Level-set percolation of the Gaussian free field on regular graphs I: Regular trees

We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $d\geq 3$. Denoting by $h_\star$ the critical value, we obtain the following results: for $h>h_\star$ we derive estimates on conditional exponential moments of the size of a fixed connected component of the level set above level $h$; for $h<h_\star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_\star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_\star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].


Introduction
In this article we investigate the Gaussian free field on d-regular trees with d ≥ 3. We focus in particular on the level sets and their behaviour in connection with level-set percolation. The goal is to obtain a good description of the nature of the level sets for levels away from the critical value of level-set percolation.
Level-set percolation of the Gaussian free field is a significant representative of a percolation model with long-range dependencies and it has attracted attention for a long time, dating back to [MS83], [LS86] and [BLM87]. More recent developments can be found for instance in [RS13], [PR15], [Szn15], [DPR18b], [DPR18a], [Nit18] and [CN18]. The particular case of Gaussian free field on regular trees was studied before in [Szn16] and [Szn19], and on general transient trees in [AS18]. Compared to the present article, the emphasis in these three papers is put on a different aspect of the Gaussian free field, namely its connection with the model of random interlacements.
Studying level-set percolation of the Gaussian free field on regular trees specifically is of intrinsic interest. The case of the regular tree comes along with strong tools based on the structure and symmetry of the graph. These allow for often very exact computations which potentially lead to especially explicit, though not at all trivial, results. They also make it one of the most promising setups for understanding level-set percolation of the Gaussian free field near criticality.
Besides the fact that the results obtained in this article are interesting in their own right, our prime motivation comes from a concrete application: in the accompanying paper [AČ19] we prove a phase transition in the behaviour of the level sets of the zeroaverage Gaussian free field on a certain class of finite d-regular graphs that are locally (almost) tree-like. This class includes d-regular expanders of large girth and typical realisations of random d-regular graphs. In a certain sense, the Gaussian free field on the d-regular tree provides the local picture of the zero-average Gaussian free field on these finite graphs, and its detailed understanding developed in the present article is a key ingredient for [AČ19].
We now describe our results more precisely. Let d ≥ 3 and denote by T d the infinite d-regular tree. On T d we consider the Gaussian free field with law P T d on R T d and canonical coordinate process (ϕ T d (x)) x∈T d so that, under P T d , (ϕ T d (x)) x∈T d is a centred Gaussian field on T d with covariance for all x, y ∈ T d , where g T d (·, ·) is the Green function of simple random walk on T d (see (1.3)). (0.1) Our main interest lies in investigating properties of the level sets of ϕ T d , i.e. of E ≥h In particular, we are interested in the connected component of E ≥h ϕ T d containing a fixed vertex o ∈ T d (called root) and denoted by (0.2) With this notation at hand we can define the critical value of level-set percolation of the Gaussian free field via We point out that there is no explicit formula for h , even though we consider the Gaussian free field on a regular tree. However, the special structure of the underlying graph allows for a crucial spectral characterisation of the critical value, as derived in [Szn16]. We recall it in details in Section 1.2. Very roughly, in this spectral characterisation one associates to any level h ∈ R a self-adjoint, non-negative operator L h on L 2 (R, B(R), ν), where ν is a certain centred Gaussian measure. The operator L h is naturally linked to the distribution of ϕ T d at a vertex conditioned on the value of ϕ T d at a neighbouring vertex and truncated below level h (see (1.12) and below it). One then considers the operator norms (λ h ) h∈R of the operators (L h ) h∈R and finds that (see [Szn16], Section 3) the map h → λ h is a decreasing homeomorphism from R to (0, d − 1) and h is the unique value in R such that λ h = 1. (0.4) Additionally, for h ∈ R one has that λ h is a simple eigenvalue of L h which is associated to a unique, non-negative eigenfunction χ h with unit L 2 -norm, vanishing on (−∞, h) and positive elsewhere. So far one very important aspect of the eigenfunctions (χ h ) h∈R was unknown, namely the precise understanding of their asymptotic behaviour. Some care is applied in [Szn16] to circumvent this lack of control (see Remark 3.4 and Remark 4.4 therein).
As a first result in this paper, we close this gap and we obtain matching upper and lower bounds on the eigenfunctions (χ h ) h∈R . In essence, we show in Proposition 2.1 that for every h ∈ R there exist c h , c h > 0 such that 0 < c h a 1−log d−1 (λ h ) ≤ χ h (a) ≤ c h a 1−log d−1 (λ h ) for all a ∈ [h, ∞) (0.5) (see also Remark 2.2 (i)). Presumably, such exact bounds might be helpful when tackling level-set percolation questions of ϕ T d near the critical value h . In this paper we will use the upper bound to show the exponential growth of |C h o | for h < h (see (0.10)). We also obtain another result related to the spectral characterisation of level-set percolation of ϕ T d . It concerns the non-negative martingale (M ≥h k ) k≥0 for h ∈ R on which the proof of the spectral characterisation of h in [Szn16] heavily relies and in which the eigenfunction χ h and the associated eigenvalue λ h appear (see (1.16) for the definition). We show in the present paper that for all h ∈ R \ {h } the probability of a non-vanishing martingale limit is equal to the 'forward percolation probability' (see where T + d ⊆ T d is the 'forward tree', that is, the subtree of T d containing the root o and in which each vertex except for the root o has d neighbours and the root o has d − 1 neighbours (the precise definition is given below (1.1)).
