterministic r(ε, d)-regular tree T, rooted at 0, with r(ε, d) comparable to d for every ε > 0, of depth depending on ε only. Here again, the "perturbative representation" of (0.12) crucially enters in allowing us to compare E ≥has(d)(1−ε) ϕ ∩ T to a certain (supercritical) Galton-Watson process on the same tree in order to derive a meaningful lower bound for the probability to see a substantial component at the origin. The precise statement is the object of Theorem 3.4, and an easy consequence is that most vertices in H are in fact either neighboring or contained in such a substantial component, see Corollary 3.7.
The second step then consists of gluing all substantial components within the hypercube to a giant one (i.e. a connected component whose closure in H contains at least (1−d −2 )2 d points), and subsequently connecting neighboring giant components, which is achieved in Theorem 3.8 by two successive sprinkling operations. Quantifying that it is highly unlikely for the substantial components not to merge after sprinkling requires isoperimetric considerations in H (as were used in the proofs of [1], Theorem 1 and [29], Theorem 4.2). However, the (wealth of) edges which are hereby deduced to be pivotal for a giant component to emerge or not, are not necessarily "well spread out" within H and might therefore influence each other rather strongly. To cope with this problem, we use a recent (conditional) decoupling result, see [21] (allowing for events which do not necessarily occur in distant parts of space) to prove that it is too costly for all pivotal sites to remain "closed" (cf. the proof of Lemma 3.9). This then completes the proof of the lower bound (0.10).
Let us now describe the organization of this article. In Section 1, we introduce some notation and review a few known results concerning simple random walk on a high-dimensional lattice and the Gaussian free field. Section 2 is devoted to the proof of Theorem 0.2. Subsection 2.1 introduces the renormalization scheme. Its main result is Proposition 2.1, which entails the induction step (one-step renormalization). Subsection 2.2 contains the local estimates on the connectivity of E ≥has(1+ε) ϕ , ε > 0. The main result, Theorem 2.5, is of independent interest, but it is needed crucially to establish the required seed estimate which enables us to trigger the renormalization scheme. Finally, in Subsection 2.3, we put these two ingredients together to complete the proof of (0.9). The proof of the lower bound (0.10) is the object of Section 3. First, we show the abovementioned finite-size criterion, which requires another (simpler) renormalization scheme. The main result is Theorem 3.1, which can be found in Subsection 3.1. Having established this reduction step, we prove that most points in the hypercube are either contained or neighboring a substantial component in the level set of interest. This is done in Subsection 3.2, see in particular Theorem 3.4 and Corollary 3.7 therein. Finally, Theorem 3.8 in Subsection 3.3 connects these substantial components, thereby completing the proof of the criterion. The lower bound (0.10) then follows readily by collecting the pieces, and Subsection 3.3 also contains the proof of Theorem 0.1, which follows straightforwardly from (0.9) and (0.10).
We conclude this introduction with a remark concerning our convention regarding constants: we denote by c, c ′ , . . . positive constants with values changing from place to place. Numbered constants c 0 , c 1 , . . . and c ′ 0 , c ′ 1 , . . . are defined at the place they first occur within the text and remain fixed from then on until the end of the article. All constants are numerical, and their dependence on any additional parameter, including, most importantly, the dimension d, will always appear in the notation. The only exceptions to this rule are the last two Sections 3.2 and 3.3, in which constants may implicitly depend on a parameter ε > 0. The Reader will be reminded of this exception in due time.

Notation and useful results
In this section, we introduce some notation to be used in the sequel, collect some important estimates related to simple random walk on a high-dimensional lattice, and review a few useful facts concerning the Gaussian free field.
We denote by N = {0, 1, 2, . . . } the set of natural numbers, and by Z the set of integers. We write R for the set of real numbers, abbreviate x ∧ y = min{x, y} and x ∨ y = max{x, y} for any two numbers x, y ∈ R, denote by ⌊x⌋ = max{n ∈ N; n ≤ x} the integer part of x, for any x ≥ 0, and let ⌈x⌉ = min{n ∈ N n ≥ x}. We consider the lattice Z d , and tacitly assume that d ≥ 3. Given a subset K of Z d , K c = Z d \ K stands for the complement of K in Z d , and |K| for the cardinality of K. Moreover, we write K + K ′ = {x + y ; x ∈ K, y ∈ K ′ } for the Minkowski sum of arbitrary sets K, K ′ ⊂ Z d .
On Z d , we respectively denote by | · | 1 , | · | 2 and | · | ∞ the ℓ 1 , Euclidean and ℓ ∞ -norms. The three norms are equivalent, and satisfy the relations For any x ∈ Z d , r ≥ 0, and p = 1, 2, ∞, we let B p (x, r) = {y ∈ Z d ; |y − x| p ≤ r} and S p (x, r) = {y ∈ Z d ; |y − x| p = r} stand for the the (closed) ℓ p -ball and ℓ p -sphere of radius r centered at x. We also note for later purposes the following bound on the cardinality of a d-dimensional ℓ 1 -sphere, which is easily computed by considering the generating function of |S 1 (0, n)|, see for example [30], Lemma 3.2 (i) for a proof. Given two arbitrary sets K, K ′ ⊂ Z d , we define their ℓ pdistance as d p (K, K ′ ) = inf{|x − y| p ; x ∈ K, y ∈ K ′ }, for p = 1, 2, ∞, and simply write d p (x, K ′ ) when K = {x} is a singleton. Moreover, we denote the interior (ℓ 1 -)boundary of K as ∂ int K = {x ∈ K; ∃y ∈ K c , |y − x| 1 = 1}, the outer boundary of K as ∂K = ∂ int (K c ), and write K = K ∪ ∂ out K for the (ℓ 1 -)closure of K and K We now introduce the (discrete-time) simple random walk on Z d . We endow the lattice Z d with the usual nearest-neighbor graph structure, and we will frequently use x ∼ y instead of |x − y| 1 = 1 to denote two neighboring vertices x, y ∈ Z d . Moreover, x, y ∈ Z d will be called * -nearest neighbors if |x − y| ∞ = 1. A (Z d -valued) nearest-neighbor path is a (finite or infinite) sequence (x n ) n of vertices in Z d satisfying x n+1 ∼ x n for all n ≥ 0. We define its length as the number of edges it traverses. When its length is infinite, we will often use the term trajectory instead of path. A * -path is defined accordingly. Let W denote the space of nearest-neighbor trajectories, and let W, (X n ) n≥0 and (θ n ) n≥0 , stand for the canonical σ-algebra, the canonical process and the canonical shifts on W , respectively. We write P x for the canonical law of the walk starting at x ∈ Z d and E x for the corresponding expectation. We denote by g(·, ·) the Green function of the walk, i.e.
