On the cost of the bubble set for random interlacements

The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{Z}}^{d}$\end{document}, d≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d \ge 3$\end{document}. We investigate the occurrence in a large box of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. The results significantly improve our findings in Sznitman (Probab. Math. Phys. 2–3:563–311, 2021). In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. A challenging difficulty revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplets for Bernoulli percolation or for the Ising model. This feature is reflected in the present context by the so-called bubble set, a possibly quite irregular random set. A pivotal progress in this work has to do with the improved construction of a coarse grained random set accounting for the cost of the bubble set. This construction heavily draws both on the method of enlargement of obstacles originally developed in the mid-nineties in the context of Brownian motion in a Poissonian potential in Sznitman (Ann. Probab. 25(3):1180–1209, 1997; Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998), and on the resonance sets recently introduced by Nitzschner and Sznitman in (J. Eur. Math. Soc. 22(8):2629–2672, 2020) and further developed in a discrete set-up by Chiarini and Nitzschner in (Commun. Math. Phys. 386(3):1685–1745, 2021).


Introduction
Both random interlacements and the Gaussian free field are models with long range dependence for which the Dirichlet energy plays an important role. The study of largely deviant events and their impact on the medium in the context of the percolation of the vacant set of random interlacements, or in the closely related level-set percolation of the Gaussian free field, has attracted much attention over the recent years. In particular, important progress has been achieved in the understanding of various instances of atypical disconnection of macroscopic bodies, see for instance [7], [8], [9], [16], [19], [22], [23], [29], [30], [31], [32], [34].
Our main focus in this article lies in the strongly percolative regime of the vacant set of random interlacements on Z d , d ≥ 3. Specifically, we investigate the occurrence in a large box centered at the origin of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. Our results significantly improve our findings in [34]. In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. As often the case, the derivation of the lower bound suggests a scheme on how to produce the largely deviant event, and a matching upper bound confers some degree of pertinence to this scheme. A challenging difficulty in the derivation of an upper bound for the present problem revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplet in the context of Bernoulli percolation or for the Ising model, see [5], [4]. In the present context this feature is reflected by the so-called bubble set. A pivotal progress in this work has to do with an improved construction of a coarse grained random set accounting for the cost of the bubble set. The bubble set is quite irregular and our construction heavily draws on a combination of ideas and techniques from the method of enlargement of obstacles in [26], [27], and from the resonance sets in [23], [9].
We now describe the results in more details. We let I u stand for the random interlacements at level u ≥ 0 in Z d , d ≥ 3, and V u = Z d I u for the corresponding vacant set. We refer to [6], [13], and Section 1 of [30] for background material. We are interested in the strong percolative regime of the vacant set, i.e. we assume that where u * and u respectively denote the critical level for the percolation and for the strong percolation of V u . When u < u * the infinite cluster C u ∞ of V u exists and is unique a.s., see [28], [25], [35], and informally u < u corresponds to the local presence and local uniqueness of the infinite cluster. We refer to (2.3) of [30] or (1.26) of [34] for the precise definition of u. It is expected (and presently the object of active research) that u * = u. The corresponding equality has been established in the closely related model of the level-set percolation of the Gaussian free field, see [15].
We write θ 0 for the percolation function ←→ / ∞} stands for the event that 0 does not belong to an infinite connected component of V a . The function θ 0 is known to be non-decreasing, left-continuous, identically equal to 1 on (u * , ∞) with a possible (although not expected from simulations) jump at u * . It is continuous elsewhere, see [35]. We also refer to [33] for the C 1 -property of θ 0 below a certain levelû, which is also expected to coincide with u * . Nevertheless, the behavior of θ 0 in the vicinity of u * is poorly understood so far, and the convexity of θ 0 below u * is presently unclear, see Remark 5.2 3). We writeθ for the function (0.3)θ(a) = θ 0 (a) 1{a < u} + 1{a ≥ u}, a ≥ 0, and denote by θ 0 the right-continuous modification of θ 0 . One has (0. 4) θ 0 ≤θ (with equality of the two functions if u and u * coincide).
We now describe the kind of excess disconnection event that we are interested in. Given N ≥ 1, we consider In addition, for r ≥ 0, we write S r = {x ∈ Z d ; x ∞ = r} for the set of points in Z d with sup-norm equal to r, and define (0.7) C u r = the connected component of S r in V u ∪ S r (so S r ⊆ C u r by convention).
Our focus lies in the set of points in D N that get disconnected by I u from S 2N , i.e. D N C u 2N . We also consider its subset D N C u N of points in the interior of D N disconnected by I u from S N . We are interested in their "excessive presence". Specifically, we consider (0. 8) ν ∈ [θ 0 (u), 1), and the excess events (where for F finite subset of Z d , F denotes the number of points in F ): An asymptotic lower bound for P[A 0 N ] was derived in (6.32) of [32]. Together with Theorem 2 of [33] it shows that for 0 < u < u * and ν ∈ [θ 0 (u), 1), with ⨏ D . . . the normalized integral 1 D ∫ D . . . , D = 2 d the Lebesgue measure of D, and D 1 (R d ) the space of locally integrable functions f on R d with finite Dirichlet energy that decay at infinity, i.e. such that { f > a} has finite Lebesgue measure for a > 0, see Chapter 8 of [20].
Our main result in this work is Theorem 4.1 that we describe further below. Its principal application is Theorem 5.1, which states that for 0 < u < u and ν ∈ [θ 0 (u), 1), (the existence of minimizers is shown as in Theorem 2 of [33]).
As often the case for large deviation asymptotics, the derivation of the lower bound involves a "scenario that produces the deviant event of interest". This is indeed the case in (0.10), which is proved through the change of probability method. In this light the quantity ( √ u + ϕ) 2 ( ⋅ N ) can heuristically be interpreted as the slowly varying level of tilted interlacements (see [19]) that appear in the derivation of the lower bound (see Section 4 and Remark 6.6 2) of [32]). The minimizers in (0.11) do not exceed the value √ u * − √ u, see Theorem 2 of [33]. The regions where the minimizers of (0.11) reach the value √ u * − √ u, if they exist, could reflect the presence of droplets secluded by the interlacements that might share the burden of creating an excess fraction of volume of disconnected points in D N . This occurrence, which does not happen when ν is small, see [34], but might take place when ν is close to 1, would exhibit some similarity to the Wulff droplet in the case of Bernoulli percolation or for the Ising model, see in particular Theorem 2.12 of [5], and also [4]. Additional information on the behavior of θ 0 near u * would be helpful in this matter. We also refer to Remark 5.2 3).
Theorem 5.1 that proves (0.12) is a significant improvement on the main Theorem 4.3 of [34]. It replaces the function θ * from [34] (defined similarly asθ with ( √ u + c 0 ( √ u − √ u)) 2 < u playing the role of u) by the smaller functionθ in the variational problem (0.13). This not only leads to a sharper asymptotic upper bound, but also if the equality u = u * holds, to the true exponential rate of decay. Informally, Theorem 5.1 replaces the dimension dependent constant c 0 ∈ (0, 1) from [34] by the value 1.
In this light Theorem 4.1 is the central result of the present article. It subsumes Theorem 3.1 of [34]. Theorem 4.1 pertains to the construction of a coarse grained random set C ω , which plays a crucial role in the assignment of a cost to the bubble set Bub, see (1.26). Following [34], the bubble set is obtained by paving D N with boxes B 1 of size L 1 ≃ N 2 d and retaining those B 1 -boxes that are not met "deep inside" by a random set U 1 , see (1.22). This random set U 1 , following [23], is obtained by an exploration starting (u, u) and should be thought of as being close to u), and have a "local level" below u (actually below β). The random set U 1 brings along a profusion of "highways" in the vacant set V u and permits to exit The bubble set is quite irregular and lacks inner depth. In Theorem 4.1 we obtain a pivotal improvement on the results of Section 3 of [34]. Given an arbitrary a ∈ (0, 1) we construct a random set C ω contained in [−4N, 4N] d . The set C ω can take at most exp{o(N d−2 )} possible shapes, further, when thickened in scale L 1 it has a small volume compared to D N , it is made of well-spaced B 0 -boxes that are (α, β, γ)-good with local level bigger or equal to β, and (see (4.5) v)): (0.15) the set of points in the bubble set Bub, where the equilibrium potential h Cω of C ω takes a value smaller than a has small volume relative to D N .