We moreover investigate the continuity properties in h of percolation probabilities like on the left hand side of (0.6) and as a third result we show in Theorem 4.1 that the percolation probability η(h) : We then turn to C h o and we obtain rather precise estimates of its cardinality in both the subcritical (h > h ) and supercritical (h < h ) phase.
We show that if h > h , then there is some δ > 0 such that for all γ > 0 we can find c h,γ , c h,γ > 0 satisfying (see Theorem 5.1) In particular, this will imply that |C h o ∩ T + d | has exponential moments. Incidentally, it also implies conditional exponential-tail estimates of |C h o | of the form with a control of the dependence of the constant c h,a > 0 on the value a = ϕ T d (o) of the field at the root (see Remark 5.2).
Finally, for h < h we prove that the number of vertices connected over distance k above level h to the root o ∈ T d grows exponentially in k with positive probability. This can be shown by using our first result (0.5) in combination with (0.6). More precisely, with S T d (o, k) denoting the sphere of radius k ≥ 0 around o in T d , we prove that (see Theorem 4.3) We remind that here λ h > 1 is the eigenvalue from (0.4).
As explained earlier, we will see in the accompanying paper [AČ19] that the Gaussian free field ϕ T d on T d in essence plays the role of the local picture of the zero-average Gaussian free field on a specific class of finite d-regular graphs that are locally (almost) tree-like. By exploiting this feature, we establish in [AČ19] a phase transition for levelset percolation of the zero-average Gaussian free field on the finite graphs which is characterised by the critical value h on the infinite tree. Roughly, the strategy of [AČ19] is to use the local picture to transfer the problem from the finite graphs to T d and then to use the new results developed in the present article. Specifically, we apply the estimate (0.8) in the proof of the subcritical phase ([AČ19], Theorem 3.1) and the two results (0.7) and (0.10) in the proof of the supercritical phase ([AČ19], Theorem 4.1). As an aside, let us mention that for the application in [AČ19] it is crucial that the exponent γ on the right hand side of (0.8) can be chosen strictly smaller than 2.
A similar approach as explained in the previous paragraph was carried out in [ČTW11] to describe a phase transition for the vacant set of simple random walk on the same class of finite graphs as considered in the accompanying paper [AČ19]. As shown in [ČTW11], the local picture in that case is given by the vacant set of random interlacements on T d . Thanks to the detailed understanding of random interlacements in the infinite model, the phase transition in the finite model can be established. An advantage of random interlacements on a tree is that the connected components of its vacant set can be described rather easily. Indeed, as observed in [Tei09], they are distributed as Galton-Watson trees with a binomial offspring distribution. Thus, properties like (0.9) or (0.10) are classical.
In contrast, the connected components of the level sets of ϕ T d , which play the corresponding role in our setup, are not Galton-Watson trees. The situation is more complicated and obtaining results like (0.9) and (0.10) is not straightforward. Instead, by the domain Markov property of the Gaussian free field, one can view C h o for h ∈ R as a certain multi-type branching process with an uncountable type space. Some of the results in this paper are similar to classical results about branching processes, though to our knowledge they are not covered by the literature. We would like to stress that our arguments rely on the special structure of the Gaussian free field on regular trees. Let us also mention that, despite the connection between the Gaussian free field and random interlacements via isomorphism theorems, we are not aware of any technique allowing to transfer the results of [Tei09] and [ČTW11] directly to our context. The structure of the article is as follows. In Section 1 we collect the main part of the notation and some known results about the Gaussian free field on T d . In particular, in Section 1.2 we recall the spectral description of the critical value h obtained in [Szn16]. In Section 2 we derive asymptotic bounds on the eigenfunctions appearing in the spectral description of h . In Section 3 we give a recursive equation for the conditional non-percolation probability. Subsequently, we analyse the behaviour of the level sets of the Gaussian free field in the supercritical phase in Section 4. This includes the continuity of the percolation probability (Section 4.1) and the geometrical growth of level sets (Section 4.3). Finally, in Section 5 we investigate the subcritical phase and show that the cardinality of the connected component of the level set containing the root has exponential moments (and more).
Acknowledgements. The authors wish to express their gratitude to A.-S. Sznitman for suggesting the problem and for the valuable comments made at various stages of the project.

Notation and useful results
We start by introducing the notation and recalling known properties of the Green function and the Gaussian free field on T d . These include a recursive construction of ϕ T d (Section 1.1) and the spectral characterisation of h (Section 1.2).
