{'fraction_non_alphanumeric': 0.0815056855313031, 'fraction_numerical': 0.0287021304404653, 'mean_word_length': 3.343817417962984, 'pattern_counts': {'":': 0, '<': 31, '<?xml version=': 0, '>': 23, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 25, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Phase transition and level-set percolation for the Gaussian free field

We consider level-set percolation for the Gaussian free field on Z^d, with d bigger or equal to 3, and prove that there is a non-trivial critical level h_* such that for h>h_*, the excursion set above level h does not percolate, and for h<h_*, the excursion set does percolate. It is known from the work of Bricmont-Lebowitz-Maes that h_* is non-negative for all d bigger or equal to 3, and finite, when d=3. We prove here that h_* is finite for all d bigger or equal to 3. In fact, we introduce a second critical parameter h_**, which is bigger or equal to h_*. We show that h_** is finite for all d bigger or equal to 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h>h_**. Finally we prove that h_*>0 in high dimension. It remains open whether h_* and h_** actually coincide, and whether h_*>0 for all d bigger or equal to 3.


Introduction
In the present work, we investigate level-set percolation for the Gaussian free field on Z d , d ≥ 3. This problem has already received much attention in the past, see for instance [11], [2], and more recently [5], [14]. The long-range dependence of the model makes this problem particularly interesting, but also harder to analyze. Here, we prove the existence of a non-trivial critical level for all d ≥ 3, and the positivity of this critical level when d is large enough. Some of our methods are inspired by the recent progresses in the study of the percolative properties of random interlacements, where a similar long-range dependence occurs, see for instance [18], [19], [22], [25].
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. Our main object of study is the Gaussian free field on Z d , with canonical law P on R Z d 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 (1.1). Note in particular the presence of strong correlations, see (1.9). For any level h ∈ R, we introduce the (random) subset of Z d sometimes called excursion set (above level h). We are interested in the event that the origin lies in an infinite cluster of E ≥h ϕ , which we denote by {0 ≥h ←→ ∞}, and ask for which values of h this event occurs with positive probability. Since (with the convention inf ∅ = ∞). A non-trivial phase transition is then said to occur if h * is finite. It is known that h * (d) ≥ 0 for all d ≥ 3 and that h * (3) < ∞ (see [2], Corollary 2 and Theorem 3, respectively; see also the concluding Remark 5.1 in [2] to understand why the proof does not easily generalize to all d ≥ 3). It is also known that when d ≥ 4, for large h, there is no directed percolation inside E ≥h ϕ , see [5], p. 281 (note that this reference studies the percolative properties of the excursion sets of |ϕ| in place of ϕ).
It is not intuitively obvious why h * should be finite, for it seems a priori conceivable that infinite clusters of E ≥h ϕ could exist for all h > 0 due to the strong nature of the correlations. We show in Corollary 2.7 that this does not occur and that (0. 5) h * (d) < ∞, for all d ≥ 3.
In fact, we prove a stronger result in Theorem 2.6. We define a second critical parameter  (2.8) for the precise definition), where the 2 n boxes are indexed by the "leaves" of a dyadic tree of depth n. The key to proving (0.11) is to provide a suitable induction step relating p n+1 (h n+1 ) to p n (h n ), for all n ≥ 0, where the increase in parameter h n → h n+1 allows to dominate the interactions ("sprinkling"). This appears in Proposition 2.2. The resulting estimates are fine enough to imply not only that h ∞ ≥ h * , but even the stretched exponential decay of the connectivity function of E ≥h∞ ϕ , thus yielding (0.8). The proof of (0.9) then only requires a small refinement of this argument. Note that the strategy we have just described is precisely the one used in [19] for the proof of a similar theorem in the context of random interlacements. We actually also provide a generalization of Proposition 2.2, which is of independent interest, but goes beyond what is directly needed here, see Proposition 2.2'. It has a similar spirit to the main renormalization step leading to the decoupling inequalities for random interlacements in [22], see Remark 2.3.
We now comment on the proof of (0.10), which has two main ingredients. The first ingredient is a suitable decomposition of the field ϕ restricted to the subspace Z 3 into the sum of two independent Gaussian fields. The first field has independent components and the second field only acts as a "small" perturbation when d becomes large, see Lemmas 3.1 and 3.2 below. The second ingredient is a Peierls-type argument, which actually enables us to deduce a stronger result than (0.10). Namely, we show in Theorem 3.3 that one can find a level h 0 > 0 and a positive integer L 0 , such that for large d and all h ≤ h 0 , the excursion set E ≥h ϕ already percolates in the two-dimensional slab (0.12) As already pointed out, some of our proofs employ strategies similar to those developed in the study of the percolative properties of the vacant set left by random interlacements, see [19], [22]. This is not a mere coincidence, as we now explain. Continuous-time random interlacements on Z d , d ≥ 3, correspond to a certain Poisson point process of doubly infinite trajectories modulo time-shift, governed by a probability P , with a non-negative parameter u playing the role of a multiplicative factor of the intensity measure pertaining to this Poisson point process (the bigger u, the more trajectories "fall" on Z d ), see [23], [21]. This Poisson gas of doubly infinite trajectories (modulo time-shift) induces a random field of occupation times (L x,u ) x∈Z d (so that the interlacement at level u coincides with {x ∈ Z d ; L x,u > 0}, whereas the vacant set at level u equals {x ∈ Z d ; L x,u = 0}). This field is closely linked to the Gaussian free field, as the following isomorphism theorem from [23] shows: (0.13) L x,u + 1 2 ϕ 2 x x∈Z d , under P ⊗ P, has the same law as It is tempting to use this identity as a transfer mechanism, and we hope to return to this point elsewhere.