= g(x − y) due to translation invariance. We further recall that (see for example [15], Theorem 1.5.4) Given K ⊂ Z d , we denote the entrance time in K by H K = inf{n ≥ 0; X n ∈ K} and the hitting time of K by H K = inf{n ≥ 1; X n ∈ K}. This allows us to define the Green function g K (·, ·) killed outside K as which is symmetric and vanishes if x / ∈ K or y / ∈ K. The relation between g and g K for any K ⊂ Z d is given by the following formula, the proof of which is a mere application of the strong Markov property (at time H K c ), We now turn to a few aspects of potential theory associated to simple random walk. For any K ⊂⊂ Z d , we write for the equilibrium measure (or escape probability) of K, and for its capacity. It immediately follows from the definitions (1.7) and (1.8) that the capacity is subadditive, i.e.
(1.9) cap(K ∪ K ′ ) ≤ cap(K) + cap(K ′ ), for all K, K ′ ⊂⊂ Z d , and one also easily infers (see for example [15], Proposition 2.2.1 (a)) that it is monotonous, i.e. that Moreover, the entrance probability in K may be expressed in terms of e K (·) as which is a mere consequence of the simple Markov property (see for example [25], Theorem 25.1, p. 300). We collect some useful estimates on these quantities in high dimension, which will be used repeatedly in the sequel. We remind the Reader of our convention regarding constants at the end of the previous section.
We now turn to the Gaussian free field on Z d , d ≥ 3, as defined in (0.1), and introduce certain crossing events involving paths of high level. On the space {0, 1} Z d endowed with its canonical σ-algebra A, let {K ←→ K ′ } (∈ A), for K, K ′ ⊂ Z d , denote the event that there exists a nearest-neighbor path connecting K and K ′ along which the configuration has value 1.
In words, this is the event that K and K ′ are connected by a nearest-neighbor path of vertices in the level set E ≥h ϕ , cf. (0.2). Note that this event is increasing upon in- . Moreover, the probability P[K ≥h ←→ K ′ ] is a decreasing function of h ∈ R. In general, we will use the notation We will also need the notion of flipping events. Let Y x , x ∈ Z d , denote the canonical coordinates on {0, 1} Z d . One defines the inversion map ι : , which will be referred to as flipped event. Note that A is decreasing whenever A is increasing. Moreover, observing that (−ϕ x ) x∈Z d has the same law as (ϕ x ) x∈Z d under P, we obtain, for all A ∈ A and h ∈ R, Next, we collect some classical results concerning the maximum of the Gaussian free field in a finite set.
We also recall the following elementary tail estimate of the normal distribution. For ξ ∼ N (0, σ 2 ) (see for example [4], Ch. 2, p. 49), We proceed with a classical fact concerning conditional distributions for the Gaussian free field on Z d , the proof of which can be found in [23] (see Lemma 1.2 therein).
Then, under P, (1.27) (In particular, ϕ K x = 0, P-a.s. whenever x ∈ K). Lemma 1.2 yields the following choice of regular conditional distributions for (ϕ x ) x∈Z d conditioned on the variables (ϕ x ) x∈K , which will prove very useful in several instances below. Namely, P-a.s., The following is an immediate corollary of Lemma 1.2, which provides a way to construct (ϕ x ) x∈K inductively from a family of independent Gaussian variables.
. . , x |K| be an enumeration of the elements of K, and K n = {x i ; 1 ≤ i < n}, for 1 ≤ n ≤ |K| (in particular, K 1 = ∅). Let ψ x , x ∈ K, be an independent family of Gaussian random variables, with and define recursively (with a slight abuse of notation) (1.30) Then (ϕ x ) x∈K has the law of Gaussian free field restricted to K.
Proof. We proceed by induction over n.

Upper bound
In this section, we show Theorem 0.2. As explained in the Introduction, this includes setting up an appropriate renormalization scheme, which is done in Subsection 2.1, and deducing suitable recursive bounds for the probability of the relevant crossing events at different scales. Proposition 2.1 entails the induction step, which is then propagated inductively in Proposition 2.2 along any increasing sequence of levels (h n ) n≥0 satisfying a mild growth condition (cf. Remark 2.8), provided a suitable bound for the probability of crossing events at the lowest scale holds. Obtaining the desired seed estimate requires a substantial amount of work (see also Remark 2.3, 2) below, which details this difficulty more quantitatively) and is the object of Subsection 2.2. The main result is Theorem 2.5 therein. Finally, Subsection 2.3 brings together the two ingredients to complete the proof of the upper bound (0.9).

Renormalization scheme
We start by developing a renormalization scheme, which, in its broad lines, is adapted from the one described in Section 2 of [23]. We start by introducing an integer parameter N ≥ 1 and a geometrically increasing sequence of length scales and corresponding renormalized lattices We are interested in the crossing events (cf. (1.17) for notation) for all x ∈ L n . The case n = 0 requires special treatment due to the following competing interests. On the one hand, the seed estimate we will need to establish forces B 0,x , x ∈ L 0 , to be not too small (this is the reason for introducing the additional parameter N ). On the other hand, B 0,x should remain sufficiently "invisible" to a simple random walk started at ℓ ∞ -distance of order L n , for some n ≥ 1, from x (see the proof of Proposition 2.1 below).