As mentioned earlier, we consider for fixed d ≥ 3 the d-regular tree T d with root o. We endow T d with the usual graph distance d T d (·, ·). For any R ≥ 0 and x ∈ T d we let denote the ball and the sphere of radius R around x, respectively. For x, z ∈ T d a path from x to z is a sequence of vertices x = y 0 , y 1 , . . . , y m = z in T d for some m ≥ 0 such that y i and y i−1 are neighbours for all i = 1, . . . , m (if m ≥ 1). It is a geodesic path from x to z if it is the path of shortest length. (1.1) x for the canonical law of the simple random walk (X k ) k≥0 on T d starting at x as well as E T d x for the corresponding expectation. Given U ⊆ T d we write T U := inf{k ≥ 0 | X k / ∈ U } for the exit time from U and H U := inf{k ≥ 0 | X k ∈ U } for the entrance time in U (here we set inf ∅ := ∞). In the special case of U = {z} we use H z in place of H {z} . Recall that (see e.g. [Woe00], proof of Lemma 1.24) (1. 2) The Green function g T d (·, ·) of simple random walk on T d is given by (see [Woe00], Lemma 1.24, for the explicit computation) The functions g T d (·, ·) and g U T d (·, ·) are symmetric and finite, and g U T d (·, ·) vanishes whenever x / ∈ U or y / ∈ U . They are related by the identity which is an easy consequence of the strong Markov property of simple random walk at time T U . In the particular case of U := U x this implies that (by using (1.2), (1.3) and that (1.5) Recall from (0.1) that (ϕ T d (x)) x∈T d is the centred Gaussian field with covariance given by g T d (·, ·). It satisfies the following domain Markov property: for U ⊆ T d define the new field Then, (1.6) The proof of this fact follows by an easy computation of covariances and (1.4).

Recursive construction of the Gaussian free field
Property (1.6) can be applied to obtain a useful recursive representation of the Gaussian free field on T d that we introduce now. We point out that this description crucially relies on the special features of the Gaussian free field when considered on a (regular) tree.
Let x ∈ T d and let {x 1 , . . . , x I } be the neighbours of x not contained in the geodesic path from x to o. In particular, I = d if x = o and I = d − 1 otherwise. We set Since T d is a tree, it can be easily seen that • for any y ∈ U , one has X T U = x on {T U < ∞} under P T d y .
Hence (1.6) together with (1.2) and (1.5) yields that and variance d d−1 . Furthermore, they are independent of (ϕ T d (y)) y∈T d \U . (1.7) Let now (Y x ) x∈T d be a collection of independent centred Gaussian variables defined on some auxiliary probability space (Ω, Then, by applying (1.7) iteratively, we see that under P, the law of ( ϕ(x)) x∈T d is P T d , (1.9) so that (1.8) can serve as an alternative construction of the Gaussian free field on T d .
The recursive representation (1.8) has many useful consequences and it will be used repeatedly throughout the paper. In particular, it gives a representation of the conditional distribution of ϕ T d given with corresponding expectation E T d a . Moreover, if we let x 1 , . . . , x d denote the neighbours of the root o ∈ T d , then from (1.8) and (1.9) it follows that for every a ∈ R, under P T d a , the random fields (ϕ T d (y)) y∈Ux i for i = 1, . . . , d are independent. Furthermore, for any event (1.11) (In the equality in (1.11) we also use that the law of Due to (1.8) and (1.9), the Gaussian free field on T d can be related to a multitype branching process with type space R. Indeed, we can view every x ∈ S T d (o, k) as an individual in the k-th generation of the branching process with an attached type ϕ T d (x) ∈ R. In this perspective (1.7) can be rephrased as: . This point of view can easily be adapted to C h o from (0.2) as well, namely by considering the same multi-type branching process but instantly killing all individuals whose type does not exceed h. In other words, C h o can be viewed as a multi-type branching process with a barrier and the percolation of C h o corresponds to the non-extinction of this branching process. However, while some of the results in this paper are reminiscent of classical results about branching processes, we would like to emphasise that the proofs make heavy use of the special structure of the Gaussian free field on a regular tree. We are going to recall one of the special features in the next section.

Spectral characterisation of the critical value
We now recall the spectral characterisation of the critical value h from [Szn16], which is central for our paper. Note that our d-regular tree T d corresponds in the notation of [Szn16] to the ( d + 1)-regular tree T with d := d − 1. Moreover in [Szn16], in the definition of the Green function g T d (·, ·) on the tree, there is an extra normalising factor equal to the degree of the tree (see [Szn16], (3.1)). This explains the differences between the formulas to come and the formulas in [Szn16].
and E Y is the expectation with respect to Y . The operator L h is closely linked to the Gaussian free field and its level set above level h. Indeed, one has The following proposition summarises the known properties of the operators (L h ) h∈R and characterises the critical value h .