We conclude this introduction by describing the organization of this article. In Section 1, we introduce some notation and review some known results concerning simple random walk on Z d and the Gaussian free field. Section 2 is devoted to proving that excursion sets at a high level do not percolate. The main results are Theorem 2.6 and Corollary 2.7. The positivity of the critical level in high dimension and the percolation of excursion sets at low positive level in large enough two-dimensional slabs is established in Theorem 3.3 of Section 3.
One final 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 , . . . are defined at the place they first occur within the text and remain fixed from then on until the end of the article. In Sections 1 and 2, constants will implicitly depend on the dimension d. In Section 3 however, constants will be purely numerical (and independent of d). Throughout the entire article, dependence of constants on additional parameters will appear in the notation.

Notation and some useful facts
In this section, we introduce some notation to be used in the sequel, and review some known results concerning both simple random walk and the Gaussian free field.
We denote by N = {0, 1, 2, . . . } the set of natural numbers, and by Z = {. . . , −1, 0, 1, . . . } 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, and denote by [x] the integer part of x, for any x ≥ 0. We consider the lattice Z d , and tacitly assume throughout that d ≥ 3. On Z d , we respectively denote by | · | and | · | ∞ the Euclidean and ℓ ∞ -norms. Moreover, for any x ∈ Z d and r ≥ 0, we let B(x, r) = {y ∈ Z d ; |y − x| ∞ ≤ r} and S(x, r) = {y ∈ Z d ; |y − x| ∞ = r} stand for the the ℓ ∞ -ball and ℓ ∞ -sphere of radius r centered at x. Given K and U subsets of Z d , K c = Z d \ K stands for the complement of K in Z d , |K| for the cardinality of K, K ⊂⊂ Z d means that |K| < ∞, and d(K, U ) = inf{|x − y| ∞ ; x ∈ K, y ∈ U } denotes the ℓ ∞ -distance between K and U . If K = {x}, we simply write d(x, U ). Finally, we define the inner boundary of K to be the set ∂ i K = {x ∈ K; ∃y ∈ K c , |y − x| = 1}, and the outer boundary of K as ∂K = ∂ i (K c ). We also introduce the diameter of any subset K ⊂ Z d , diam(K), as its We endow Z d with the nearest-neighbor graph structure, i.e. the edge-set consists of all pairs of sites {x, y}, x, y ∈ Z d , such that |x − y| = 1. A (nearest-neighbor) path is any sequence of vertices γ = (x i ) 0≤i≤n , where n ≥ 0 and x i ∈ Z d for all 0 ≤ i ≤ n, satisfying |x i − x i−1 | = 1 for all 1 ≤ i ≤ n. Moreover, two lattice sites x, y will be called * -nearest neighbors if |x − y| ∞ = 1. A * -path is defined accordingly. Thus, any site x ∈ Z d has 2d nearest neighbors and 3 d − 1 * -nearest neighbors. We now introduce the (discrete-time) simple random walk on Z d . To this end, we let W be the space of nearest-neighbor Z d -valued trajectories defined for non-negative times, and let W, (X n ) n≥0 , stand for the canonical σ-algebra and canonical process on W , respectively. Since d ≥ 3, the random walk is transient. Furthermore, 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. Given U ⊂ Z d , we further denote the entrance time in U by H U = inf{n ≥ 0; X n ∈ U }, the hitting time of U by H U = inf{n ≥ 1; X n ∈ U }, and the exit time from U by T U = inf{n ≥ 0; X n / ∈ U } = H U c . This allows us to define the Green function g U (·, ·) killed outside U as It vanishes if x / ∈ U or y / ∈ U . The relation between g and g U for any U ⊂ Z d is the following (we let K = U c ): The proof of (1.3) is a mere application of the strong Markov property (at time H K ).
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 (1.4) and (1.5) that the capacity is subadditive, i.e.