Similarly to the approach taken in Section 2 of [23], instead of looking at A h n,x directly, we will consider the trace left by the crossing path above level h in 2 n "well-separated" sets at scale n = 0, indexed by the leaves of a binary tree. To this end, we make the following "graphical cantor set" construction. We write T (k) = {1, 2} k for all k ≥ 0 (with the convention {1, 2} 0 = ∅), and T n = 0≤k≤n T (k) for the canonical dyadic tree of depth n. For arbitrary n ≥ 0, we call a map T : We denote by Λ n,x the set of proper embeddings of T n in Z d with root at x ∈ L n , for n ≥ 0. One easily infers that for some constant c 2 ≥ 1. Moreover, on account of (2.1), (2.4) and (2.5), if T ∈ Λ n,x for some n ≥ 1 and x ∈ L n , and if m ∈ T (k) for some 0 ≤ k < n with descendants m 1 , m 2 ∈ T (k+1) , then By elementary geometric considerations, and using (2.7), one then deduces that for all n ≥ 1, see Figure 1 below, and inductively that Accordingly, we introduce the quantity Figure 1: Any nearest-neighbor path in E ≥h ϕ (represented by the dashed line) connecting B∞(0, Ln) to S∞(0, 3Ln), for some n ≥ 1, must cross a box Bn−1,x 1 , for some x1 ∈ S∞(0, Ln) ∩ Ln−1, as well as Bn−1,x 2 , for some x2 ∈ S∞(0, 2Ln) ∩ Ln−1, thus giving rise to the simultaneous occurrence of A h n−1, which does not depend on x ∈ L n due to translation invariance, and is a decreasing function of h ∈ R. We will later apply a union bound in (2.8) in order to bound P[A h n,x ]. Thus, estimates for p n (h) will have to be strong enough to overcome the combinatorial complexity |Λ n,x | coming from the number of trees one can choose. To begin with, the following proposition provides recursive bounds for p n (h n ), n ≥ 0, along a suitable non-decreasing sequence (h n ) n≥0 .
There exists a constant c 3 such that, defining given any positive sequence (α n ) n≥0 satisfying and any increasing, real-valued sequence (h n ) n≥0 satisfying The proof of Proposition 2.1 is similar to that of Proposition 2.2 in [23] (see also Lemma 2.6 in [8] and Proposition 4.1 in [22]), with certain modifications. In particular, note that the constants appearing in these references all implicitly depend on d. In the present situation however, it is imperative to have a good control over the "sprinkling" condition for the sequence (h n ) n≥0 , cf. (2.12), in terms of d as d grows to infinity.
Proof. Let T be a proper embedding in Λ n+1,0 , for some n ≥ 0. For i = 1, 2, we denote by T i ∈ Λ n,T (i) the proper embedding of depth n with root at T (i) "induced" by T , i.e. determined by T i (m) = T (im), for all m ∈ T n (where im ∈ T n+1 stands for the concatenation of {i} ∈ T (1) and m), and define the sets (2.14) On account of (2.3) and (2.8), we see that We introduce a parameter α > 0, set α ′ = c 1 α (cf. (1.21) for the definition of c 1 ), and write, abbreviating max K 1 ϕ = max y∈K 1 ϕ y , (2.15) By (1.28), the conditional probability appearing in the last line of (2.15) can be rewritten as (1.18)). In particular, A T 2 is measurable with respect to the canonical coordinates in K 2 . We will now estimate the random shift µ K 1 x , for x ∈ K 2 , which will involve the bounds (1.15), (1.16). To this end, we first note that, by construction, for all x ∈ L 0 in the second line), which, together with the constraints L 0 ≥ d and l 0 ≥ 20( √ d + N ), immediately yields (2.17). Thus, for arbitrary x ∈ K 2 and α > 0, on the event {max K 1 ϕ ≤ α ′ }, where (1.15) applies due to (2.17) and (1.16) because ( √ d + N )L 0 > d by assumption, and where the last line in (2.18) defines the constant c 3 appearing in the statement of Proposition 2.1 above. In particular, on the event {max K 1 ϕ ≤ α ′ } and for any x ∈ K 2 , the inequality Hence, on the event where A T 2 denotes the flipped event (recall the notation from above (1.19)), and where we have used in the second line that ϕ K 1 and − ϕ K 1 have the same law under P, cf. (1.27), and that P does not act on µ K 1 . Inserting this bound into (2.15), we obtain 19) and the second step is due to (1.19) and the FKG-inequality for the free field (see for example [10], Ch. 4), which applies and yields an upper bound, because the event = α fulfilling (2.11) and all h ′ ≥ h, The claim (2.13) now readily follows upon taking suprema over all T ∈ Λ n+1,0 on both sides, = h ′ , so that requiring h n+1 = h ′ ≥ h = h n + γ, by virtue of (2.18), is precisely the condition (2.12). This concludes the proof of Proposition 2.1.
We will now propagate the bounds (2.13) inductively along a suitable sequence (h n ) n≥0 , and select to this end for some parameter k 0 > 0 to be specified later, and with m n (d, L 0 , N ) as defined in (2.10). In particular, note that (2.20) satisfies the condition (2.11) for every choice of k 0 > 0.
and let the sequence (h n ) n≥0 satisfy (2.12) with (α n ) n≥0 as defined in (2.20). Then, On account of Proposition 2.1 and the choice of α n , n ≥ 0, in (2.20), (compare this to Proposition 2.2 and (2.51) in [23], respectively), the proof of Proposition 2.2 is completely analogous to that of Proposition 2.4 in [23]. We therefore omit it. Remark 2.3. 1) Even though we will not need this below, note that the conclusions of Propositions 2.1 and 2.2 continue to hold for arbitrary increasing (seed) events A 0,x ⊂ {0, 1} Z d , for x ∈ L 0 , which are measurable with respect to the canonical coordinates in B 0,x as defined in (2.4

), upon letting
, for all x ∈ L 0 and h ∈ R, defining events A h T , with T ∈ Λ n,x for arbitrary n ≥ 0 and x ∈ L n , as in (2.8), and taking an additional supremum over x ∈ L n in the definition (2.9) of p n (h) (no translation invariance required). Moreover, by symmetry, if instead all events A 0,x , x ∈ L 0 , are decreasing, under the assumptions of Proposition 2.1, the conclusion (2.13) holds for the sequence (−h n ) n≥0 in place of (h n ) n≥0 .