Proposition 1.1 ([Szn16], Propositions 3.1 and 3.3). For every h ∈ R the operator L h is self-adjoint, non-negative and its operator norm is a simple eigenvalue of L h . Moreover, there is a unique, non-negative eigenfunction χ h ∈ L 2 (ν) of L h , corresponding to the eigenvalue λ h , with χ h L 2 (ν) = 1. The function χ h is continuous and positive on [h, ∞), and vanishing on (−∞, h). Additionally, the map h → λ h is a decreasing homeomorphism from R to (0, d − 1) and h is the unique value in R such that λ h = 1.
In Proposition 2.1 in Section 2 we will give matching upper and lower bounds on the eigenfunctions χ h .
On the way, we recall the following hypercontractivity estimate which is a direct consequence of the hypercontractivity property of the Ornstein-Uhlenbeck semigroup (see [Szn16], (3.14)): for 1 < p < ∞ and q = ( (1.14) We will use the estimate (1.14), with its precise relation between the parameters p and q, several times. Especially, it will be applied to prove Proposition 3.6 which computes the Fréchet derivative of a certain operator. This will be a key ingredient for showing the existence of conditional exponential moments of |C h o | in the subcritical phase in Section 5.
then (see [Szn16], (3.31) and (3.35)) defines a non-negative martingale under P T d with respect to the filtration (F k ) k≥0 given by (1.17) In particular, M ≥h k converges P T d -almost surely to some M ≥h ∞ ≥ 0 as k → ∞ and (see [Szn16], proof of Proposition 3.3) Note that there is a direct relation between the probability of a non-vanishing martingale limit M ≥h ∞ and the forward percolation probability from (0.7). We only need to observe that . In Section 4 we will see that this inequality is actually an equality, at least when h = h (Proposition 4.2). As a last observation, note that by a union bound and the symmetry of T d we obtain that η + (h) ≤ η(h) ≤ d · η + (h) for the percolation probability η from (0.7). Hence by (0.3) one has η + (h) = 0 for h > h and η + (h) > 0 for h < h .
A final word on the convention followed concerning constants: by c, c , . . . we denote positive constants with values changing from place to place and which only depend on the dimension d. The dependence of constants on additional parameters appears in the notation.

Asymptotic behaviour of the eigenfunctions
The main result of this section are the matching bounds on the eigenfunctions (χ h ) h∈R from Proposition 1.1 collected in Proposition 2.1 below (corresponding to (0.5)). The upper bound will be used later to show that connected components of supercritical level sets grow exponentially with positive probability (Theorem 4.3 in Section 4). The corresponding lower bound is not used further but it is included for completeness.
Remark 2.2. (i) From Proposition 1.1 recall that χ h is continuous and strictly positive on [h, ∞). Therefore, by adjusting the constant c, Proposition 2.1 implies (0.5).
(ii) By Proposition 1.1 one has λ h ∈ (0, d − 1). Hence, the exponent . By iteratively using (1.12) and the fact that χ h is the eigenfunction of L h with eigenvalue λ h , we obtain for every a ∈ R and k ≥ 1 where the expectation on the right hand side of (2.3) is taken with respect to Y 1 , . . . , Y k . Note that for any k ≥ 1 the random variable Hence, if we denote by f µ,τ 2 the density of the normal distribution N (µ, τ 2 ) and a k : where in ( * ) we apply the Cauchy-Schwarz inequality. Note that for all k ≥ 1 where we use σ 2 1 ≤ σ 2 k ≤ σ 2 (see (2.4)). By combining (2.3) with (2.5) and (2.6) we obtain for any a ∈ R and k ≥ 1 which concludes the proof of part (i).

Define the intervals
which form a non-disjoint decomposition of [h, ∞). (To see that all left boundaries of the intervals are larger than h use the assumption on K as well as d ≥ 3.) The intervals are such that a d − 1 − (k + 1) ∈ I k−1 and a d − 1 + (k + 1) ∈ I k−1 for a ∈ I k , k ≥ 1. (2.10) Indeed, for k ≥ 1 and d ≥ 3 one has 4 k+2 d−1 + (k + 1) ≤ 4(k + 1) and hence for every a ∈ I k a d−1 + (k + 1) ≤ (d − 1) K+k + 4 k+2 d−1 + (k + 1) ≤ (d − 1) K+k + 4(k + 1), By using once more that χ h is the eigenfunction of L h , we have for every a ∈ I k with k ≥ 1 that The repeated application of (2.12) implies that By the exponential Markov inequality with c ∈ (0, 1) independent of k ≥ 1.
The inequality (2.2) is now an easy consequence of (2.13) and (2.14). First note that χ h (a) ≥ inf b∈I 0 χ h (b) > 0 for a ∈ [h, (d − 1) K+1 ) since χ h is continuous and strictly positive on [h, ∞) by Proposition 1.1, and In particular a ∈ I k(a) and therefore from (2.13) and (2.14) we obtain that This shows (2.2) and concludes the proof of Proposition 2.1.