Moreover, the entrance probability in K may be expressed in terms of e K (·) (see for example [20], Theorem 25.1, p. 300) as from which, together with classical bounds on the Green function (c.f. (1.9) below), one easily obtains (see [19], Section 1 for a derivation) the following useful bound for the capacity of a box: We next review some useful asymptotics of g(·). Given two functions f 1 , Proof. For (1.9), see [10], Theorem 1.5.4, for (1.10), see [15], pp. 246-247. In order to prove (1.11), we assume that d ≥ 6 and define π : = π • X n , for all n ≥ 0, is a "lazy" walk on Z d−3 starting at the origin. Clearly, refers to the first return to 0 for the walk Y . Hence, We now turn to the Gaussian free field on Z d , as defined in (0.1). Given any subset K ⊂ Z d , we frequently write ϕ K to denote the family (ϕ x ) x∈K . For arbitrary a ∈ R and K ⊂⊂ Z d , we also use the shorthand {ϕ | K > a} for the event {min{ϕ x ; x ∈ K} > a} and similarly {ϕ | K < a} instead of {max{ϕ x ; x ∈ K} < a}. Next, we introduce certain crossing events for the Gaussian free field. To this end, we first consider the space Ω = {0, 1} Z d endowed with its canonical σ-algebra and define, for arbitrary disjoint subsets K, K ′ ⊂ Z d , the event (subset of Ω) For any level h ∈ R, we write Φ h for the measurable map from R Z d into Ω which sends ϕ ∈ R Z d to 1{ϕ x ≥ h} x∈Z d ∈ Ω, and define (a measurable subset of R Z d endowed with its canonical σ-algebra F), which is the event that K and K ′ are connected by a (nearest-neighbor) path in E ≥h ϕ , c.f. (0.2). Denoting by Q h the image of P under Φ h , i.e. the law of 1{ϕ x ≥ h} x∈Z d on Ω, we have that . We proceed with a classical fact concerning conditional distributions for the Gaussian free field on Z d . We could not find a precise reference in the literature, and include a proof for the Reader's convenience. We first define, for U ⊂ Z d , the law P U on R Z d of the centered Gaussian field with covariance with g U (·, ·) given by (1.2). In particular, ϕ x = 0, P U -almost-surely, whenever x ∈ K = U c . We then have where µ x is the σ(ϕ x ; x ∈ K)-measurable map defined as Then, under P, Proof. Note that for all x ∈ K, ϕ x = 0 (since µ x = ϕ x for x ∈ K, by (1.16)) and that for all x ∈ K, ϕ x = 0, P U -almost surely. Hence, it suffices to consider ( ϕ x ) x∈U . We first show independence. By (1.16), ( ϕ x ) x∈U , (ϕ y ) y∈K , are centered and jointly Gaussian. Moreover, they are uncorrelated, since for x ∈ U , y ∈ K, Thus, ( ϕ x ) x∈U , (ϕ y ) y∈K , are independent. To conclude the proof of Lemma 1.2, it suffices to show that where F U stands for the canonical σ-algebra on R U . Furthermore, choosing some ordering (x i ) i≥0 of U , by Dynkin's Lemma, it suffices to assume that A has the form ..,xn} , for some n ≥ 0 and A x i ∈ B(R), i = 0, . . . , n.
We fix some A of the form (1.19), and consider a subset V such that K ∪{x 0 , . . . , x n } ⊆ V ⊂⊂ Z d (we will soon let V increase to Z d ). We let P V x , x ∈ V , denote the law of simple random walk on V starting at x killed when exiting V (its Green function corresponds to g V (·, ·)), and define ϕ V x for x ∈ V as in (1.15) but with P V x replacing P x in the definition (1.16) of µ x . It then follows from Proposition 2.3 in [24] (an analogue of the present lemma for finite graphs) that with P V , P V \K as defined in (1.14) and , hence by dominated convergence that both sides of (1.20) converge towards the respective sides of (1.18), thus completing the proof.
Remark 1.3. Lemma 1.2 yields a choice of regular conditional distributions for (ϕ x ) x∈Z d conditioned on the variables (ϕ x ) x∈K , which is tailored to our future purposes. Namely, P-almost surely, where µ x , x ∈ Z d is given by (1.16), P does not act on (µ x ) x∈Z d , and ( ϕ x ) x∈Z d is a centered Gaussian field under P, with ϕ x = 0, P-almost surely for x ∈ K. Lemma 1.2 also provides the covariance structure of this field (namely g U (·, ·), with U = K c ), but its precise form will be of no importance in what follows. Note that conditioning on (ϕ x ) x∈K produces the (random) shift µ x , which is linear in the variables ϕ y , y ∈ K.
The explicit form of the conditional distributions in (1.21) readily yields the following result, which can be viewed as a consequence of the FKG-inequality for the free field (see for example [6], Chapter 4).
where the left-hand side is defined by the version of the conditional expectation in (1.21).
Intuitively, augmenting the field can only favor the occurrence of A, an increasing event.
Proof. On the event ϕ | K ≥ α , we have, for µ x , x ∈ Z d , as defined in (1.16), with equality instead on the event ϕ | K = α . Since A is increasing, this yields, with a slight abuse of notation, Integrating both sides with respect to the probability measure ν(·) def.
This completes the proof of Lemma 1.4.