2) The condition (2.21) is quite strong. Indeed, by (2.8) and (2.9), we have, for all h ∈ R, n ≥ 0 and x ∈ L n , If the bound (2.22) were to hold for some value of k 0 ≥ b, it would yield, together with the estimate (2.6) on the combinatorial complexity |Λ n, Thus k 0 will have to satisfy the requirement e k 0 ≥ cl for a certain constant c > 0 in order for this bound to be of any use. Substituting this into (2.21) yields the condition . For the purpose of proving Theorem 0.2, h 0 will be at most h as (1 + ε), for some ε > 0. We will therefore essentially have to prove that the crossing event A has(1+ε) 0,0 decays roughly like exp(−cd log d) for all d ≥ c ′ (ε) (recall that l 0 is required to grow at least like √ d, cf. (2.1)).

The local picture
In this subsection, we provide an estimate regarding the local connectivity properties of the level set E ≥has(1+ε) ϕ , for arbitrary ε > 0, in high dimension. Here, "local" means that we only investigate connectivity within an ℓ 1 -ball having a radius of order at most d around a given point on the lattice. Specifically, we will consider the probability of a nearest-neighbor path in E ≥has(1+ε) ϕ connecting the center of such a ball to its boundary. The resulting bound, cf. Theorem 2.5, is of independent interest, but it will be crucial in establishing the seed estimate (2.21), thus enabling us to launch the renormalization. We begin with the following simple lemma, which will be useful in several instances below.
There exists a constant c > 0 such that, for all ℓ > 0, x ∈ Z d , all sets K, U satisfying U ⊆ K ⊂⊂ Z d \ {x} and |U | ≤ ℓd, and all ε > 0, with p K x,y as defined in (1.24).
We compute

(2.25)
Observe that for all y ∈ U and d ≥ 3, where the second step follows because |U | ≤ ℓd by assumption. Substituting this into (2.24) and optimizing over λ, which occurs for the choice λ = εdh as /cℓ(ℓ + 1) (with c the constant appearing in the bound for Var(λζ x ) above), yields for all d ≥ 3. On account of (0.8), and since g(0) ≥ 1, this implies (2.23), and thus completes the proof of Lemma 2.4.
We proceed to the main result of this section. For convenience, we introduce the shorthand (2.28) h (ε) as = h as (1 + ε), for ε > 0.
There exist constants c 4 (ε) ≥ 1 and c 5 > 0 such that for all N ≥ c 4 (ε) and d ≥ c(ε, N ), We briefly outline the proof of Theorem 2.5, which essentially comprises three steps. First, instead of considering the whole path connecting the origin to S 1 (0, N d) directly, we look at the "local traces" it leaves after first visiting each of N 0 = cN (for suitable c ∈ (0, 1)) concentric ℓ 1 -annuli around the origin having a width of order d (similarly to what was done in [30] in the context of interlacement percolation). We are led to consider the probability that N 0 "wellseparated" paths of high level of length ⌊ℓd⌋ each, for some 0 < ℓ ≤ 1, all occur simultaneously (which competes against a suitable combinatorial complexity). The parameter ℓ will have to be carefully chosen a posteriori. In a second step, we discover the field along a fixed collection of paths "dynamically," using Lemma 1.3, as alluded to in the introduction. Finally, Lemma 2.4 will provide good controls on the error terms that arise.
Let ε > 0 be fixed and assume N ≥ c 6 (stronger conditions on N will follow). Instead of working with N directly, it will be convenient to use We also introduce a parameter to be specified later. Given a vertex x ∈ Z d , we denote by Π ℓ (x) the set of all self-avoiding nearest-neighbor paths of length ⌊ℓd⌋ starting in x (by convention, if ⌊ℓd⌋ = 0, the set consists only of the vertex x itself). By construction, any path connecting the origin to S 1 (0, N d) intersects all spheres S 1 (0, n c 6 d), for 0 ≤ n ≤ N 0 . Moreover, any self-avoiding path π ∈ Π ℓ (x), with x ∈ S 1 (0, n c 6 d) for some 0 ≤ n < N 0 and 0 < ℓ ≤ 1, satisfies range(π) ⊂ B 1 (0, N d). Thus, for all N 0 ≥ 1 (i.e. N ≥ c 6 ) and 0 < ℓ ≤ 1 (see also Figure 2 below). For later reference, we note that given two paths π n ∈ Π ℓ (x n ) and π m ∈ Π ℓ (x m ) for some 0 < ℓ ≤ 1, with 0 ≤ n < m < N 0 , x n ∈ S 1 (0, nc 6 d) and x m ∈ S 1 (0, mc 6 d), Since |Π ℓ (x)| ≤ (2d) ⌊ℓd⌋ for all d ≥ 3, 0 < ℓ ≤ 1 and x ∈ Z d , and using the estimate (1.2) connecting the origin to S1(0, N d), obtained by considering the first ⌊ℓd⌋ steps after first hitting each of the spheres S1(0, nc6d), for 0 ≤ n < N0 (when viewed as starting from 0). The paths πn, 0 ≤ n < N0, are in fact "well-separated," as captured by (2.34). on the cardinality of ℓ 1 -spheres, the number of terms appearing in the double sum of (2.33) is bounded by for all N 0 ≥ 1, 0 < ℓ ≤ 1 and d ≥ c(ε, N 0 , ℓ). Returning to (2.33), this yields as ϕ ⊇ range(π n ), 0 ≤ n < N 0 ], (2.35) for all N 0 ≥ 1, 0 < ℓ ≤ 1 and d ≥ c(ε, N 0 , ℓ). We now focus on the probability appearing on the right-hand side of (2.35). Thus, we fix N 0 self-avoiding paths π n = (x n,k ) 0≤k≤⌊ℓd⌋ ∈ Π ℓ (x n,0 ), with x n,0 ∈ S 1 (0, nc 6 d) for all 0 ≤ n < N 0 , and define We further denote by ≺ the lexicographic order on I. This induces an ordering of the points in K. Finally, we also set (2.37) The event is measurable with respect to the σ-algebra generated by ϕ x , x ∈ K, and we will now construct the field (ϕ x ) x∈K using Lemma 1.3, by adding one variable at a time, according to the ordering ≺ (thus, K i as defined in (2.37) denotes the set of points at which the field has been discovered "before time i"). Specifically, we proceed as follows. Let ψ i , i ∈ I, be an independent family of Gaussian random variables, with , for all i ∈ I (see (1.5) for the definition of the killed Green function), and recall from (1.24) that p K i x,y = P x [H K i < ∞, X H K i = y], for all x, y ∈ Z d . We define recursively By Lemma 1.3, the law of (ϕ x i ) i∈I as defined above is precisely that of the Gaussian free field restricted to K. By (2.34), the main contribution to the sum on the right hand side of (2.40) will come from the points which belong to the same path as x i (i.e. to π i 1 ). Accordingly, we introduce the sets and obtain the decomposition, where the above sums are understood to vanish identically whenever summation is over the empty set. Now, observe that for all i ∈ I, on the event (2.43) In order to bound the hitting probability P x i [H K i \U i < ∞], we note that, by construction, cf.