Recursive equation for the non-percolation probability
In this section we adopt the perspective of multi-type branching processes and show that a certain function (see (3.1)) closely related to the forward percolation probability from (1.19) is the unique solution to a recursive equation (Theorem 3.1). This fact will be used to derive the results on the supercritical behaviour of the level sets of ϕ T d in Section 4, in particular the continuity of the percolation probability and its equality with the probability of a non-vanishing martingale limit (Theorem 4.1 and Proposition 4.2). At the end of the section we compute the Fréchet derivative of the operator involved in the recursive equation (Proposition 3.6). This will be an important ingredient to estimate exponential moments of |C h o | in Section 5 (Theorem 5.1). The proofs in this section partially follow the lines from multi-type branching processes (see e.g. [Har63], Chapter III). However, a lot depends on the special structure of the Gaussian free field on regular trees.
We introduce for every h ∈ R the conditional forward extinction probability (see (1.10) for notation) The function q h is closely related to the value of the forward percolation probability from (1.19) at h. Indeed, since the distribution ν above (1.12) is the distribution of In particular, this shows by the comment at the end of Section 1 that q h is identically 1 if h > h and is not identically 1 if h < h . Now, recall the space L 2 (ν) defined in (1.12) and let us define for every h ∈ R the (non-linear) operator R h on L 2 (ν) through: for f ∈ L 2 (ν) and a ∈ R, where Y ∼ N (0, d d−1 ) as in (1.12). To see that indeed R h f ∈ L 2 (ν), abbreviatef (a) : for a ∈ R and apply the hypercontractivity estimate (1.14) with p = 2 and q = (d − 1) 2 + 1 to find that f d−1 . We are actually only interested in the operator R h for h ∈ R on the subset By definition we directly have R h : S h → S h . In Theorem 3.1 we prove that q h is essentially the unique solution in S h to the equation From the multi-type branching process perspective, the operator R h can be used to write recurrence relations for generating functionals related to C h o , cf. for example [Har63], Section III.7. In particular, by using the notation from (1.10), (1.15) and by applying (1.7), we see that where the empty product is interpreted as being equal to 1. This identity can be extended: define iteratively Then one can prove by induction on k ≥ 0 and using (1.7) that We come to the main result of this section. The proof of the theorem is broken into several steps stated as Lemmas 3.2-3.5. The first one is a classical observation from the theory of multi-type branching processes.
Lemma 3.2. Let h ∈ R. The function q h satisfies q h ∈ S h and solves the equation Proof. The fact that q h ∈ S h is clear. To prove the second statement, denote S + T d (o, 1) =: {x 1 , . . . , x d−1 } and recall the notation from (1.1). Then, for every a ∈ R, completing the proof.
Next, we give various necessary properties of solutions to R h f = f . Proof. For the continuity and positivity we note that for a ≥ h one can write The right hand side is continuous in a by the dominated convergence theorem and it is also positive since f is non-negative and equal to 1 on (−∞, h). We will now show that one even has sup [h,∞) f (a) < 1. Consider the intervals I k ⊆ [h, ∞), k ≥ 0, from (2.9). Since f < 1 and f is continuous on [h, ∞) and I 0 is compact, we have ∆ := max{ 1 9 , sup a∈I 0 f (a)} < 1. If we show by induction on k ≥ 0 that sup a∈I 0 ∪..
This shows that sup a∈I k+1 f (a) ≤ ∆, which together with the induction hypothesis implies sup a∈I 0 ∪..
Hence lim a→∞ f (a) = 0 because f is non-negative.
The third step of the proof of Theorem 4.1 is the following statement of 'transience'.
Lemma 3.4. For K ≥ 1 and Λ ≥ h let (see (1.15) for the notation) Then for every a ∈ R, K ≥ 1 and Λ ≥ h one has Proof. Observe that the events B k := A K,Λ k ∩ n≥k+1 (A K,Λ n ) c for k ≥ 0 are disjoint. Furthermore, denoting S + T d (o, k + 1) ∩ U y =: {y 1 , . . . , y d−1 } for y ∈ Z h k and recalling the definition of F k from (1.17), it holds that for a ∈ R and k ≥ 0 Thus for a ∈ R we have The claim then follows by the Borel-Cantelli lemma.