We now introduce the canonical shift τ z on , for all A ∈ F (the canonical σ-algebra on R Z d ), by translation invariance of g(·, ·) (see below (1.1)), and has the following mixing property: , for all A, B ∈ F (one first verifies (1.24) for A, B depending on finitely many coordinates with the help of (1.9) and the general case follows by approximation, see [3], pp.157-158). The following lemma gives a 0-1 law for the probability of existence of an infinite cluster in E ≥h ϕ , the excursion set above level h ∈ R, c.f. (0.2).
ϕ contains an infinite cluster ], for arbitrary h ∈ R. One then has the following dichotomy: Proof. This follows by ergodicity, which is itself a consequence of the mixing property (1.24). Remark 1.6. When Ψ(h) = 1, in particular in the supercritical regime h < h * , the infinite cluster in E ≥h ϕ is P-almost surely unique. This follows by Theorem 12.2 in [9] (Burton-Keane theorem), because the field 1{ϕ x ≥ h} x∈Z d is translation invariant and has the finite energy property (see [9], Definition 12.1).

Non-trivial phase transition
The main goal of this section is the proof of Theorem 2.6 below, which roughly states that h * * (d) (and hence h * (d), c.f. Corollary 2.7) is finite for all d ≥ 3, and that the connectivity function of E ≥h ϕ , c.f. (0.2), has stretched exponential decay for arbitrary h > h * * . The proof involves a certain renormalization scheme akin to the one developed in [19] and [22] in the context of random interlacements. This scheme will be used to derive recursive estimates for the probabilities of certain crossing events, c.f. Proposition 2.2, which can subsequently be propagated inductively, c.f. Proposition 2.4. The proper initialization of this induction requires a careful choice of the parameters occurring in the renormalization scheme. The resulting bounds constitute the main tool for the proof of the central Theorem 2.6. In addition, an extension of Proposition 2.2 can be found in Remark 2.3 2).
We begin by defining on the lattice Z d a sequence of length scales where L 0 ≥ 1 and l 0 ≥ 100 are both assumed to be integers and will be specified below. Hence, L 0 represents the finest scale and L 1 < L 2 < . . . correspond to increasingly coarse scales. We further introduce renormalized lattices and note that L k ⊇ L n for all 0 ≤ k ≤ n. To each x ∈ L n , we attach the boxes and L ≥ 1 (not to be confused with B(x, L)). Moreover, we let so that {B n,x ; x ∈ L n } defines a partition of Z d into boxes of side length L n for all n ≥ 0, and B n,x , x ∈ L n , is simply the union of B n,x and its * -neighboring boxes at level n. Moreover, for n ≥ 1 and x ∈ L n , B n,x is the disjoint union of the l d 0 boxes {B n−1,y ; y ∈ B n,x ∩ L n−1 } at level n − 1 it contains. We also introduce the indexing sets and given (n, x) ∈ I n , n ≥ 1, we consider the sets of labels Note that for any two indices (n − 1, At last, given x ∈ L n , n ≥ 0, we introduce Λ n,x , a family of subsets T of 0≤k≤n I k (soon to be thought of as binary trees) defined as Λ n,x = T ⊂ n k=0 I k ; T ∩ I n = (n, x) and every (k, y) ∈ T ∩ I k , 0 < k ≤ n, has two 'descendants' (k − 1, y i (k, y)) ∈ H i (k, y), i = 1, 2, such Hence, any T ∈ Λ n,x can naturally be identified as a binary tree having root (n, x) ∈ I n and depth n. Moreover, the following bound on the cardinality of Λ n,x is easily obtained, where c 0 ≥ 1 is a suitable constant. We now consider the Gaussian free field ϕ = (ϕ x ) x∈Z d on Z d defined in (0.1) and introduce the crossing events (c.f. (1.13) for the notation) Three properties of the events A h n,x will play a crucial role in what follows: n,x is increasing (in ϕ) (see the discussion below (1.13)), (2.11) 2). Next, we provide a lemma which separates the combinatorial complexity of the number of crossings in A h n,x from probabilistic estimates, using Λ n,x as introduced in (2.7). This separation will be key in obtaining estimates fine enough to yield the desired stretched exponential decay. Albeit being completely analogous to Lemma 2.1 in [21], we repeat its proof, for it comprises an essential geometric observation concerning the events A h n,x .
Proof. We use induction on n to show that for all (n, x) ∈ I n , from which (2.13) immediately follows. When n = 0, (2.14) is trivial. Assume it holds for all (n − 1, y) ∈ I n−1 . For any (n, x) ∈ I n , a path in E ≥h ϕ starting in B n,x and ending in ∂ i B n,x must first cross the box B n−1,y 1 for some (n − 1, y 1 ) ∈ H 1 (n, x), and subsequently B n−1,y 2 for some (n − 1, y 2 ) ∈ H 2 (n, x) before reaching ∂ i B n,x , c.f. Figure 1 below. Thus, Upon applying the induction hypothesis to A h n−1,y 1 and A h n−1,y 2 separately, the claim (2.14) follows.