The crucial difference, when compared to Section 3 of [34], is that we can now choose a in (0.15) arbitrarily close to 1, whereas in [34] a was at most equal to the dimensional constant c 0 ∈ (0, 1) of [34]. We refer to the discussion below Theorem 4.1 for an outline of its proof. Quite interestingly the construction of the coarse grained random set C ω brings in full swing both the method of enlargement of obstacles, specifically the capacity and volume estimates from Chapter 4 §3 of [27], or Section 2 of [26], as well as the resonance sets of [23], [9]. We refer to Sections 2 and 3 for an implementation of these concepts in the present context.
Coming back to Theorem 5.1, its proof can be adapted to the case where one replaces C u 2N by C u mN with m > 2 integer in the definition of A N (hence leading to a larger event). However, the case where C u 2N is replaced by C u ∞ (the infinite cluster) remains open, see Remark 5.2 2). Further, Theorem 5.1 has a direct application to the simple random walk. Informally, the simple random walk corresponds to the singular limit u → 0 for random interlacements, see Section 7 of [29] or the end of Section 6 of [30]. In Corollary 5.3 we obtain an asymptotic upper bound on the exponential rate of decay of the probability that the trajectory of the simple random walk disconnects a positive fraction ν of sites of D N from S 2N . It is an open question whether a matching asymptotic lower bound holds as well, see Remark 5.4.
We should also mention that combined with [29] the methods developed here and in [34] ought to be pertinent to handle similar questions in the context of the level-set percolation of the Gaussian free field. Denoting by φ the Gaussian free field on Z d , d ≥ 3, and for h in R by C ≥h 2N the connected component of S 2N in {φ ≥ h} ∪ S 2N , one would now look at the points of D N that get disconnected from S 2N by the sub-level set {φ < h}, namely D N C ≥h 2N . In this model, the critical value h * for the percolation of {φ ≥ h} lies in (0, ∞), see [11], and the value h ≤ h * from [29] corresponding to the strong percolation of {φ ≥ h} when h < h, is known to coincide with h * by [15]. Thus for h < h * and where θ G 0 stands for the right-continuous modification of θ G 0 (which only possibly differs from θ G 0 at h * , see Lemma A.1 of [1], and is expected to coincide with θ G 0 ). We will now describe the organization of the article. Section 1 collects some notation and recalls various facts about the simple random walk, potential theory, and random interlacements. It also recalls some lemmas from [34], in particular Lemmas 1.1 and 1.2 which will be used in Chapter 4 in the construction of C ω (Lemma 1.2 is actually based on [2]). Section 2 develops capacity and volume estimates for certain rarefied boxes originating from the method of enlargement of obstacles, see Chap. 4 §3 of [27] or Section 2 of [26]. Section 3 contains an adaptation to our set-up of controls attached to the resonance sets developed in [23] and [9]. Section 4 is the main body of the article. It develops the construction of the coarse grained random set C ω , see Theorem 4.1. Then Section 5 builds on Theorem 4.1 and the results in Section 4 of [34]. Theorem 5.1 proves the main asymptotic upper bound corresponding to (0.12). The application to the simple random walk is contained in Corollary 5.3. The Appendix sketches the proof of Proposition 3.1, which pertains to the resonance sets considered here.
Finally, let us state our convention concerning constants. Throughout the article we denote by c,c, c ′ positive constants changing from place to place that simply depend on the dimension d. Numbered constants such as c 0 , c 1 , c 2 , . . . refer to the value corresponding to their first appearance in the text. The dependence on additional parameters appears in the notation.
Acknowledgements: The author wishes to thank Alessio Figalli for his very helpful comments about the minimizers of the variational problem in (0.11) and the issue of knowing whether they reach the maximum value √ u * − √ u for ν close to 1, see also Remark 5.2 3).

Notation, useful results, and random sets
In this section we introduce further notation. We also collect several facts concerning random walks, potential theory, and random interlacements. We introduce the length scales L 0 and L 1 see (1.9), (1.10), as well as the random sets U 1 , U 0 = Z d U 1 , see (1.22), (1.23), and the "bubble set" Bub, see (1.26). They play an important role in this article. We also recall Lemmas 1.1 and 1.2 from [34] (Lemma 1.2 is in essence due to [2]): they will enter the construction of the random set C ω in Section 4. We begin with some notation. We denote by N = {0, 1, 2, . . . } the set of non-negative integers and by N * = {1, 2, . . . } the set of positive integers. For (a n ) n≥1 and (b n ) n≥1 positive sequences, a n ≫ b n or b n = o(a n ) means that lim n b n a n = 0. We write ⋅ 1 , ⋅ , and ⋅ ∞ for the ℓ 1 -norm, the Euclidean norm, and the supremum norm on R d . Throughout we tacitly assume that d ≥ 3. Given x ∈ Z d and r ≥ 0, we let B(x, r) = {y ∈ Z d ; y − x ∞ ≤ r} stand for the closed ball of radius r around x in the supremum distance. Note that D N in (0.5) coincides with B(0, N). Given L ≥ 1 integer, we say that a subset B of Z d is a box of size L if B = x + Z d ∩ [0, L) d for an x in Z d (which actually is unique). We write x B for this x and refer to it as the base point of B. We sometimes write ∃y ∈ Z d A; y −x = 1} the boundary and the inner boundary of A. When f, g are functions when the sum is absolutely convergent. We also use the notation ⟨ρ, f ⟩ for the integral of a function f (on an arbitrary space) with respect to a measure ρ when this quantity is meaningful.
Concerning connectivity properties, we say that x, y in Z d are neighbors when y−x = 1 and we call π ∶ {0, . . . , n} → Z d a path when π(i) and π(i − 1) are neighbors for 1 ≤ i ≤ n. For A, B, U subsets of Z d we say that A and B are connected in U and write A U ←→ B when there is a path with values in U, which starts in A and ends in B. When no such path exists we say that A and B are not connected in U and we write A U ←→ / B. We then recall some notation concerning the continuous-time simple random walk. For U ⊆ Z d , we write Γ(U) for the set of right-continuous, piecewise constant functions from [0, ∞) to U ∪∂U with finitely many jumps on any finite interval that remain constant after their first visit to ∂U. We denote by (X t ) t≥0 the canonical process on Γ(U). For U ⊂⊂ Z d the space Γ(U) conveniently carries the law of certain excursions contained in the trajectories of the interlacements. We also view the law of P x of the continuous-time simple random walk on Z d with unit jump rate, starting at x ∈ Z d , as a measure on Γ(Z d ).
We write E x for the corresponding expectation. We denote by (F t ) t≥0 the canonical rightcontinuous filtration, and by (θ t ) t≥0 the canonical shift on Γ(Z d ). Given U ⊆ Z d , we write H U = inf{t ≥ 0; X t ∈ U} and T U = inf{t ≥ 0; X t ∉ U} for the respective entrance time in U and exit time from U. Further, we letH U stand for the hitting time of U, that is, the first time after the first jump of X . when X . enters U.
We write g(⋅, ⋅) for the Green function of the simple random walk and g U (⋅, ⋅) for the Green function of the walk killed upon leaving U(⊆ Z d ): Both g(⋅, ⋅) and g U (⋅, ⋅) are known to be finite and symmetric, and g U (⋅, ⋅) vanishes if one of its arguments does not belong to U. When f is a function on Z d such that in an analogous fashion with g U (⋅, ⋅) in place of g(⋅, ⋅).
Due to translation invariance g(x, y) = g(x − y, 0) and one knows that Theorem 1.5.4, p. 31 of [17]). We denote by c * the positive constant We will also use the fact that for a suitable dimension dependent constant c ∆ > 0 (1.4) for all R ≥ 1 and x, y ∈ B(0, R), g(x, y) ≥ g B(0,2R) (x, y) ≥ c ∆ g(x, y) (this follows for instance by adapting the proof of Theorem 4.26 a) on p. 121 of [3]).
Given A ⊂⊂ Z d , we write e A for the equilibrium measure of A, and cap(A) for its total mass, the capacity of A.
, for x ∈ Z d , and e A is supported by the inner boundary of A. Further, one knows that At the end of Section 4 we will use an identity generalizing (1.5) to the case of the simple random walk killed outside U ⊆ Z d and A finite subset of U. Then, one has h A, In the case of a box B = [0, L) d one knows (see for instance [17], p. 31) that as well as (see (2.16) on p. 53 of [17]) We will also need the constant We now introduce some length-scales. Apart from the macroscopic scale N that governs the size of the box D N in (0.5), two other length-scales play an important role: , and (1.9) We call B 0 -box (or sometimes L 0 -box) any box of the form We often write B 0 to refer to a generic box B 0,z , z ∈ L 0 (so z = x B 0 will be the base point of B 0 ). Likewise we call B 1 -box (or L 1 -box) any box of the form , (1.10)), and denote by B 1 a generic box B 0,z , z ∈ L 1 (so that z = x B 1 is the base point of B 1 ).