(2.34) and (2.41), Thus, we can find c > 0 such that (with the convention sup ∅ = 0, which occurs if K i \ U i = ∅, i.e. i 1 = 0). Thus, for all i ∈ I and d ≥ c(ε, N 0 , ℓ), and therefore Going back to our initial event in (2.38), by a union bound, we obtain, for all N 0 ≥ 1, 0 < ℓ ≤ 1 and d ≥ c(ε, N 0 , ℓ), where we also used independence of the variables ψ i , i ∈ I, see above (2.39). We consider each of the three terms in (2.46) separately. Recall the definition of the combinatorial complexity C d above (2.35).  Proof of Lemma 2.6. First, observe that < c 2 1 , for all i ∈ I and d ≥ 3.
Remark 2.7. The only feature of the function f (ε, N ) which will be of crucial importance below is that lim N →∞ f (ε, N ) = ∞ for every ε > 0. For the sole purpose of proving Theorem 0.2, any other function f (ε, N ) with this property would have sufficed.

Dénouement
Finally, we complete the proof of Theorem 0.2. This will involve the renormalization scheme introduced in Section 2.1. As noted in Remark 2.3, 2), the seed estimate (2.21) needed to initiate the renormalization is rather strong, and Theorem 2.5 will be of crucial use to establish it.
Proof of Theorem 0.2. Let ε > 0 be fixed. In order to establish (0.9), we have to show that To begin with, we select the parameters L 0 , l 0 and N in the renormalization scheme from Section 2.1, cf. (2.1). We let (recall the statement of Theorem 2.5 for the definition of c 4 (ε), c 5 ), so that the constraint l 0 ≥ 20( √ d + N (ε)) in (2.1) holds for all d ≥ c(ε), and choose k 0 appearing in (2.20) as (see (2.6) for the definition of c 2 ). Lastly, we define the increasing sequence (h n ) n≥0 recursively as , for all n ≥ 0, (2.54) with α n given by (2.20). In particular, (h n ) n≥0 satisfies the "sprinkling" condition (2.12 . This is more than enough to deduce (2.55) (we comment on this in Remark 2.8 below). In order to complete the proof of Theorem 0.2, we apply Proposition 2.2. Except for the condition (2.21), all its requirements are clearly satisfied by our choices of parameters in (2.52), (2.53) and (2.54), whenever d ≥ c(ε). To deduce the necessary seed estimate (2.21), we use the connectivity bound from Theorem 2.5. Observing that the choice of N = N (ε) in (2.52) guarantees that c 5 f (ε/2, N ) ≥ 6, for f and c 5 as appearing in (2.29), it follows that for all d ≥ c ′ (ε). On the other hand, (2.53) impies that e −k 0 ≥ d −3d for all d ≥ c, hence the condition p 0 (h 0 ) ≤ e −k 0 in (2.21) holds for alld ≥ c ′ (ε). We may thus apply Proposition 2.2, thereby obtaining with (h n ) n≥0 as defined in (2.54). Hence, for all n ≥ 1 and d ≥ c(ε), In particular, this implies that lim inf L→∞ P[B ∞ (0, L) Remark 2.8. Our renormalization scheme is somewhat asymmetrical. On the one hand, the sprinkling condition (2.12) for the sequence (h n ) n≥0 turns out to be very mild, as it only costs an "additive" ε, cf. (2.57) (obtaining h ∞ ≤ h 0 (1 + ε), for all d ≥ c(ε) would have sufficed for the purpose of proving Theorem 0.2). On the other hand, the scheme relies on the strong seed estimate (2.21), and establishing it is what prevents us from obtaining a more precise result than (0.9).

Lower bound
We proceed to show the lower bound, Theorem 0.3. As described in the Introduction, the proof comprises two main steps. The first one, which is the subject of Subsection 3.1, reduces the problem of constructing an infinite cluster to a local statement. The assertion is roughly the following (see Theorem 3.1 below): given δ > 0, if the level h is such that with high probability, the set E ≥h+δ ϕ possesses a ubiquitous component in each of the d-dimensional hypercubes 2x + {0, 1} d , for x ∈ Z 2 (⊂ Z d ) and |x| 1 ≤ 1, which are all connected, then E ≥h ϕ percolates whenever d is sufficiently large (depending on δ).
In order to prove the lower bound (0.10), it then suffices to verify that this criterion holds when h = h as (1 − ε), for arbitrary ε > 0. This step is split again into two parts, to which Subsections 3.2 and 3.3 are respectively devoted. First, we construct a wealth of substantial components in the hypercube, with cardinality growing polynomially in d, and show that most vertices in {0, 1} d are either contained in or neighboring such a component. The main result is entailed in Corollary 3.7 below. The second part consists of patching together these substantial components to form a "giant" one, and to then connect the latter to the ubiquitous components contained in the neighboring translates of {0, 1} Z d . This is achieved in Theorem 3.8. All ingredients are put together at the end of Subsection 3.3 to complete the proof of (0.10).

Local ubiquity and connectivity are sufficient
We begin by establishing the finite-size criterion that guarantees percolation, Theorem 3.1 below. To cope with the long-range interactions, we use a renormalization argument on Z 2 , which bears some resemblance to the one developed in Section 2.1 above, together with a standard duality argument.