Before proceeding, we remark that the events A h T , with h ∈ R and T ∈ Λ n,x for some (n, x) ∈ I n , n ≥ 0, defined in (2.13) inherit certain properties from the events A h 0,y , (0, y) ∈ T ∩ I 0 . Namely, it follows from (2.11) and (2.12) that (2.15) A h T is an increasing event (in ϕ), and that, for any two levels h, h ′ ∈ R, Further, given any n ≥ 0, (n, x) ∈ I n , and T ∈ Λ n,x , we define the set Hence, K T is the disjoint union of 2 n boxes of side length 3L 0 each, and K T ⊂ B n,x . It then immediately follows from the definition of A h T in (2.13) that Finally, upon introducing We now derive the aforementioned "recursive bounds" for the probabilities p n (h n ), c.f. (2.24) below, along a suitable increasing sequence (h n ) n≥0 (one-step renormalization). These estimates will be key in proving Theorem 2.6 below.
There exist positive constants c 1 and c 2 such that, defining then, given any positive sequence (β n ) n≥0 satisfying and any increasing, real-valued sequence (h n ) n≥0 satisfying The main idea of the proof is to "decouple" the event A hn T ′ ∩ A hn T ′′ , where T ′ and T ′′ are the (binary) subtrees at level n of some given tree T ∈ Λ n+1,x , x ∈ L n+1 , using the increase in parameter h n → h n+1 to dominate the interactions ("sprinkling").
Proof. We let n ≥ 0, consider some m = (n+1, x) ∈ I n+1 and some tree T ∈ Λ m . We decompose where (n, y i (m)), i = 1, 2 are the two descendants of m in T and that is T n,y i (m) is the (sub-)tree consisting of all descendants of (n, y i (m)) in T . Thus, the union in (2.25) is over disjoint sets. Note in particular that T n,y i (m) ∈ Λ n,y i (m) . By construction, the subsets K T n,y i (m) ⊂ B n,y i (m) , for i = 1, 2, see (2.17), satisfy K T n,y 1 (m) ∩ K T n,y 2 (m) = ∅. For sake of clarity, and since m and T will be fixed throughout the proof, we abbreviate In order to estimate the probability of the event (2.13), we introduce a parameter α > 0 and write where max K 1 ϕ = max{ϕ x ; x ∈ K 1 } and the second line follows because A h T 1 ∩ max K 1 ϕ ≤ α is measurable with respect to σ(ϕ K 1 ), c.f. (2.18). We begin by focusing on the conditional probability P[A h T 2 |ϕ K 1 ] in (2.28). Using (1.21) and (2.18) applied to A h T 2 , and with a slight abuse of notation, we find which is deterministic and linear in α. Moreover, we can bound m x (α) as follows. By virtue of (1.7), P Since K 1 consists of 2 n disjoint boxes of side length 3L 0 , c.f. (2.27) and (2.17), its capacity can be bounded, using (1.6) and (1.8), as cap(K 1 ) ≤ c2 n L d−2 0 . By (1.9), (2.1) and the observation that |x − y| ≥ c ′ L n+1 whenever x ∈ K 1 and y ∈ K 2 , it follows that where c 1 is the constant from (2.23), and the factor (2g(0)) −1/2 is kept for later convenience.

Remark 2.3.
1) The bound (2.38), which we have derived using an elementary argument, also follows from a more general (and stronger) estimate. One knows that (in fact, this holds for a large class of Gaussian fields, c.f. [1], Theorem 1.3.3), denotes the smallest number of closed balls of radius ε in this metric covering K, and C is a universal constant. Clearly, N (ε) ≤ |K| for all ε ≥ 0. Moreover, sup x,y,∈Z d d(x, y) ≤ 2g(0) by virtue of (0.1). Inserting this into (2.43) immediately yields the bound (2.38).
2) We mention a generalization of Proposition 2.2, which is of independent interest, but will not be needed in what follows. Consider integers L 0 ≥ 1, l 0 ≥ 100, and a collection D x , x ∈ L 0 , of events in Ω (= {0, 1} Z d , see above (1.12)), such that where Y z , z ∈ Z d , stand for the canonical coordinates on Ω.
Given h ∈ R, n ≥ 0, x ∈ L n , and T ∈ Λ n,x , we replace A h T in (2.13) by (see below (1.12) for the notation) If instead, Proof. The arguments employed in the proof of Proposition 2.2 yield the first statement (with D x , x ∈ L 0 , increasing events). To derive the second statement (when D x , x ∈ L 0 , are decreasing events), one argues as follows. One introduces the inversion ι : Ω −→ Ω such that Y z • ι = 1 − Y z , for all z ∈ Z d , and the collection of "flipped" events D x = ι −1 (D x ) = ι(D x ), x ∈ L 0 . One defines D h T as in (2.45) with D y , y ∈ L 0 , in place of D y , y ∈ L 0 . Now observe that (−ϕ x ) x∈Z d has the same law as (ϕ x ) x∈Z d under P, and that for any h ∈ R, 1{ϕ x < −h} x∈Z d has the same law Q h as 1{ϕ x ≥ h} x∈Z d under P. From this, we infer that for all h ∈ R, x ∈ L n , and T ∈ Λ n,x , (2.50) When (2.49) holds, the events D x , x ∈ L 0 , satisfy (2.47), and thanks to the identity (2.50), the second statement of Proposition 2.2' is reduced to the first statement. This concludes the proof of Proposition 2.2'.