The next lemma proves Theorem 3.1. Before that, we introduce for every h ∈ R the functions It can be easily seen that q k h ∈ S h for k ≥ 0 and 1 (−∞,h) (a) = q 0 h (a) ≤ q 1 h (a) ≤ q 2 h (a) ≤ . . . for a ∈ R. In particular, lim k→∞ q k h (a) = q h (a) for all a ∈ R by (3.1). In addition, applying (3.4) to the function f = 1 (−∞,h) implies that Proof. From Lemma 3.2 we know that q h ∈ S h and R h q h = q h . The same is of course true for the constant 1 function. We first claim that every solution in 0 by (3.3) and (3.8). By letting k tend to infinity we find f ≥ q h , proving the claim. In particular, if q h ≡ 1 (e.g. when h > h , see below (3.2)), then we have f ≡ 1 and thus R h f = f has a unique solution. Now assume that q h ≡ 1 (e.g. when h < h , see below (3.2)) and that f ≡ 1 is a solution to R h f = f . We claim that f = q h . As we have already shown f ≥ q h , it remains to prove f ≤ q h . To see this, observe that by Lemma 3.3 we know that δ := sup a∈[h,∞) f (a) ∈ (0, 1). Let m ≥ 0 be such that δ := sup a∈[h,∞) f (a) ∈ 1 2 m+1 , 1 2 m . Then for a ∈ R and k ≥ 0 one has Note that for the events appearing on the right hand side of (3.9) one has with K n := log 1/δ (2 n+1 ) and Λ n := sup{a ∈ R | f (a) ≥ 1 2 n+1 }. We observe that for n ≥ m it holds K n ≥ 1 since then 1/δ ≤ 2 m+1 ≤ 2 n+1 . Moreover, h ≤ Λ n < ∞ since f ∈ S h (so f = 1 on (−∞, h)) and lim a→∞ f (a) = 0 by Lemma 3.3. As a consequence, Lemma 3.4 and Fatou's lemma imply lim k→∞ P T d a [A Kn,Λn k ] = 0. Therefore, by using the dominated convergence theorem, for a ∈ R one has f (a) This implies that f = q h , completing the proof.
As last result of Section 3 we compute the Fréchet derivative of the operators (R h ) h∈R defined in (3.3). This technical result is one of the main ingredients for proving the existence of exponential moments of |C h o | in the subcritical phase (Section 5). Incidentally, let us mention that its proof is based on the hypercontractivity estimate (1.14) and that the precise relation between p and q in the estimate is vital (for p = 2).
To compute the Fréchet derivative of R h note that for f, g ∈ L 2 (ν) one has (3.14) where is the function defined in (3.11) and the operator E f h : L 2 (ν) → L 2 (ν) is given by Note that the map A f h is linear and also bounded since This is the case because implying (3.16). Thus A f h is the Fréchet derivative of R h at f . From (3.11) and (1.12) we directly see that A 1 h g = L h g if g ∈ L 2 (ν) vanishes on (−∞, h). It remains to show the second part of the statement. We have by (1.13). Moreover, the formula b d−2 − 1 = (b − 1)(1 + b + . . . + b d−3 ) and the triangle inequality imply and therefore A f by (3.13). All in all we showed Now let ε > 0 and take r > 0 such that (d−1)((1+r) d−2 −1) ≤ ε. Then if f −1 L 2 (ν) ≤ r, and hence also f L 2 (ν) ≤ 1 + r, we have A f h g L 2 (ν) ≤ (λ h + ε) g L 2 (ν) by (3.17). This concludes the proof.

Behaviour of the level sets in the supercritical phase
In this section we study the behaviour of the level sets of the Gaussian free field on T d for h < h . The main goal is to show that the percolation probabilities η and η + are continuous functions of the level h on the interval (−∞, h ) (Theorem 4.1, corresponding to (0.7)) and to prove that |C h o | grows exponentially in the radius with probability bounded away from zero when h < h (Theorem 4.3, corresponding to (0.10)). Along the way we also show the equivalence of the probabilities of forward percolation and of a non-vanishing martingale limit (Proposition 4.2, corresponding to (0.6)). These results essentially come as an application of Theorem 3.1 from Section 3. For this section recall the measure ν defined above (1.12).

Continuity of the percolation probability
In this section we analyse the continuity properties of the percolation probabilities η and η + , and show (0.7) in Theorem 4.1. Recall the functions q h , h ∈ R, introduced in (3.1) and their relation with η + reported in (3.2). Proof. Note that is a continuous function. Therefore by (4.1), η + is a decreasing limit of continuous functions and hence upper semicontinuous. As η + is a non-increasing function, it is thus left-continuous. With the obvious changes in (4.1) one can also show the left-continuity of η.
To show the right-continuity on R \ {h } observe first that if h > h , then η(h) = η + (h) = 0 by definition and the comment at the end of Section 1. So it remains to prove the right-continuity on (−∞, h ). Fix h < h and assume (h ) ≥0 is a sequence satisfying h ↓ h and h < h for all ≥ 0. We will show that lim →∞ η + (h ) = η + (h) and lim →∞ η(h ) = η(h). Observe that by (3.2) and the dominated convergence theorem the former follows from the claim lim →∞ q h (a) = q h (a) for a ∈ R \ {h}.
Define the two auxiliary functions q h and q h on R by (4.4) We will now apply Theorem 3.1 to show q h = q h . From this the claim (4.2) follows by (4.4) and (4.3).