We recall two lemmas from [34]. They play an important role in the proof of the main Theorem 4.1, where the coarse grained random set C ω is constructed. The combination of isoperimetric controls and the first lemma will typically enable us to extract well-spaced (good) B 0 -boxes in the boundary of the random set U 1 (see below (1.23)), which carry substantial capacity, see (4.79), (4.81), (4.84). In the statement of the next lemma, and throughout the article, the terminology coordinate projection refers to any of the d canonical projections on the respective hyperplanes of points with vanishing i-th coordinate.
Lemma 1.1. Given K ≥ 100d, a ∈ (0, 1), then for large N, for any box B of size L ≥ L 1 and any set A union of B 0 -boxes contained in B such that for a coordinate projection π one has one can find a subsetÃ of A, which is a union of B 0 -boxes having base points with respective π-projections at mutual ⋅ ∞ -distance at least K L 0 (with K = 2K + 3), and such that The second lemma corresponds to Lemma 1.2 of [34], and is a direct application of Theorem 1.4 of [2]. It will be used at the end of the proof of Proposition 4.2 to extract small collections of B 0 -boxes with substantial capacity, see (4.97).
For K, N ≥ c 1 , whenÃ is a union of B 0 -boxes with base points that are at mutual ⋅ ∞distance at least KL 0 , then there exists a union of B 0 -boxes A ′ ⊆Ã such that We will now collect some notation and facts concerning random interlacements. We refer to [6], [13], and the end of Section 1 of [30] for more details. The random interlacements I u , u ≥ 0 and the corresponding vacant sets V u = Z d I u , u ≥ 0 are defined on a probability space denoted by (Ω, A, P). In essence, I u corresponds to the trace left on Z d by a certain Poisson point process of doubly infinite trajectories modulo time-shift that tend to infinity at positive and negative infinite times, with intensity proportional to u. As u grows V u becomes thinner and there is a critical value u * ∈ (0, ∞) such that for all u < u * , P-a.s. V u has a unique infinite component C u ∞ , and for u > u * all components of V u are finite, see [28], [25], [35], as well as the monographs [6], [13].
In this work we are mainly interested in the strongly percolative regime of V u that corresponds to (1.16) u < u, where we refer to (2.3) of [30] or (1.26) of [34] for the precise definition of u. Informally, (1.16) corresponds to a regime of local presence and uniqueness of the infinite cluster C u ∞ in V u . One knows by [14] that u > 0 and that u ≤ u * , see (2.4), (2.6) in [30]. The equality u = u * is expected but presently open. In the closely related model of the level-set percolation for the Gaussian free field, the corresponding equality has been shown in the recent work [15].
We now introduce some additional boxes related to the length scale L 0 , which take part in the definition of the important random set U 1 (see (1.40) of [34], and also (4.27) of [23], as well as (1.22) below). Throughout K implicitly satisfies We consider the boxes Given a box B 0 as above and the corresponding D 0 , we consider the successive excursions in the interlacements that go from D 0 to ∂U 0 (see (1.41) of [30]) and write (see (1.42) and (2.14) of [30]): N v (D 0 ) = the number of excursions from D 0 to ∂U 0 in the interlacements (1.19) trajectories with level at most v, for v ≥ 0.
the notion of an (α, β, γ)-good box B 0,z plays an important role in the definition of U 1 . We refer to (2.11) -(2.13) of [30], see also (1.38) of [34] for the precise definition. The details of the definition will not be important here. In essence, one looks at the (naturally ordered) excursions from D 0,z to ∂U 0,z in the trajectories of the interlacements. For an (α, β, γ)-good box B 0,z the complement of the first α cap(D 0,z ) excursions leaves in B 0,z at least one connected set with sup-norm diameter at least L 0 10, which is connected to any similar components in neighboring boxes of B 0,z via paths in D 0,z avoiding the first β cap(D 0,z ) excursions from D 0,z to ∂U 0,z . In addition, the first β cap(D 0,z ) excursions spend a substantial "local time" on the inner boundary of D 0,z of an amount at least γ cap(D 0,z ). We refer to B 0 -boxes that are not (α, β, γ)-good as (α, β, γ)-bad boxes.
We now fix a level u as in (0.1), that is Following (4.27) of [23] or (1.40) of [34], we introduce the important random set which are all, except maybe for the last one, (α, β, γ)-good and such that N u (D 0,z i ) < β cap(D 0,z i ).
We then define We use the notation ∂ B 0 U 1 to refer to the (random) collection of B 0 -boxes that are not contained in U 1 but are neighbor of a B 0 -box in U 1 . Note that when B 0 is (α, β, γ)-good and belongs to ∂ B 0 U 1 , then necessarily N u (D 0 ) ≥ β cap(D 0 ) (otherwise B 0 would belong to U 1 ). Although we will not need this fact here, let us mention that the random set U 1 provides paths in V u going from any Such paths necessarily meet the inner boundary S 2N of B(0, 2N), see below (1.40) of [34].
We then proceed with the definition of the bubble set. Given an L 1 -box B 1 , we denote by Deep B 1 the set (see (1.46) of [34]) which in essence is obtained by "peeling off" a shell of depth 3L 0 from the surface of B 1 , thus only keeping B 0 -boxes such that the corresponding D 0 is contained in B 1 .
One then defines the bubble set Figure 2.

Volume estimates for rarefied boxes
In this section we bring into play a notion of rarefied boxes. With the help of capacity and volume estimates originally developed in the context of the method of enlargement of obstacles, see Chapter 4 §3 of [27], or Theorem 2.1 of [26], we derive volume controls in Proposition 2.1. They play an important role in the proof of Theorem 4.1, when constructing the coarse grained random set C ω accounting for the bubble set. The volume estimates of Proposition 2.1 are applied in Section 4 to boxes of so-called Types b and B, see (4.13), (4.16). Informally, they correspond to certain nearly macroscopic boxes "at the boundary of U 0 " that intersect the bubble set. The estimates in the present section enable us to discard the so-called rarefied boxes of Types b and B in Section 4, see (4.37), (4.49). On the other hand, the substantial (i.e. non-rarefied) boxes of Types b and B fulfill a kind of Wiener criterion, which ensures that (α, β, γ)-good boxes of ∂ B 0 U 1 are "present on many scales", see (4.36), (4.48). As an aside, the notion of rarefied boxes that we consider in this section is substantially more refined than that which was used in Section 3 of [34].
We first introduce an "M-adic decomposition of Z d " where L 1 , see (1.10), corresponds to the smallest scale, and N roughly to the largest scale. More precisely, we consider a dyadic integer M > 4, solely depending on d, such that Having M a dyadic number will be convenient in the next section when discussing resonance sets (see also the Appendix).
As mentioned above, the smallest scale under consideration corresponds to L 1 and the largest scale corresponds to M ℓ N L 1 , where We view things from the point of view of the top scale and 0 ≤ ℓ ≤ ℓ N labels the depth with respect to the top scale. For such ℓ we set (hopefully there should be no confusion with the notation for random interlacements, where the level always appear as a superscript, see above (0.1)) I ℓ = the collection of M-adic boxes of depth ℓ, i.e. of boxes of the form The collections I ℓ , 0 ≤ ℓ ≤ ℓ N are naturally nested; I ℓ N corresponds to the collection of B 1 -boxes and I 0 to boxes of approximate size N. Given ℓ as above and B ∈ I ℓ , the "tower above B" stands for the collection of boxes We consider some depth as well as a set For the applications given in Section 4, the depth q will correspond top in (4.9) and F will correspond to ⋃ j∈G B ′ j in the context of Type b boxes, see (4.35), (4.13), and to ⋃ 1≤j≤J B ∆ ′ j in the context of Type B boxes, see (4.47), (4.16). We then introduce the dimension dependent constants Given a box B ∈ I q , we say that where the sum runs over the boxesB in the tower above B that are contained in B (1) (see (2.4) for notation).