= H 0 , so that H x , x ∈ Z 2 , forms a partition of H + Z 2 . The following definition is essentially borrowed from [30], Section 2 (this will in particular enable us to reinject certain isoperimetric controls which follow from the results in [5] (see Lemma 4.3 in [30]) directly at a later stage.) For arbitrary h ∈ R and x ∈ Z 2 , we introduce the (local) event x occurs, for arbitrary h ∈ R, and h-bad otherwise. By Lemma 2.1 of [29], it follows that for all d ≥ c and x ∈ Z 2 , any connected component C x of H x ∩ E ≥h ϕ whose closure in H x has cardinality at least (1 − d −2 ) 2 d is necessarily unique. We will thus refer to it as the giant component of H x ∩ E ≥h ϕ . It is then plain from (3.2) that for all d ≥ c and h ∈ R, i.e. percolation of h-good sites in Z 2 implies percolation of E ≥h ϕ (in H + Z 2 ). We are now ready to proceed to the main result of this section, which has a similar flavor as Theorem 2.2 of [29]. Given any sequence (h(d)) d≥3 such that Proof. By (3.3) and the definition of h * in (0.3), in order to prove (3.5), it suffices to show that for this implies P[0 . The proof of (3.6) involves a renormalization scheme, which we now describe. We introduce the integer parameter and define an increasing sequence of length scales (L n ) n≥0 recursively as (3.8) L n+1 = l n L n , l n = 20⌈L a n ⌉, for n ≥ 0 (this should not be confused with the geometric sequence of length scales appearing in Section 2.1), together with corresponding renormalized lattices We also introduce, for arbitrary n ≥ 0 and x ∈ L (2) n , the (bad) events ∞ (x, 3L n ) by a * -path of h-bad vertices (in Z 2 )} (3.10) (recall that y ∈ Z 2 is called h-bad if (G h y ) c occurs). By (3.2), the event D h n,x is decreasing (in ϕ). Moreover, by translation invariance, the function is well-defined (i.e. independent of x) and non-decreasing in h, for every n ≥ 0. The key to establishing (3.6) will be to show that, if L 0 is a suitable (increasing) function of d, and (3.4) holds for some ε > 0, the probability q n (h(d)) decays sufficiently rapidly to 0 as n → ∞, for all d ≥ c(ε), see (3.29) below. Together with a straightforward (planar) duality argument, which will be detailed below (3.29), this will then yield (3.6).
To obtain good estimates for q n (h(d)), n ≥ 0, we first develop "recursive bounds" relating the functions q n+1 (·) and q n (·), for arbitrary n ≥ 0 (similar in spirit to what was done in Proposition 2.1 above, but simpler). To this end, we let n ≥ 0 and x ∈ L (2) n+1 be fixed, and introduce the sets By geometric considerations similar to those leading to (2.8), see also Figure 1, and on account of (3.8), we deduce that (3.13) q n+1 (h) ≤ P for a suitable constant c ′ 0 ≥ 1 and all h ∈ R. We consider the probability on the right-hand side of (3.13), for fixed x i ∈ S i , i = 1, 2 and h ∈ R. Abbreviating (3.14) 1 (y,1) H z and observing that D h n,x i ∈ σ(ϕ y ; y ∈ K i ), for i = 1, 2, due to (3.2) and (3.10), we write for some cut-off value β n > 0 to be selected below. By (1.28), , P-a.s.
We now complete the proof of Theorem 3.1. Let ε > 0 and (h(d)) d≥3 be a sequence of levels such that (3.4) holds. The latter implies that By the choice of scales in (3.8) and (3.16), this yields, for all d ≥ c(ε), where we have also used in the penultimate step that G h 0 is decreasing (in ϕ) and that l n > L a n ≥ L a(1+a) n 0 for all n ≥ 0. This completes the proof of (3.6), hence of Theorem 3.1. ) c ] = 0, for some δ > 0 (using a choice of a in (3.7) depending on δ). However, this will not be of importance, as our proof will show that for the relevant choice of h(d) = h as (d)(1 − 8ε), with ε > 0 arbitrary (the factor of 8 is just for convenience), the above probability decays to 0 as d → ∞ faster than any polynomial, see Theorem 3.8 below.

Constructing substantial components
With Theorem 3.1 at hand, in order to prove the lower bound (0.10), we will check that condition (3.4) holds at level h(d) = h as (d)(1 − 8ε), for arbitrary ε > 0 (the factor 8 is immaterial, and merely reflects the fact that a few more sprinkling operations will be performed on the way). This will involve showing that H contains a giant component above this level with sufficiently high probability, cf. (3.2). We will construct this component by gluing together smaller building blocks, so-called substantial components, which, by definition, have cardinality growing like a polynomial in d of sufficiently high degree. In this subsection, we show that most vertices in H are either neighboring or contained in such a substantial component with high probability, see Corollary 3.7 below for the precise statement. The substantial component neighboring a given point in H will be built as a connected subset of a (large) deterministic tree embedded in H and rooted at this point. In particular, the "perturbative" representation of Lemma 1.3 will enable us to show that, conditionally on an event of high probability, the law of the restriction of E ≥has(1−2ε) ϕ to this tree dominates a Galton-Watson process (with suitable binomial offspring distribution) on the same tree, cf. Lemma 3.5 below.
Let ε > 0. For the sake of clarity, in the remainder of this article, the dependence of constants on ε will be kept implicit. We introduce a parameter (3.30) b = b(ε) = 1 + 11 ε (≥ 1).
Given K ⊂ Z d , we call C a substantial component of K if C is a connected subset of K containing at least ⌊d ε /b⌋ b−1 points.