Since h < h , one has q h ≡ 1 for all ≥ 0 (see below (3.2)). This implies q h ≡ 1 by (4.3) (being a decreasing limit) and hence also q h ≡ 1 by (4.4). Moreover if a < h, then a < h for all ≥ 0, which yields q h (a) = 1 for all ≥ 0. This implies q h (a) = 1 for a < h by (4.3) and (4.4). Thus q h ∈ S h . Finally, for a > h and ≥ 0 such that h ≤ a, one finds by Lemma 3.2 and (3.3) that q h (a) = E Y q h ( a d−1 + Y ) d−1 . If we let tend to infinity on both sides, then (4.3) and the dominated convergence theorem give q h (a) = E Y q h ( a d−1 + Y ) d−1 for all a > h. This together with (4.4) shows q h = R h q h .
By Theorem 3.1 we conclude that q h = q h . The proof is complete.

Percolation probability and non-triviality of the martingale limit
Recall the martingale (M ≥h k ) k≥0 from (1.16). We now apply Theorem 3.1 from Section 3 to show in Proposition 4.2 the equivalence (0.6) between the probability of non-vanishing of the martingale limit and η + (h). From the discussion at the end of Section 1.2 we already know that P T d M ≥h ∞ > 0 = η + (h) = 0 for h > h . We now prove that the first equality remains true also if h < h .

Proposition 4.2. One has
Proof. For every h ∈ R we introduce the function m h (a) := P T d a M ≥h ∞ = 0 for a ∈ R, where P T d a is the conditional probability defined in (1.10). We note that (1.10)

Geometrical growth of |C h o | in the supercritical phase
We come to the proof of (0.10), essentially that for h < h the number of vertices in T + d connected over distance k above level h to the root o ∈ T d grows exponentially in k with positive probability. Recall the notation from (1.15).
Theorem 4.3. Let h < h (so that λ h > 1, see Proposition 1.1). Then Proof. Note that one directly has lim sup Thus we only have to find a corresponding lower bound. By Fatou's lemma where we introduced the event (4.8) To deal with the event (A h k ) c note that by Proposition 2.1 and Remark 2.2 (here h < h ) one has χ h (a) ≤ c h a for a ≥ h and χ h (a) = 0 for a < h. Thus, for y ∈ T d and for k ≥ 0 , where in the last step we use the exponential Markov inequality. Hence, by a union bound, for k ≥ 0 From (4.7), (4.8) and (4.9) we have that lim inf k→∞ P T d |Z h k | ≥ λ k h k 2 ≥ η + (h) and the proof of Theorem 4.3 follows.

Exponential moments of |C h o | in the subcritical phase
This section proves that for every h > h the cardinality of the connected component of the level set of ϕ T d above level h in T + d containing the root o ∈ T d has exponential moments and actually, as a function of the value of ϕ T d (o), these exponential moments do not grow too fast. This is the content of Theorem 5.1 below (corresponding to (0.8)). In its proof we will use Proposition 3.6 from Section 3.
To state the result, we define for every h ∈ R and δ > 0 the (potentially infinite) function for a ∈ R, (5.1) where we use the notation for the conditional distribution of ϕ T d given ϕ T d (o) = a defined in (1.10). Observe that (recall ν from above (1.12)) (1.10) Note that if q h (a) < 1 for q h from (3.1) (in particular this is the case in the supercritical phase h < h for a ≥ h), then g h,δ (a) is infinite. The main goal of this section is to show that in the subcritical phase h > h there exists δ > 0 such that the right hand side of (5.2) is finite and such that g h,δ (a) does not grow too fast as a tends to infinity. Recall the space L 2 (ν) defined in (1.12).
Theorem 5.1. Let h > h . Then there exists δ h > 0 such that Moreover, g h,δ h equals 1 on (−∞, h) and g h,δ h (a) is finite for all a ∈ R. Finally, g h,δ h is continuous on [h, ∞) and for all γ > 0 there exist c h,γ > 0 and c h,γ > 0 such that In particular, (5.2) and (5. Remark 5.2. Note that (0.9) follows from Theorem 5.1 by the exponential Markov inequality. More precisely, for h < h and a ∈ R take say γ = 1 in (5.4). Then , thus (0.9).
The proof of Theorem 5.1 is split into various lemmas. The first one characterises g h,δ as a monotone limit of functions in L 2 (ν) which are obtained via iterated applications of a certain operator R h,δ (see (5.5)) to the constant 1 function (Lemma 5.3). The second lemma shows that for h > h we can choose δ > 0 such that the operator R h,δ is a strict contraction on a closed subset of L 2 (ν) including the constant 1 function (Lemma 5.4). This is an application of the technical Proposition 3.6 from Section 3. The combination of these two results will quickly lead to (5.3) via the Banach-Caccioppoli fixed-point theorem and to the other properties of g h,δ stated in Theorem 5.1 except for (5.4). This is the content of Corollary 5.5. It then remains to prove (5.4). We first show a weaker statement in which γ = 1 on the right hand side (Lemma 5.6). It implies a recursive bound on g h,δ (Lemma 5.7) which subsequently can be used to show the stronger statement (Lemma 5.8).