The main object of this section is Proposition 2.1. Assume that q and F are as in (2.5), (2.6) and that Then, for any box ∆ ∈ I 0 , one has (2.10) Proof. The next lemma contains the key control that will then be iterated over scales. In the words of Remark 3.3 of p. 172 of [27], it reflects growth (with M 2 3 d +1 > 1) and saturation (corresponding to the truncation byη 0 ) in the evolution of the properly normalized capacities from one scale to the next.

Lemma 2.2. (under (2.9))
Consider 0 ≤ ℓ ≤ ℓ + 1 ≤ q and B ∈ I ℓ . Denote byB ⊆ B a generic subbox in I ℓ+1 and set (with c * as in (1.3)) Then one has Proof. Consider B ∈ I ℓ as above. For eachB ⊆ B,B ∈ I ℓ+1 , setting L = B 1 d as a shorthand notation, we first show the existence of Observe that by (2.9) and (2.11) To constructB ′ we note that: 14)) and we remove fromB ∩ F one point at a time, decreasing at each step the capacity by an amount at most 1 g(0, 0) (i.e. the capacity of a point in Z d ) until the first time when the resulting capacity is smaller or equal to η 0 L d−2 . The set thus obtained has now capacity at most ). So choosing this set aŝ B ′ , the claim (2.13) holds as well.
So, for eachB ⊆ B withB ∈ I ℓ+1 with pickB ′ ⊆B ∩ F satisfying (2.13). Next, we consider the measure (with hopefully obvious notation) which is supported by the set Then, for x ∈ S we denote by Σ 1 the sum over the boxesB containing x or neighbor (in the sup-norm sense) of the box containing x, and by Σ 2 the sum over the remaining boxeŝ B ⊆ B. We then have It follows that Gν(x) ≤ 3 d + 1 for each x ∈ S. Since ν is supported by S, we find that 14), (2.11), the claim (2.12) follows. This proves Lemma 2.2.
We now proceed with the proof of Proposition 2.1. We define the quantities We note that the Y B satisfy (with hopefully obvious notation) We can then apply Lemma 3.4, p. 173 of [27] and obtain as on p. 175 of the same reference the capacity estimate (where in the notation of [27], c 6 =ĉ, (and c ′ corresponds to c 6 c 8 c 9 in the notation of (3.38) on p. 175 of [27]). Further, one knows by (3.36) of [34] Since for all B ∈ I q , M dq B = ∆ , the claim (2.10) follows. This completes the proof of Proposition 2.1.
Remark 2.3. Incidentally let us mention thatĉ >η 0 > δ (in the notation of (1.8), (2.11), (2.7)). The second inequality is immediate. As for the first inequality, setting a n = cap([0, M n L 1 ) d ) (M n L 1 ) d−2 for n ≥ 0, and assuming N large enough so that (2.9) holds, Lemma 2.2 with F = Z d shows that a n+1 ≥ M 2 3 d +1 (a n ∧η 0 ) for all n ≥ 0. Since the sequence a n is bounded byĉ and M 2 (3 d + 1) > 1, this implies that for some n 0 ≥ 0 the inequality a n 0 ≤η 0 occurs so thatĉ ≥ a n 0 +1 ≥ M 2 3 d +1η 0 >η 0 , and this yields the first inequality. ◻ We close this section with some notation. We set so that R ℓ is the size of the boxes in I ℓ , for 0 ≤ ℓ ≤ ℓ N . Also we denote by

Resonance sets
In this section we introduce certain resonance sets, where on many well-separated scales a finite set U 0 and its complement U 1 (which up to translation play the role of U 0 and U 1 in (1.23), (1.22)) occupy a non-degenerate fraction of volume. The main Proposition 3.1 comes as an adaptation to our set-up of the results of [23] in the case of Brownian motion, and of [9] in the case of random walks among random conductances. It shows that when the simple random walk starts at a point where on many well-separated scales U 0 occupies at least half of the relative volume, see (3.11), then it enters the resonance set with high probability. Proposition 3.1 plays an important role in the proof of Theorem 4.1, specifically for the treatment of boxes of Type A (informally, they correspond to certain nearly macroscopic boxes "well within U 0 " that intersect the bubble set, see (4.17)). Proposition 3.1 provides us with a quantitative bound on the probability that the random walk starting on such a box avoids a certain set Res J , see (4.62), (4.56). The main steps of the proof of Proposition 3.1 are sketched in the Appendix.
We now introduce the set-up. We first consider an integer, which will parametrize the strength of the resonance as well as the integer which will control the separation between dyadic scales. We keep here similar notation as in [23] and [9]. In this section and in the Appendix, L does not refer to a spatial scale but to a separation between dyadic scales (hopefully no confusion should occur). Further, we have a separation between (dyadic) scales and a finite set A such that for some non-negative integer h (we will assume that U 0 occupies at least half of the volume of the balls B(0, 2 m L 1 ) for each m ∈ A). We also consider the enlarged set A * where intermediate scales have been added: The next Proposition 3.1 states a uniform control showing that it is hard for the simple random walk to avoid the resonance set in (3.7), when A is sufficiently large, and U 0 occupies more than half the volume of the sup-norm balls centered at its starting point with radius 2 m L 1 , m ∈ A. More precisely, we have There exists a doubly indexed non-negative sequence γ I,J , I, J ≥ 1, such that and a sequence of positive integers They have the property that for any I, J ≥ 1, when for any U 0 , U 1 as in (3.3), L ≥ L(J) as in (3.4), and A satisfying (3.5) for which one has the bound (see the beginning of Section 1 for notation) The proof of Proposition 3.1 is similar to the proofs in [23] and [9]. The main steps are sketched in the Appendix. One actually has a more quantitative statement than (3.8), see (A.43): (3.14) for each J ≥ 1, lim In the proof of Theorem 4.1 in the next section, we will apply Proposition 3.1 to the random sets U 0 , U 1 shifted at a point where on many well-separated scales 2 m L 1 , U 0 occupies at least half of the relative volume of the sup-norm ball of radius 2 m L 1 centered at this point.

Coarse graining of the the bubble set
This section contains Theorem 4.1, which is the central element of this article. It constructs a coarse grained object, namely a random set C ω of low complexity whose equilibrium potential is close to 1 on most of the bubble set Bub. This random set is made of (α, β, γ)-good B 0 -boxes for which the corresponding D 0 -boxes (see (1.18)) have a "local level" N u (D 0 ) cap(D 0 ) at least β. Its purpose is to quantify the cost induced by the bubble set (about its specific use we refer to the proof of Proposition 4.1 in [34]). The challenge in the construction of C ω lies in the fact that the bubble set may be very irregular with little depth apart from its constitutive grains of size L 1 . Theorem 4.1 constitutes an important improvement on Theorem 3.1 of [34]: loosely speaking, it shows that the c 0 of [34] can be chosen arbitrarily close to 1.
We now specify the set-up. We assume that (see (1.16)) Further, with c 1 as in Lemma 1.2 and c 2 (α, β, γ) as in Lemma 1.3, we assume that We recall that the asymptotically negligible bad event B N is defined in (1.24) and that K = 2K + 3. Here is the main result.
We refer to the discussion below Theorem 3.1 of [34] for an informal description of the use of the conditions in (4.5). Here, the crucial novelty is that a in v) can be chosen arbitrarily close to 1. In Section 5 this leads to the improved asymptotic upper bound on the probability P[A N ] of an excess of disconnected points in D N , see Theorem 5.1. If u and u * coincide (as expected), this asymptotic upper bound matches in principal order the asymptotic lower bound in (0.10), and thus yields the exponential rate of decay of P[A N ], see (0.14).
At this point it is perhaps helpful to provide an informal description of the proof of Theorem 4.1. Loosely speaking, one considers nearly macroscopic boxes B at depth p (i.e. in I p , see (2.3)) that meet the bubble set Bub, where p as chosen in (4.11), eventually only depends on d, a and ε, see (4.99). One classifies these boxes into Types b, B, and A, see (4.13), (4.16), and (4.17). When N is large, all B 1 -boxes contained in Bub are almost contained in U 0 , see (1.26), (1.23). The much bigger (nearly macroscopic) boxes B of Types b or B loosely speaking correspond to "boundary boxes of U 0 ", where in the case of Type b, U 1 occupies more than half the volume of B, see (4.13), and in the case of Type B something similar takes place at an intermediate scale not much bigger than that of B, see (4.16). The boxes B of Type A correspond to "inner boxes of U 0 ", where on quite a few well-separated intermediate scales above that of B, the random set U 0 occupies more than half of the volume.