3) There is an analogy between Proposition 2.2' and the main renormalization step Theorem 2.1 of [22], for the decoupling inequalities of random interlacements (see Theorem 2.6 of [22]). Note however, that unlike condition (2.7) of [22] (see also (2.70) in [22]), (2.22) and (2.23) tie in the finest scale L 0 to the sequence (h n ) n≥0 . This feature has to do with the role of the cut-off level α we introduce in (2.28) and the remainder term it produces.
We now return to Proposition 2.2 and aim at propagating the estimate (2.24) inductively. To this end, we first define, for all n ≥ 0, where K 0 > 0 is a certain parameter to be specified below in Proposition 2.4 and later in (2.59). Note in particular that condition (2.22) holds for this choice of (β n ) n≥0 . In the next proposition, we inductively derive bounds for p n (h n ), n ≥ 0 given any sequence (h n ) n≥0 satisfying the assumptions of Proposition 2.2, provided the induction can be initiated, see (2.52) below.

Remark 2.5.
Although we will not need this fact in what follows, let us point out that a straightforward adaptation of Proposition 2.4 holds in the context of Proposition 2.2'.
We will now state the main theorem of this section and prove it using Proposition 2.4. To this end, we select K 0 appearing in Proposition 2.4 as follows: Moreover, we will solely consider sequences (h n ) n≥0 with , for all n ≥ 0, so that condition (2.23) is satisfied. We recall that β n is given by (2.51), which now reads (2.61) β n = (log 2) 1/2 + c 2 log(2 n (3L 0 ) d ) 1/2 + 2 (n+1)/2 n 1/2 + (log(2c 0 l 2(d−1) 0 where we have substituted M (n, L 0 ) from (2.21) and K 0 from (2.59). Note that L 0 , l 0 and h 0 are the only parameters which remain to be selected. We finally proceed to the main Moreover, for all d ≥ 3 and h > h * * (d), there exist positive constants c(h), c ′ (h) and 0 < ρ = ρ(h, d) < 1 such that In particular, the connectivity function P 0 ≥h ←→ x of the excursion set above level h has stretched exponential decay, i.e. there exists c ′′ (h) > 0 such that and h * ≤ h * * readily follows. As for (2.66), it is an immediate consequence of Lemma 1.5.
Proof of Theorem 2.6. To prove (2.62), it suffices to construct an explicit levelh with 0 <h < ∞ such that P B(0, L) ≥h ←→ S(0, 2L) decays polynomially in L, as L → ∞. In fact, we will even show that P B(0, L) ≥h ←→ S(0, 2L) has stretched exponential decay. We begin by observing that the sequence (h n ) n≥0 defined in (2.60) has a finite limit h ∞ = lim n→∞ h n for every choice of L 0 , l 0 and h 0 . Indeed, β n as given by (2.61) satisfies β n ≤ c(L 0 , l 0 )2 n+1 for all n ≥ 0, hence we have for suitable c, c ′ > 0 and 0 < ρ < 1. To this end, we note that p 0 (h 0 ) defined in (2.20) is bounded by for all n ≥ 0.
Therefore, we find that for all n ≥ 0 and x ∈ L n , We now set ρ = log 2/ log l 0 , whence 2 n = l nρ 0 = (L n /L 0 ) ρ . Given L ≥ 1, we first assume there exists n ≥ 0 such that 2L n ≤ L < 2L n+1 . Then, since and the number of sets contributing to the union on the right-hand side is bounded by cl d−1 0 , (2.72) readily implies (2.70), and by adjusting c, c ′ , (2.70) will hold for L < 2L 0 as well. It follows thath ≥ h * * , which completes the proof of (2.62).
We now turn to the proof of (2.63). Let h be some level with h * * < h < ∞, and define h 0 = (see (2.20) for the definition of p 0 (·)). Moreover, we let so that l 0 ≥ 100 is an integer, as required. From (2.61), it is then easy to see, using (2.74), that β n ≤ c(h) log(l 0 )2 n+1 , for all n ≥ 0. Hence, the limit h ∞ = lim n→∞ h n of the increasing sequence (h n ) n≥0 defined in (2.60) satisfies Thus, (2.74) and (2.75) imply that h ∞ ≤ h whenever L 0 ≥ c(h). Moreover, where the last inequality follows by (2.73). We thus select L 0 = c(h) so that both (2.76) and h ∞ ≤ h hold. Since condition (2.52) is satisfied, Proposition 2.4 yields for all n ≥ 0, from which point on one may argue in the same manner as for the proof of (2.62) to infer (2.72) (with h in place ofh) and subsequently deduce (2.63). In particular, this involves defining ρ = log 2/ log l 0 , which depends on h (and d) through l 0 . The stretched exponential bound (2.64) for the connectivity function of E ≥h ϕ is an immediate corollary of (2.63), since P 0 ≥h ←→ x ≤ P B(0, L) ≥h ←→ S(0, 2L) ≤ c(h)e −c ′′ (h)|x| ρ whenever 2L ≤ |x| ∞ < 2(L + 1).