Let us introduce for every h ∈ R and δ > 0 the operator R h,δ on L 2 (ν) through: for f ∈ L 2 (ν) and a ∈ R, where, as usual, Y ∼ N (0, d d−1 ). By the same observations as below (3.3) one can check that indeed R h,δ f ∈ L 2 (ν) for f ∈ L 2 (ν). Note also that R h,δ for δ > 0 can be expressed in terms of the operator R h from (3.3) via R h,δ = (1 (−∞,h) + 1 [h,∞) · (1 + δ))R h . The role of R h,δ can be seen from the following lemma.
Since B h,r is a closed subset of L 2 (ν), it is a complete metric space.
Proof. Let δ > 0 and consider f, g ∈ L 2 (ν). By the relationship between R h,δ and R h explained below (5.5) one has (3.14) (5.9) Since h > h (and thus λ h < 1 by Proposition 1.1), we can choose ε h > 0 such that λ h + 2ε h < 1. Now on the one hand, by Proposition 3.6 there exists s h > 0 such that (5.10) Moreover, since R h,δ 1 = 1 + 1 [h,∞) δ by (5.5), one also has for f ∈ B h,r h that (5.11) With Lemma 5.3 and Lemma 5.4 at hand, we can readily show the first half of Theorem 5.1.
Proof. Consider δ h > 0, r h > 0 and f ∈ L 2 (ν) from Lemma 5.4. We start by showing that ν-almost everywhere f = g h,δ h and hence g h,δ h ∈ L 2 (ν). Note that by Lemma 5.4 one has 1 ≤ f and thus by (5.8) also g k h,δ h = R k+1 h,δ h 1 ≤ R k+1 h,δ h f = f for all k ≥ 0. By (5.7) and the monotone convergence theorem this shows Now since 1 ∈ B h,r h , we know from Lemma 5.4 that lim k→∞ R k h,δ h 1 − f L 2 (ν) = 0. Hence lim k→∞ g k h,δ h − f L 2 (ν) = 0 by (5.12) and so ν-almost everywhere f = g h,δ h by (5.7). By (5.1) and (1.10) it is obvious that g h,δ h equals 1 on (−∞, h). Now if we take k to infinity on both sides of the equation g k+1 h,δ h = R δ h ,h g k h,δ h from (5.8), we obtain g h,δ h = R h,δ h g h,δ h (5.13) by (5.7) and the monotone convergence theorem. The right hand side of (5.13) satisfies (R h,δ h g h,δ h )(a) < ∞ for all a ∈ R by (5.5) and (5.3). Hence g h,δ h (a) < ∞ for all a ∈ R. With (5.13) established, we can show the continuity of g h,δ h on [h, ∞) in the same way as the continuity of f in Lemma 3.3. Hence the proof of Corollary 5.5 is complete.
It remains to prove (5.4). In the next lemma we show a weaker statement by applying results from Corollary 5.5.
Lemma 5.6. Let h > h and consider the function g h,δ h ∈ L 2 (ν) from Corollary 5.5. For all ζ > 0 there exists c h,ζ > 0 such that g h,δ h (a) ≤ c h,ζ exp(ζa 2 ) for a ≥ h. (5.14) Proof. We will first show that g h,δ h ∈ L q (ν) for all q ≥ 1, which will then imply (5.14). Let us define We prove by induction that g h,δ h ∈ L p i (ν) for all i ≥ 0 by using the hypercontractivity estimate (1.14). For i = 0 we have p 0 = 2 and hence g h,δ h ∈ L p 0 (ν) as seen in Corollary 5.5. Now assume g h,δ h ∈ L p i (ν) for i ≥ 0. Since g h,δ h = R h,δ h g h,δ h by Corollary 5.5, it follows that to prove g h,δ h ∈ L p i+1 (ν) it is enough to showĝ d−1 h,δ h ∈ L p i+1 (ν), where we abbreviatedĝ h,δ h (a) := E Y g h,δ h ( a d−1 + Y ) . And indeed we have, using (5.15), (1.14) and the induction hypothesis, that ĝ d−1 h,δ h L p i+1 (ν) = ĝ h,δ h d−1 L p i (ν) < ∞. Next we show that the sequence (p i ) i≥0 diverges to infinity as i tends to infinity. To see this, note that r i := d−2 d−1 (d − 1) i + 1 d−1 + 1, i ≥ 0, solves the recursion for (p i ) i≥0 given in (5.15) and clearly r i i→∞ − −− → ∞. This implies that g h,δ h ∈ L q (ν) for all q ≥ 1 since we can take i ≥ 0 such that q < p i . Then g h,δ h ∈ L p i (ν) ⊆ L q (ν).