The boxes of Types b and B are then classified as either rarefied or substantial, see Using isoperimetric controls going back to [10] one can infer the substantial "surface-like presence" of (α, β, γ)-good B 0 -boxes of ∂ B 0 U 1 in many scales above B. As observed below (1.23), the inequality N u (D 0 ) ≥ β cap(D 0 ) necessarily holds for these B 0 -boxes. In the case of boxes B of Type A, the methods of Section 3 (concerning resonance sets) apply instead. They show that when starting in B the simple random walk enters with a "high probability" (specifically at least 1 − (1 − a) 10) a certain resonance set Res J , see (4.56), (4.62). On this resonance set again a similar procedure can be performed to extract on many scales a "surface-like presence" of (α, β, γ)-good B 0 -boxes with N u (D 0 ) ≥ β cap(D 0 ). Then, in Proposition 4.2, we extract a collection of (α, β, γ)good B 0 -boxes with N u (D 0 ) ≥ β cap(D 0 ), having base points at mutual distance at least Proof of Theorem 4.1: We recall the assumptions (4.1) -(4.4) on the parameters. We assume that we are on the event (4.6) We recall the dyadic integer M (solely depending on d) from (2.1), as well as the naturally nested collections of boxes I ℓ (at depth ℓ), for 0 ≤ ℓ ≤ ℓ N , see (2.3). The size of a box in I ℓ is R ℓ = M ℓ N −ℓ L 1 , see (2.23). We also recall that when B ∈ I ℓ , then for 0 ≤ ℓ ′ ≤ ℓ the notation B (ℓ ′ ) refers to the unique box of I ℓ ′ that contains B, see (2.4).
We will later choose J as a function of d ≥ 3 and a ∈ (0, 1), see (4.99). Having in mind the estimates on the resonance set from Proposition 3.1, we pick in the notation of (3.8) (and view for the time being I as a function of d, a, and J). Next, with the volume estimates for rarefied boxes from Proposition 2.1 in mind, we choose an integer, to be later interpreted as a depth: ii)p ≥ 2ĉ J δ, withĉ, δ as in (1.8), (2.7).
With L(J) as in (3.2) we also define the depth From now on we implicitly assume N large enough so that (in the notation of (3.9), (2.9), (2.2), and (1.25)): We will now classify the boxes B ∈ I p intersecting Bub (⊆ D N ), see (1.26), into three types (namely b, B, and A). Informally, the boxes of Types b and B correspond to "boundary boxes in U 0 ", whereas boxes of Type A correspond to "inner boxes in U 0 ". More precisely, we say that We now describe Types B and A. To this end, for B ∈ I p and 0 ≤ ℓ ≤ p we define (4.14) where we recall that x B stands for the base point of B (see the beginning of Section 1) and R ℓ is defined in (2.23). We note that with B and ℓ as above We then say that  Finally, we say that Note that the three types are mutually exclusive. Also for 1 ≤ ℓ ≤ p, one has 4R ℓ ≤ 4N M < N, so that by (4.15) ii) (and Moreover, one has the inclusion and the above three sets in parentheses are pairwise disjoint. Each type will require a different treatment. We start with the case of boxes of Type b. Our next goal is to establish (4.39) (4.40).
We note that when B 1 is an L 1 -box and B ∈ I p is of type b with B 1 ⊆ B ∩ Bub, then by (4.12) we have Moreover, we note that (4.23) when B is of Type b, then as B 1 ⊆ Bub ∩ B varies, the corresponding B ′ (B 1 ) are pairwise disjoint or comparable for the inclusion relation.
As an aside the boxes B ′ (B 1 ) can be "very small" with a size close to L 1 . In particular, they can be much smaller than B (∈ I p ). We then introduce an arbitrary enumeration

By (4.22) we know that
So, by the isoperimetric controls (A.3) -(A.6), p. 480-481 of [10], we see that when N is large on the event Ω ε,N in (4.6) , b -columns, with base points at distance at least c 5 L 1 from ∂B ′ j .
(We recall that M in (2.1) is viewed as a dimension dependent constant).
We then introduce the set of good indices: As remarked below (1.23), when B 0 is an (α, β, γ)-good box in ∂ B 0 U 1 , then N u (D 0 ) ≥ β cap(D 0 ) holds as well. We thus find that  We then define for 1 ≤ j ≤ J b , Then, in the same spirit as below (3.26) of [34], one finds that where in the last step we have used that Ω ε,N ⊆ B c N (with B N defined in (1.24)). We now introduce the event In essence, on the complement ofΩ ε,N in Ω ε,N , see (4.6), we will simply "discard" the whole set Bub∩(⋃ Type b B) to achieve (4.39), (4.40). The main work pertains to the event Ω ε,N ∩Ω ε,N . On this event we find by (4.31) that Taking the d d−1 -th power of the above inequality, we find in view of (4.30) and the definition of L 0 in (1.9) that for large N on Ω ε,N ∩Ω ε,N We now classify the boxes of Type b as either rarefied or substantial. We recall the definition of δ in (2.7), as well as that ofp in (4.9). We work on the event Ω ε,N from (4.6). We say that a box B of Type b is rarefied if where the above sum runs over the boxesB in the tower above B (p) (see (2.4) for notation) that are contained in B (1) . Further, we say that B of Type b is substantial if Thus, B of Type b is rarefied when B (p) is rarefied in the sense of (2.8) with the choices q =p and F = ⋃ j∈G b B ′ j . By (4.12) the condition (2.9) is satisfied and the controls from Proposition 2.1 now yield that We then set We thus find by (4.25), (4.38) that for large N on the event Ω ε,N in (4.6) We now turn to the treatment of boxes of Type B, see (4.16). Our next goal is to establish in (4.51), (4.52) an analogue of (4.39), (4.40).
In either case (whether the set above (4.42) is empty or not) we have a . Next, we note that ∆ ′ (B) has size at least 4R p (and at most 4Rp +1 ), and the number of columns of B 0 -boxes in any given coordinate direction that are contained in ∆ ′ (B) is We can apply isoperimetric considerations (as in (4.27), (4.29)), where we now leverage the above mentioned rarity of (α, β, γ)-bad boxes, to obtain that for large N, on the event Ω ε,N in (4.6), for any 1 ≤ j ≤ J B , one can find a coordinate projection π ′ j , B and at least c 6 ( ∆ ′ , and with base points at distance at least c 7 L 1 from ∂∆ ′ j . Recall δ andp from (2.7) and (4.9). We then say that a box B of Type B is rarefied when (with analogous notation as in (4.35), (4.36)): and that it is substantial when The volume controls of Proposition 2.1 (with q =p and F = ⋃ 1≤j≤J B ∆ ′ j ), noting that (2.9) holds due to (4.12), now yield We thus set and find that We finally turn to the treatment of boxes of Type A. Our goal is to establish (4.62). To this end we have in mind to apply the results of Section 3 concerning resonance sets. This first requires some preparation. We introduce the notation (with U 1 as in (1.22)) (4.53) , for x ∈ Z d and r ≥ 0 integer, as well as the J-resonance set (recallp, p from (4.9), (4.11), and R ℓ from (2.23)), Note that when B ∈ I p and x, y ∈ B, then x − y 1 ≤ d R p and for each 0 ≤ ℓ ≤ p − 4, one has Thus, when B ⊆ I p intersects Res J , we can apply (4.55) to y = x B (the base point of B) and x in B ∩ Res J to find that (4.56) As we now explain, the results of Section 3 show that when N is large, the simple random walk starting in a box B of Type A enters Res J (and therefore Res J as well) with "high probability".
To this end we recall that M = 2 b , see (2.1), and we set (see (4.16)) (4.57) ℓ ∈ S} in the notation of (4.16)), and as in (3.6) with L = b L(J), we set (4.58) Consider now B ∈ I p , a box of Type A, so that, see (4.17), ∆ B,ℓ ∩ U 0 ≥ 3 4 ∆ B,ℓ for all ℓ ∈ S. When N is large, this implies that (see (4.14)) , for each ℓ ∈ S, and by a similar bound as in (4.55) it follows that Thus, for x ∈ B, setting U 0 = U 0 −x (and U 1 = U 1 −x) in (3.3), and noting that B(0, 2 m L 1 )∩ U 1 < 1 2 B(0, 2 m L 1 ) for all m ∈ A by (4.60) and the second line of (4.57), we find by Proposition 3.1 (note that (3.10) holds by (4.12)):   Taking (4.56) into account, this shows that for large N In addition, when B ∈ I p is contained in Res J , then for some ℓ ∈ (p, p − 4], B(x B , 8R ℓ ) intersects U 0 and U 1 and hence B(0, 3N + L 0 ), see (1.23), so that As a result, for large N, We then define the following set, which is a union of boxes of I p : We can now collect (4.39), (4.40) for Type b, (4.51), (4.52) for Type B, and (4.62) for Type A, to find for large N on the event Ω ε,N in (4.6) The next step on the way to the proof of Theorem 4.1 is (we recall (4.7), (4.9), (4.11) for notation).