Remark 2.8.
An important open question is whether h * equals h * * or not. In case the two differ, the decay of Hence P 0 ≥h ←→ S(0, L) decays to zero with L, but with an at most polynomial decay for h ∈ (h * , h * * ). However, for h > h * * , P 0 ≥h ←→ S(0, L) has a stretched exponential decay in L, by (2.63).

Positivity of h * in high dimension
The main goal of this section is the proof of Theorem 3.3 below, which roughly states that in high dimension, for small but positive h, the excursion set E ≥h ϕ contains an infinite cluster with probability 1. We will prove the stronger statement that percolation already occurs in a two-dimensional slab The proof essentially relies on two main ingredients. The first ingredient is a suitable decomposition, for large d, of the free field ϕ restricted to Z 3 (viewed as a subset of Z d ), into the sum of two independent Gaussian fields ψ and ξ (c.f. (3.12) and (3.13) below for their precise definition). The field ψ is i.i.d. and the dominant part, while ξ only acts as a "perturbation." The key step towards this decomposition appears in Lemma 3.1.
The second ingredient is a Peierls-type argument, which comprises several steps: first, the sublattice Z 3 is partitioned into blocks of side length L 0 , which are declared "good" if certain events defined separately for ξ and ψ occur simultaneously. Roughly speaking, these events are chosen in a way that suitable excursion sets of the dominant field ψ percolate well and the perturbative part ξ doesn't spoil this percolation (see (3.26), (3.27) and (3.28) for precise definitions). Moreover, * -connected components of bad blocks are shown to have small probability, see Lemma 3.5. This ensures that the usual method of Peierls contours is applicable, which in turn allows the conclusion that an infinite cluster of E ≥h ϕ exists within the above-mentioned slab with positive probability (and with probability 1 by ergodicity). We note that ξ doesn't have finite-range dependence, which renders impractical the use of certain well-known stochastic domination theorems (see for example [13], [17]).
Moreover, given any level h ∈ R, we define the (random) sets and E ≥h ξ , E <h ξ in analogous manner. A crucial point is that the field ξ acts only as a small perturbation when d is large, which is entailed in (3.3) and more quantitatively in the following lemma, the proof of which uses ideas developed in [16] (see in particular Theorem 2.4 therein).
This completes the proof of Lemma 3.2.
We are now ready to introduce a central quantity before proceeding to the main theorem of this section. Namely, we define, for all d ≥ 6, h ∈ R and positive integers L 0 , contains an infinite cluster .
We first construct certain families of "good" events for the Gaussian fields ψ 0 and ξ. To this end, we define boxes B We begin with ψ 0 and define C x (ω), for any x ∈ L and ω ∈ {0, 1} Z 3 , to be the open cluster in B x (2L 0 ) containing the most vertices. If several such clusters exist, we choose one according to some given prescription (say using lexicographic order). For arbitrary x ∈ L , we introduce the event F x (in the canonical σ-algebra of {0, 1} Z 3 ) as follows: given some ω ∈ {0, 1} Z 3 , we let ω ∈ F x if and only if i) C x (ω) is a crossing cluster for B (3) x (2L 0 ) in the first two axes-directions, i.e. for i = 1, 2, there exists an open path γ i in C x (ω) with endvertices y (i) , z (i) satisfying y i (i) = x i and x (2L 0 ) having the property diam(C) ≥ L 0 − 1.
Having introduced F x for all n ≥ 0 and x ∈ L , we define, for h > 0, the measurable map for all h > 0 and x ∈ L .
We now turn to the ("good") events for ξ, set for all x ∈ L , h > 0, , for all x ∈ L , by translation invariance of g ′ (·, ·), see (3.2). With "good" events for ψ 0 and ξ at hand (see (3.26) and (3.27)), for any h > 0, we define a vertex x ∈ L to be h-good if the event occurs (under P ψ 0 ⊗ P ξ ), and h-bad otherwise. A sequence γ = (x i ) 0≤i≤m , m ∈ N ∪ {∞}, in L such that |x i+1 − x i | = L 0 for all 0 ≤ i < m will be called a nearest-neighbor path in L (we will refer to m as the length of the path). Similarly, a * -nearest-neighbor path in L is any sequence in L subject to the weaker condition |x i+1 − x i | ∞ = L 0 for all 0 ≤ i < m.
A crucial property is that percolation of h-good sites in L implies percolation of E ≥2h ψ 0 ∩E ≥−h ξ (in the slab Z 2 × [0, 2L 0 )), which we state as a Lemma 3.4.