Proof. By symmetry, it suffices to consider the case x = 0. We begin with some notation. Let ε ∈ (0, 1/3). We assume without loss of generality that d ≥ b(ε) = b, with b as defined in (3.30) (it suffices to show Theorem 3.4 for d ≥ c ′ , since the remaining cases can be taken care of by adapting the constant c appearing in (3.31)), and introduce b consecutive subsets of {1, 2, . . . , d}. We will interpret a part of H as a tree by considering where e i denotes the canonical unit vector in the i-th direction, for 1 ≤ i ≤ d. Given x ∈ T, we refer to j in the (unique) decomposition of x = j k=1 e i k , with i k ∈ I k , 1 ≤ k ≤ j, as the generation of x, and to the set {x + e i ; i ∈ I j+1 } (understood as ∅ if j = b) as the children of x. Thus, every vertex in T in generation smaller than b has precisely ⌊d/b⌋ children, and b corresponds to the depth of T. Furthermore, it will be convenient to set T 0 = T ∪ {0}. We now show that for all d ≥ c, which implies (3.31). To this end, we enumerate the elements of T as a sequence x n , 1 ≤ n ≤ |T| in a hierarchical way, i.e. such that |x n | 1 ≤ |x n+1 | 1 for all 1 ≤ n < |T|, and set K n = {x 1 , . . . , x n−1 }, for 1 ≤ n ≤ |T|. We construct the field (ϕ x ) x∈T using Lemma 1.3 according to this ordering. Thus, introducing a family (ψ x ) x∈T of independent random variables with ψ xn ∼ N (0, g K c n (x n , x n )), for all 1 ≤ n ≤ |T|, we define the field (ϕ x ) x∈T in terms of (ψ x ) x∈T as in (1.30). Moreover, letting with M and its complement, respectively, we deduce that in order to prove (3.33), it suffices to show that holds for d ≥ c. We now show that under P[ · |M ], the variables (1{ϕ x ≥ h as (1 − 2ε)}) x∈T stochastically dominate a Galton-Watson chain on T with suitable (binomial) offspring distribution. More precisely, we prove the following. Let b x , x ∈ T, denote a family of independent Bernoulli variables under some auxiliary probability measure P such that P Then, for all d ≥ c, Proof of Lemma 3.5. We begin with the following remark. From (1.30), one immediately infers that ϕ xn = n k=1 α n,k ψ xn , for 1 ≤ n ≤ |T|, where α n,n = 1, α n,k = n−1 l=k p Kn xn,x l α l,k , for 1 ≤ n ≤ |T| and 1 ≤ k < n, with p Kn xn,x l , 1 ≤ l < n, as defined in (1.24). We claim that for all 1 ≤ n ≤ |T|. Since the elements of the tree T are enumerated in a hierarchical fashion, K n ⊂ B 1 (0, |x n−1 | 1 ) for all 1 < n ≤ |T|, and therefore (by construction, |z| 1 ≤ b, for z ∈ T, and the depth b does not depend on d, cf. (3.30)). In particular, this probability is less than 1/2, uniformly in n for all 1 < n ≤ |T|, whenever d ≥ c. Inserting this into (3.39) and a trivial inductive argument yield (3.38). We proceed to the proof of (3.37). We claim that for all d ≥ c and 1 ≤ n ≤ |T|,  To this end, fix some 1 < n ≤ |T|, and let A be an increasing subset of Ω n def.
= {0, 1} K n+1 (with K |T|+1 = T). For x ∈ K n+1 , we define the event {x is pivotal for A} as consisting of all ω ∈ Ω n such that 1 A (ω x ) = 1 A (ω x ), where ω x and ω x are obtained from ω by forcing ω(x) to be 1 and 0, respectively. Defining A x piv = A ∩ {x is pivotal for A}, abbreviating χ x = 1{ϕ x ≥ h as (1 − 2ε)}, for x ∈ T, we split (3.43) and consider each of these two sums separately. By monotonicity of A, the event A xn piv contains all ω ∈ A with ω(x n ) = 1 and ω xn / ∈ A. Hence, the first sum on the right-hand side of (3.43) equals where we have used that χ x i is a (measurable) function of ψ x k , 1 ≤ k ≤ i, cf. (1.30). By (3.41), which implies that {χ xn = 1} ∩ M n ⊇ {ψ xn ≥ h as (1 − 3 2 ε)} ∩ M n since the events M n are decreasing, and by the independence of the variables ψ x , x ∈ K n+1 , we have for all d ≥ c (for the first step, note also that M n = M n−1 ∩ M n and M n ⊃ {ψ xn ≥ h as (1 − 3 2 ε)} by definition, cf. (3.35)). Substituting this into the previous equation, we obtain that the first sum on the right-hand side of (3.43) is bounded from below by for all d ≥ c. Moreover, since A\A xn piv contains all ω ∈ Ω n such that both ω xn ∈ A and ω xn ∈ A, the second sum on the right-hand side of (3.43) can be rewritten as where (w ′ , 1) ∈ Ω n refers to the concatenation of 1 with w ′ , and where we have also used independence of the events {(χ y ) y∈Kn = ω ′ } and M n , so that conditioning on M n can be replaced by conditioning on M n−1 . Putting things together, we obtain that for all d ≥ c and 1 < n ≤ |T|, the law of (1{ϕ x ≥ h as (1 − 2ε)}) x∈K n+1 under P[ · |M n ], stochastically dominates the law of ((1{ϕ , where (ψ x ) x∈T is an independent copy of (ψ x ) x∈T under P. Noting that ϕ x 1 = ψ x 1 , cf. (1.30), a straightforward induction then yields (3.42).
We continue with the proof of (3.36). We introduce, for x ∈ T 0 ∩ B 1 (0, b − 1), the variable which can be interpreted as the number of "existing" children of x in the Galton-Watson chain (b x ) x∈T . We will need the following estimate.

Connecting substantial components
In this subsection, we finally prove Theorem 0.3. We will achieve this by verifying the finite-size criterion (3.4) when h(d) = h as (d)(1−8ε), for all sufficiently small ε > 0. In order to deduce that the event G has(1−8ε) 0 appearing in this context occurs with sufficiently high probability, we will use isoperimetry considerations to patch together the substantial components of E . This is the object of Theorem 3.8. The lower bound (0.10) then follows readily, by virtue of Theorem 3.1.