Once we show Proposition 4.2 it will be a quick step to complete the proof of Theorem 4.1. It may be helpful at this stage to provide a brief outline of the proof of Proposition 4.2.
In a first step we will define the set L B for each B ∈ I p contained in A, leveraging the fact that such a B is substantial when it is of Type b or B, and that otherwise it is included in Res J . The selected levels will be such that for each ℓ ∈ L B there is a collection of disjoint sub-boxes within B(x B , 8R ℓ ) such that their union has a nondegenerate capacity in B(x B , 8R ℓ ), and within each such sub-box there is a "surface-like" presence of (α, β, γ)-good B 0 -boxes from ∂ B 0 U 1 (hence such that N u (D 0 ) ≥ β cap(D 0 ), see below (1.23)), having disjoint projection in some coordinate direction.
In a second step, with the help of Lemma 1.1, we will extract for each B and ℓ ∈ L B as above, a collectionC B,ℓ of (α, β, γ)-good B 0 -boxes with N u (D 0 ) ≥ β cap(D 0 ), contained in B(x B , 8R ℓ ), with base points at mutual distance at least H K L 0 (with H "large" and solely depending on d, α, ε, J, see (4.78)), so that the union of the B 0 -boxes fromC B,ℓ has a non-degenerate capacity in B(x B , 8R ℓ ), see (4.85), (4.86). We will then consider the unionC of these collectionsC B,ℓ of B 0 -boxes. At this stage the mutual distance between B 0 -boxes inC might be smaller than KL 0 (andC need not satisfy (4.66)).
In the third and last step we will introduce an equivalence relation withinC, for which two B 0 -boxes ofC lie in the same equivalence class if they can be joined by a path of boxes inC with steps of ⋅ ∞ -size at most K L 0 . We will show that the equivalence classes have a "small size", see (4.90). Then, we will select a representative in each equivalence class, and for each B ∈ I p , B ⊆ A, and ℓ ∈ L B consider the collection of the representatives of the boxes inC B,ℓ . With the help of Lemma 1.2 (which in essence goes back to [2]), we will extract the desired collections C B,ℓ , so that (4.66) -(4.69) hold.
Proof of Proposition 4.2: Our first step is to define L B , for B ∈ I p contained in A, see (4.64). We start with the case where B is of Type b, substantial, then we proceed with the case where B is of Type B, substantial, and finally we handle the case where B is contained in Thus, we first consider B of Type b, substantial. By (4.36) we know that µ (1.8)), so that µĉ ≥ δ 2p and hence We then turn to the case of a box B of Type B, substantial. Using (4.48) in place of (4.36), a similar argument as above shows that and for B of Type B, substantial, we define (4.74) L B as the collection of the J largest integers ℓ in [1,p] such that In addition, as we now explain, for B and ℓ ∈ L B as in (4.74), we can extract a subcollection J B,ℓ ⊆ {1, . . . , J B } such that j denotes the closed ball in supremum-distance with triple radius and same center as ∆ ′ j (we refer to the unique x ∈ R p Z d and r ≥ 1 such that ∆ ′ j = x + [−r, r) d as the "center" and the "radius" of ∆ ′ j , and recall that the size of ∆ ′ j , i.e. 2r, is at most 4Rp +1 , see (4.44)). To prove (4.75) we use a routine cover argument. We list the ∆ ′ j , 1 ≤ j ≤ J B intersecting B (ℓ) by decreasing size. We first consider the first such ∆ ′ j , thus of largest size, and delete from the list all the other ∆ ′ k that intersect ∆ ′ j . They are all contained in∆ ′ j . If the remaining list is empty, we are done. Otherwise, we proceed with the next ∆ ′ k in the list, which by construction does not intersect the first chosen ∆ ′ j , and proceed similarly until coming to an empty list.
We thus find that for B of Type B, substantial, and ℓ ∈ L B , We have now defined L B for any B ∈ I p contained in A through (4.72), (4.74), (4.77) and thus completed the first step of the proof of Proposition 4.2.
In the second step we are going to introduce for each B and ℓ ∈ L B as above a col-lectionC B,ℓ of B 0 -boxes, which are (α, β, γ)-good with N u (D 0 ) ≥ β cap(D 0 ), contained in B(x B , 8R ℓ ), with base points at mutual ⋅ ∞ -distance at least H K L 0 , where H is defined in (4.78) below, and such that ⋃C B,ℓ B 0 has a non-degenerate capacity in B(x B , 8R ℓ ), see (4.85), (4.86).
We begin with the case of B of Type b, substantial. We recall that by (4.29), for any j ∈ G b , there is a coordinate projection π ′ j , b and at least 1 , b -columns, (α, β, γ)-good and such that N u (D 0 ) ≥ β cap(D 0 ), contained in B ′ j , with base points at distance at least c 5 L 1 from ∂B ′ j . We can now apply Lemma 1.1, so that for large N on the event Ω ε,N in (4.6): for each j ∈ G b one can find a collection C ′ ,b j of (α, β, γ)-good B 0 -boxes contained in B ′ j (⊆ D N , see (4.25)), such that N u (D 0 ) ≥ β cap(D 0 ), at distance at least c 5 L 1 from ∂B ′ j , with base points having π ′ j , b -projection at mutual ⋅ ∞ -distance at least H KL 0 , and so that cap We then define for each B of Type b, substantial, and ℓ ∈ L B the collection of B 0 -boxes: , and the B ′ j , j ∈ G b are pairwise disjoint, see (4.21), (4.25)).
We then turn to the definition ofC B,ℓ for B of Type B, substantial, and ℓ ∈ L B . We combine (4.46) for j ∈ J B,ℓ and Lemma 1.1 to find that for large N on the event Ω ε,N in (4.6): for any B of Type B, substantial, ℓ ∈ L B , and j ∈ J B,ℓ , one can find a collection C We then define for B of Type B, substantial, and C ∈ L B the collection of B 0 -boxes and we see that for large N on the event Ω ε,N , for each B ∈ I p , B ⊆Â, and ℓ ∈ L B , we can find a collection of at least Once again we apply Lemma 1.1 and find that for large N on the event Ω ε,N in (4.6): for any B ∈ I p contained inÂ, and ℓ ∈ L B , there is a collectionC B,ℓ of (α, β, γ)- , 3N] d , with base points having π B,ℓ -projections at mutual ⋅ ∞ -distance at least H KL 0 and such that cap Collecting (4.80), (4.82), (4.84), for large N on the event Ω ε,N in (4.6), we have defined C B,ℓ for each B ∈ I p contained in A and ℓ ∈ L B . In particular for such B and ℓ In addition, as we now explain (4.86) for any B, ℓ as above, cap ⋃ Indeed, when B is of Type b, substantial, and ℓ ∈ L B , we know by (4.72) that cap( for the j ∈ G b such that B (ℓ) ∩B ′ j = φ (and hence such that B ′ j ⊆ B (ℓ) ), see (4.80), we find by the strong Markov property and the repeated application of the second line of (1.5) that the simple random walk starting in B (ℓ) enters ⋃C B,ℓ B 0 (⊆ B (ℓ) ) with a probability bounded below by a constant. In this lower bound, integrating the starting point of the walk with the equilibrium measure e B (ℓ) , and using (1.5) as well as h B (ℓ) = 1 on B (ℓ) ⊇ ⋃C B,ℓ B 0 , we find that cap (⋃C When B is of Type B, substantial, and ℓ ∈ L B , we know by (4.73), (4.75) that cap ( by (4.81). Thus, by a similar argument as in the previous paragraph, the simple random walk starting in B (ℓ) enters ⋃ j∈J B,ℓ∆ ′ j , and hence ⋃ j∈J B,ℓ ∆ ′ j , and therefore the ⋃ j∈J B,ℓ ⋃ C ′ ,B j,B,ℓ B 0 = ⋃C B,ℓ B 0 with a probability bounded below by a constant. A similar lower bound naturally holds true when the walk starts in B(x B , 8R ℓ ) ⊇ B (ℓ) . This bigger set contains ⋃C B,ℓ B 0 , see (4.85), and as in the previous paragraph it Finally, when B ∈ I p is contained inÂ (see (4.70)) and ℓ ∈ L B , we know by (4.84) that This completes the proof of (4.86).