Let h > 0, γ = (x i ) 0≤i≤m , m ∈ N ∪ {∞}, be a nearest-neighbor path in L and assume that all vertices x i , 0 ≤ i ≤ m, are h-good. Then, the corresponding clusters C x i (ψ 0 ) (pertaining to the events F 2h , which are all connected within the set m i=0 B (3) x i (2L 0 ).
In particular, Lemma 3.4 implies that if (x i ) i≥0 is an infinite nearest-neighbor path of hgood vertices in L which is unbounded, then the set ∞ i=0 B (3) Proof of Lemma 3.4. It suffices to consider the case m = 2. The general case then follows by induction on m. Thus, let x 1 , x 2 ∈ L , |x 1 − x 2 | = L 0 , be both h-good. The following holds for i = 1, 2: by definition of F 2h It remains to show the clusters C x 1 and C x 2 are connected within B (3) x 2 (2L 0 ) by condition ii) in the above definition of the events F x . This concludes the proof of Lemma 3.4.
We now carry on with the proof of Theorem 3.3, and select the parameters h 0 , L 0 and d 0 . First note that p site c (Z 3 ), the critical level for Bernoulli site percolation on Z 3 , satisfies p site c (Z 3 ) < 1/2 (see [4], Theorem 4.1). We may thus choose h 0 > 0 such that which means that the Bernoulli site percolation model on Z 3 associated to choosing sites x ∈ Z 3 where ψ 0 x ≥ 2h 0 , is supercritical. By the site-percolation version of Theorem 7.61 in [8], we thus obtain that (3.30) lim Moreover, (see [8], Theorem 7.65), there exists a non-decreasing function π : 2h 0 x } x∈L stochastically dominates a family of independent Bernoulli random variables indexed by L , with success parameter π(δ).
Having fixed h 0 and L 0 , we choose, in the notation of Lemma We now proceed to the last step of the proof, which mainly encompasses a Peierls argument. To this end, we introduce, for x ∈ L and N a multiple of L 0 , the event H h 0 (x, N ) that x is connected to {y ∈ L ; |y − x| ∞ = N }, the restriction to L of the ℓ ∞ -sphere of radius N centered around x, by a * -path of h 0 -bad vertices in L . In the following lemma, we show that this event has small probability.
This completes the proof of Lemma 3.5.
We can now conclude the proof of Theorem 3.3. For arbitrary d ≥ d 0 , we consider the event that the set L ∩[0, 2L 0 ) 2 is surrounded by a * -circuit (i.e. a self-avoiding * -path except for the end point which coincides with the starting point) of h 0 -bad vertices in L . Considering a point of this circuit on the first axis with largest coordinate, we see that the probability of this event is bounded by If this event does not occur, then by planar duality (c.f. [8], Section 11.2), there exists an infinite self-avoiding nearest-neighbor path γ = (x i ) i≥0 of h 0 -good vertices in L . Lemma 3.4 (see in particular the remark following it) then implies that the set contains an infinite cluster. By (3.41), this event happens with positive probability, so that It then follows by (3.24) and ergodicity (see Lemma 1.5) that the value of Ψ (slab) (d, h 0 , L 0 ) is in fact one. This proves (3.20) and thus concludes the proof of Theorem 3.3.
1) We show in Theorem 3.3 that E ≥h ϕ percolates in a two-dimensional slab for small but positive h when d is sufficiently large. However, it should be underlined that E ≥h ϕ ∩Z 2 does not percolate for any h ≥ 0, as we now briefly explain. Indeed, the conditions of Theorem 14.3 in [9] (which is itself a variant of [7] when the finite energy condition holds) are met for the law of 1{ϕ x ≥ 0} x∈Z 2 on {0, 1} Z 2 under P (in particular, positive correlations, see Definition 14.1 in [9], follow from the FKG-inequality). Hence E ≥0 ϕ ∩ Z 2 and its complement in Z 2 cannot both have infinite clusters. Observe that 1{ϕ x ≥ 0} x∈Z 2 and 1{ϕ x < 0} x∈Z 2 have the same law under P, by symmetry. If E ≥0 ϕ ∩ Z 2 percolated (with probability one, by ergodicity), the same would hold true as well for E <0 ϕ ∩ Z 2 , leading to a contradiction. Therefore, E ≥0 ϕ ∩ Z 2 does not percolate.
2) With Lemma 3.2 at hand, one may immediately apply the criterion of Molchanov and Stepanov (c.f. [16], Theorem 2.1) to infer that E ≥−h ξ (c.f. (3.15) for notation) percolates strongly in Z 3 when h 2 d is sufficiently large, i.e. not only does E ≥−h ξ percolate, but in addition E <−h ξ doesn't, P ξ -almost surely. However, we note that E ≥h ψ does not percolate strongly, since for all h > 0, we have p c = p site c (Z 3 ) < p(h, σ(d)) = P ψ [ψ 0 ≥ h] < 1/2, hence in particular p(h, σ(d)) ∈ (p c , 1 − p c ), where both E ≥h ψ and its complement possess an infinite cluster with positive probability (in fact with probability one). 3