Proof. Fix ε ∈ (0, 1/3). We start by showing for all d ≥ 3 and x ∈ Z 2 . By translation invariance, it suffices to consider the case x = 0. We denote by S = (S 1 , . . . , S N subst ) (with S = ∅ if N subst = 0) the collection of substantial components of H ∩ E ≥has(1−2ε) ϕ (we assume for sake of definiteness that S 1 , . . . , S N subst are enumerated according to a specified procedure, e.g. using the lexicographic order induced by the points closest to the origin in each component). With a slight abuse of notation, S will also be used to denote the set N subst i=1 S i , but the meaning will always be clear from the context. By definition, see (3.30), the random sets S i satisfy |S i | ≥ d 10 , for all 1 ≤ i ≤ N subst , whenever d ≥ c, and S H i ∩ S j = ∅ for all i = j. We define the following set of partitions of S, ←→ K ′ } (with the convention that H is the whole space R Z d whenever P(S) = ∅) occurs with sufficiently high probability. In words, H is the event that for any partition of the substantial components of H ∩ E ≥has(1−2ε) ϕ into two "sizeable" classes (in the sense that the cardinality of the respective aggregate unions of substantial components is at least d −4 |H|), one can find a substantial component in each class such that the two are connected in E with µ U c x = z∈U c P x [H U c < ∞, X H U c = z]ϕ z , for x ∈ U , as in (1.25). Since a non-vanishing contribution to the previous sum arises only from the points in ∂ out U , and since |U | ≤ 2, Lemma 2.4 applies (with ℓ = 4 and K, U appearing therein both equal to ∂ out U here), thus yielding We also introduce the (good) event (recall (3.45) for the definition of B ε ). On account of (3.46) and (3.53), a union bound yields, for all d ≥ 3, It will be convenient to specify configurations of the level set above h as (1−2ε) in the hypercube. Thus, given K + ⊂ H, we abbreviate C(K + ) = {E  for all d ≥ c, where the set S K + in the second and third line refers to the (deterministic) family of substantial components associated to the configuration C(K + ). In order to bound the conditional probability appearing on the right-hand side of (3.56), we rely on isoperimetry considerations for subsets of the hypercube by Bollobás and Leader [5] and a recent (conditional) decoupling result for the Gaussian free field by Popov and Ráth [21]. Thus, let K + ⊂ H be such that P(S K + ) is non-empty and {K, K ′ } ∈ P(S K + ) be fixed. First, observe that B ε is a deterministic set under P[ · | C(K + )]. By construction, On account of (3.57) and (3.58), Lemma 4.3 in [29] (itself a consequence of Corollary 4 in [5]) yields that for all d ≥ c, there exist disjoint subsets U ′ 1 , . . . , U ′ m ′ in Π ∩ (H \ B ε ) with m ′ ≥ cd −6 |H| such that for all 1 ≤ k ≤ m ′ , either U ′ k = {y k } with y k ∈ ∂ out K ∩ ∂ out K ′ , or U ′ k = {y k , z k } with y k ∈ ∂ out K, z k ∈ ∂ out K ′ (and |y k − z k | 1 = 1). (3.59) Among the family U ′ k , 1 ≤ k ≤ m ′ , we may select m ≥ cd −7 |H| sets U 1 , . . . , U m satisfying the additional assumption Next, we consider a single factor P[F k | G, F l , k < l ≤ m, C(K + )] in this product, and estimate how rapidly (in terms of d) it converges to 1 as d → ∞. By definition of G U k , cf. (3.52), and on account of (3.60), the event G is measurable with respect to the σ-algebra generated by ϕ z , z ∈ H \ 1≤k≤m U k , (H denotes the ℓ 1 -closure of H), hence for every 1 ≤ k ≤ m, D k = { G, F l , k < l ≤ m, ϕ | K + ≥ h as (1 − 2ε), ϕ | H\(K + ∪U k ) < h as (1 − 2ε)} ∈ σ(ϕ z ; z ∈ H \ U k ) (here and in what follows, we use the shorthand {ϕ | K ∈ B} = x∈K {ϕ x ∈ B}, for any Borel set B ⊂ R and K ⊂ Z d ). Since all elements y ∈ U k lie on the exterior boundary of a substantial component of E x each contain at most d −2 |H| elements. Moreover, the joint occurrence of C 0 and C x implies immediately that for all d ≥ 3, there exist disjoint sets U k = {y k , z k } ⊂ (x + {0, 1} Z d ), for 1 ≤ k ≤ m, with m ≥ cd −1 (|H|/2 − 2d −2 |H|) (≥ c ′ d −1 |H|), and such that x ∩ H 2 , |x k − y k | 1 = 1, for all 1 ≤ k ≤ m and d 1 (U k , U l ) ≥ 2 for 1 ≤ k < l ≤ m. For U k not to form a path joining C 0 and C x in the level set E ≥has(1−8ε) ϕ , at least one of the two sites in U k must have a field value smaller than h as (1 − 8ε), i.e. settingF k = y∈U k {ϕ y < h as (1 − 8ε)}, we obtain P[Ĝ, C 0 and C x exist and C 0 C x in E ≥has(1−8ε) Now, on account of (3.69), an analysis similar to the one below (3.61) yields that this last quantity is bounded from above by 2 −cd −3 |H| , for all d ≥ c, which is more than enough to deduce (3.68). This completes the proof of Theorem 3.8.
We now conclude the proof of the lower bound. Proof of Theorem 0.1. The claim (0.7) is a direct consequence of (1.23) and the fact that h * (d) = h as (d)(1 + o(1)) as d → ∞, which follows immediately from (0.9) and (0.10). 1) A thorough review of the proofs reveals that the long-range dependence present in the model is a serious impediment and considerably hinders any efforts to obtain more precise results than the leading exponential order given by Theorem 0.1.
2) The heuristic paradigm requiring for this model to exhibit strong similarities to a corresponding one on the (2d)-regular tree is evidently inherent to many of the above proofs. Specifically, this behavior manifests itself when we investigate the local connectivity of the level set, essentially because the "tree-like" structure of the lattice determines the behavior of the random walk at short range, see the proofs of Lemma 2.4 and Theorem 2.5 (which crucially rely on (1.14)) for the upper bound, and the proof of Theorem 3.4 for the lower bound. Accordingly, we would like to compare (0.7) to the critical density of the same model on this tree. We hope to come back to this point in the future.