We are now ready to start the third (and last) step of the proof of Proposition 4.2. In view of (4.85), (4.86) we introduce the collection of B 0 -boxes (4.87)C = ⋃ B,ℓ∈L BC B,ℓ (the union runs over all B ∈ I p , B ⊆ A, and ℓ ∈ L B ).
The collectionC may contain B 0 -boxes with base points at mutual ⋅ ∞ -distance smaller than KL 0 , and (4.66) need not hold forC. We will eventually extract sub-collections C B,ℓ fromC for each B ⊆ I p contained in A and ℓ ∈ L B , so as to fulfill the requirements of Proposition 4.2, see (4.97), (4.98).
With this in mind, we first observe that for each B 0 ∈C and eachC B,ℓ there is at most one box ofC B,ℓ with base point at ⋅ ∞ -distance smaller than HKL 0 2 from the base point of B 0 . As a result, we find that for large N on Ω ε,N : As a consequence of (4.88), we see that for any B 0 ∈C PSfrag replacements Fig. 3: Two distinct componentsC B 0 ,C B ′ 0 depicted in blue and green colors, with respective representativesB 0 andB ′ 0 inĈ corresponding to the hatched boxes.
We then find thatC = ⋃ B 0 ∈ĈC B 0 , that for each B 0 ∈C,C B 0 ∩Ĉ = {B 0 }, and that , ii) the base points of distinct boxes ofĈ have mutual ⋅ ∞ -distance bigger than KL 0 , iii) for any B ∈ I p , B ⊆ A, and ℓ ∈ L B , the x B 0 for B 0 ∈C B,ℓ have mutual ⋅ ∞ -distance at least HKL 0 (see (4.85)).
We then introduce the dimension dependent constant (4.93) c * * = sup{g(0,ŷ) g(0, y); for y,ŷ ∈ Z d such that ŷ ∞ ≥ 1 2 The following lemma states a version of the informal principle "a lower bound on the contraction of a map induces a lower bound on the capacity of the image of a set under this map". More precisely, one has Lemma 4.3. If F andF are two finite subsets of Z d for which there is a bijectionf : Proof. We consider the equilibrium measure e F of F and its imageê underf . So,ê is supported byF and for each x ∈ F ,ê (f (x)) = e F (x). Then, for eachx =f (x) inF , we have (see (1.2) for notation) since Ge F = 1 on F . So Gê ≤ c * * onF andê is supported byF , hence we have The claim (4.94) now follows.
Given B ∈ I p contained in A and ℓ ∈ L B , we can apply Lemma 4.3 to F = ⋃C B,ℓ B 0 and F = ⋃C B,ℓB 0 , takingf to be the map from F ontoF that sends each B 0 ⊆ F toB 0 ⊆F through translation by the vector xB x, y belong to the same B 0 ∈C B,ℓ , and by (4.92) i) and iii) when x, y belong to distinct L 0 -boxes inC B,ℓ , In addition, by (4.92) i) and the fact that the L 0 -boxes inC B,ℓ are contained in B(x B , 8R ℓ ), see (4.85), we find that when N is large on Ω ε,N (see (4.6)): (4.96) for any B ∈ I p contained in A and ℓ ∈ L B , ⋃C B,ℓB 0 ⊆ B(x B , 10R ℓ ).
In view of (4.95) and (4.92) ii) we can now apply Lemma 1.2, so that for large N on Ω ε,N in (4.6), and such that cap As we now explain, for B, ℓ as above, the simple random walk starting in B(x B , 10R ℓ ) enters ⋃ C B,ℓ B 0 before exiting B(x B , 10R ℓ−1 ) with a non-degenerate probability. For this purpose we write g B,ℓ (⋅, ⋅) for the Green function of the walk killed outside B(x B , 10R ℓ−1 ). We know by (1.4) that g B,ℓ (x, y) ≥ c ∆ g(x, y) when x, y ∈ B(x B , 10R ℓ ). In addition, setting , the equilibrium measure of F for the walk killed outside B(x B , 10R ℓ−1 ) is supported by F and dominates e F , see below (1.5). It now follows that We proceed with the proof of Theorem 4.1. We will now combine (4.65) and Proposition 4.2. Thus, for large N on the event Ω ε,N in (4.6), we see that except maybe on a set of at most ε 5 D N points in the bubble set Bub, the simple random walk starting at x ∈ Bub enters the set A in (4.64) with probability at least 1 − (1 − a) 10 and once in A enters ⋃ C B 0 with probability at least 1 − (1 − c 9 ) J (using the strong Markov property at the successive times of exit of B(x B , 10R ℓ−1 ), 1 ≤ ℓ ≤ p, and (4.69), if B stands for the box of I p contained in A where the walk first enters A).
We can now choose J as a function of a ∈ (0, 1) via We have now obtained that for large N on the event Ω ε,N in (4.6)  4N, 4N) d , so that the base points of all these boxes as B and ℓ ∈ L B vary, keep a mutual distance at least KL 0 , and so that denoting their union by Thus, for large N we set C ω = φ, on B c N Ω ε,N , and on Ω ε,N construct a measurable choice L B,ω of L B for B ∈ I p , B ⊆ [−4N, 4N] d and C B,ℓ,ω of C B,ℓ when ℓ ∈ L B,ω , and set C ω = ⋃ B,ℓ∈L B,ω ⋃ C B,ℓ,ω B 0 . The conditions (4.5) i), ii), v) are fulfilled. Concerning (4.5) iv), note that the 2KL 1 -neighborhood of C ω has volume at most c K d L d by (1.9), (1.10), so that (4.5) iv) holds as well.
There remains to check (4.5) iii). For each B ∈ I p included in [−4N, 4N] d , there are at most (p J +1) possibilities for L B,ω , and for each L B,ω , which is not empty, and ℓ ∈ L B,ω , one has at most exp{c

The asymptotic upper bound
In this section we derive the main asymptotic upper bound on the exponential rate of decay of the probability of an excess of disconnected points corresponding to the event . With Theorem 4.1 now available, most of the remaining task has already been carried out in Section 4 of [34]. Importantly, if as expected the identity u = u * holds, the asymptotic upper bound of Theorem 5.1 below actually governs the exponential decay of P[A N ], see Remark 5.2 1). An application of Theorem 5.1 to the simple random walk is given in Corollary 5.3.
We assume that (see (1.16)): and we define the function (see (0.2) for notation): and for ν ∈ [θ 0 (u), 1), we set The existence of a minimizer for (5.3) is established by the same argument as in the case of J u,ν , see Theorem 2 of [33], which corresponds to the function θ 0 (≤θ) in place ofθ, see (0.11). In addition for u and ν as above, since θ 0 ≤θ, one has (5.4)Ĵ u,ν ≤ J u,ν (and these quantities coincide if u = u * ).
Remark 5.2. 1) If the identity u = u * holds (this is the object of active research, and the corresponding identity in the closely related model of the level-set percolation of the Gaussian free field has been established in [15]), then Theorem 5.1 and the lower bound (0.10) from [32] and [33] show that for 0 < u < u * and ν ∈ [θ 0 (u), 1) one has in the notation of (0.9) 2) The event A N in Theorem 5.1 pertains to an excessive fraction of points in D N disconnected by I u from S 2N . One can replace 2 by an arbitrary integer m ≥ 1 and instead consider the events { D N C u mN ≥ ν D N }, which correspond to an excessive fraction of points in D N that are disconnected by I u from S mN (these events are non-decreasing in m). The proof of Theorem 5.1 (based on Section 4 and on [34]) can straightforwardly be adapted to show that for arbitrary m ≥ 1, 0 < u < u, and ν ∈ [θ 0 (u), 1) one has (5.11) lim sup In particular, if as expected u and u * coincide, this upper bound combined with the lower bound (0.10), proves that for any m ≥ 1, 0 < u < u * and ν ∈ [θ 0 (u), 1) (this extends (5.10)).
However, it remains open whether for the larger events { D N C u ∞ ≥ ν D N } corresponding to an excessive fraction of points in D N disconnected by I u from infinity, one has a similar asymptotics. Namely, is it the case that for all 0 < u < u * and ν ∈ [θ 0 (u), 1) In the context of the Wulff droplet for super-critical Bernoulli percolation, we refer to Theorem 2.12 of [5] for a corresponding result (the leading rate of decay of the asymptotics is in that case N d−1 , i.e. "surface like", and not N d−2 , i.e. "capacity like", as here).
3) It is a natural question whether for ν close to 1 the minimizers ϕ for J u,ν in (0.11) do reach the maximum possible value √ u * − √ u on a set of positive Lebesgue measure. As explained below (0.14) this occurrence could reflect the presence of droplets secluded by the interlacements and contributing to the excess fraction of disconnected points when A N happens. If θ 0 is discontinuous at u * (a not very plausible possibility), the minimizers for J u,ν are easily seen to reach the value √ u * − √ u on a set of positive measure when ν is close to 1, see Remark 2 of [33]. But otherwise the situation is unclear, for the behavior of θ 0 close to u * is very poorly understood. The same is true in the case of the percolation function θ G 0 for the level-set percolation of the Gaussian free field on Z d , see above (0.16). Interestingly, in the case of the cable graph on Z d the corresponding functionθ G 0 is explicit. The critical level is 0 andθ G 0 (h) = 2Φ(h ∧ 0), for h ∈ R, where Φ denotes the distribution function of a centered Gaussian variable with variance g(0, 0), see Corollary 2.1 of [12]. However, in the case of Z d , d = 3, the simulations in Figure 4 of [21] suggest a behavior of θ G 0 close to the critical level h * different from that ofθ G 0 near the critical level 0. Coming back to the original question whether for ν close to 1 the minimizers for J u,ν reach the maximal value √ u * − √ u on a set of positive measure, let us mention that the question has a similar flavor to the problem concerning the existence of dead core solutions for semilinear equations 1 2d ∆v = f (v), i.e. non-negative solutions in a bounded domain U of R d , which vanish on a relatively compact open subset V in U and are positive in U V , see [24]. Quite informally, assuming θ 0 to be C 1 to simplify the discussion, the link goes via the consideration of v = √ u * − √ u − ϕ, where ϕ minimizer for J u.ν satisfies an Euler-Lagrange equation − 1 2 ∆ϕ = λη ′ (ϕ) 1 D , with λ > 0 a Lagrange multiplier and η(b) = θ 0 (( √ u + b) 2 ), see for instance Lemma 5 of [33]. Under suitable assumptions on f the reference [24] provides an integral criterion, which characterizes the existence of dead core solutions, see Theorems 1.1, 7.2 and 7.3 of [24]. Brought into our context, these results raise the question: does the convergence of ∫ The above Theorem 5.1 also has an immediate application to a similar upper bound, where the simple random walk replaces the random interlacements. Informally, this corresponds to taking the singular limit u → 0 in (5.5). Specifically, we denote by I the set of points in Z d visited by the simple random walk, and by C 2N the connected component of S 2N in (Z d I) ∪ S 2N . One has The existence of a minimizer for (5.15) is established by a similar argument as in the case of J u,ν in Theorem 2 of [33]. Also, if as expected, the identity u = u * holds, thenθ coincides with θ 0 .
Proof. Without loss of generality, we can assume that ν > 0 and choose u ∈ (0, u) such that θ 0 (u) < ν. As in Corollary 7.3 of [29] or Corollary 6.4 of [30], one has a coupling P of I u under P[⋅ x ∈ I u ] and of I under P x , so that P -a.s., I ⊆ I u . Then P -a.s., the points in D N disconnected by I from S 2N are also points disconnected by I u from S 2N , so that P -a.s., As a result of Theorem 5.1 it follows that (5.17) lim sup By direct inspectionĴ u,ν is non-increasing in u andĴ u,ν ≤Ĵ ν . As we now explain The argument is similar to the proof of (5.9). We consider a sequence u n in (0, u) decreasing to 0 with θ 0 (u n ) < ν for all n, and let ϕ n be a minimizer forĴ un,ν . Once again, by Theorem 8.6 and Corollary 8.7 of [20], we can extract a subsequence ϕ n ℓ , ℓ ≥ 1, converging in L 2 loc (R d ) and a.e. to ϕ element of D 1 (R d ) such that This shows thatĴ ν ≤ lim nĴun,ν and hence completes the proof of (5.18). The claim of Corollary 5.3 now follows by letting u tend to 0 in (5.17).

Remark 5.4.
It remains an open question whether a matching asymptotic lower bound for (5.14) holds as well. The consideration of tilted random walks as in [18] (which provides the "right" asymptotic lower bound for the disconnection of D N by I), and similar ideas as in Section 4 and Remark 6.6 of [32] might be helpful. ◻

A Appendix: Resonance sets and I-families
In this appendix we recall some results concerning resonance sets and I-families developed in [23] and [9], and we sketch the proof of Proposition 3.1.
First some notation. For x ∈ Z d and r ≥ 0 integer we write (A.1) m x,r for the normalized counting measure on B(x, r) ∩ Z d , and ⟨f ⟩ B(x,r) for the integral of f with respect to m x,r .
We consider an integer and (as in (4.20) of [9], with δ = 200J) a length scale r min (J) such that and such that for suitable constantsč 0 (J),č 1 (J), c ′ (J) ∈ (0, 1), for all integers r ≥ r min where q t,B(x,r) (⋅, ⋅) stands for the transition kernel of the simple random walk with unit jump rate killed outside B(x, r).
We assume that N is large so that in the notation of (1.10) We then consider (A.6) U 0 a finite non-empty subset of Z d , U 1 = Z d U 0 , and we define the density functions for x ∈ Z d and m ≥ 0 integer, , and The next two lemmas are straightforward adaptations of Lemmas 1.1 and 1.2 of [23] in the R d -case and of Lemmas 4.3 and 4.4 of [9] in the discrete case. We have: Further, we have (and the δ below is unrelated to (2.7) but follows the notation of [23], [9]): Lemma A.2. For x ∈ Z d , 0 ≤ m ′ < m, setting β ′ = ⟨σ m ′ ⟩ B(x,2 m L 1 ) , then for all 0 ≤ δ ≤ β ′ ∧ (1 − β ′ ) at least one of i) or ii) below holds true: The next proposition corresponds to Proposition 4.5 of [9] in our set-up (with the choice δ = 1 (200J)), see also Proposition 1.3 of [23]. , 0 ≤ j ≤ J, together with the non-decreasing sequence of stopping times (A.16) γ 0 = H {σm 0 ∈I 0 } , and γ j+1 = γ j + H {σm j+1 ∈I j+1 } ○ θ γ j , for 0 ≤ j < J (we refer to the beginning of Section 1 for notation).
The next proposition corresponds to Proposition 4.6 of [9] in the set-up of random walks among random conductances, and to Proposition 1.4 of [23] in the Brownian case.
Proposition A.4. Recall (A.5), (A.6). Assume that J ≥ 1 and that the non-negative integers m j , 0 ≤ j ≤ J, satisfy (A.14). Let E in the notation of (A.16), (A.12) denote the event Then, for any x ∈ Z d such that In addition, on the event E, we have (see (A.7) for notation) sup{ X s − X γ j ∞ ; γ j ≤ s ≤ γ J } ≤ Let us mention that the constraint on x stated in (A.18) is more restrictive (i.e. less general) than the constraint σ m 0 (x) ∈ [ 1 2 − r min (J) −1 , 1 2 + r min (J) −1 ] corresponding to Proposition 4.6 of [9], but more convenient in the present context.
We then turn to the crucial notion of I-family that will provide the tool which permits to bound the probability that the walk avoids the resonance set, when starting at a point around which the local density of U 0 is at least 1 2 on many well-separated scales 2 m L 1 . Similar to (2.11) of [23] and (A.2) of [9], we consider for all m ∈ A.
Note that at each jump of the walk the functions σ m (⋅) vary at most by an amount (2 m L 1 ) −1 , see (A.8). Since the walk is transient and U 0 finite, P 0 -a.s., σ m (X s ) is equal to 1 for large s, and hence P 0 -a.s. all the functions σ m (X s ), s ≥ 0, m ∈ A, visit the interval , for some m ∈ L , , for some m ∈ L {m 1 } ,  The interest of the quantities Γ (J) k (I) stems from a basic recursive inequality, which they satisfy, see Lemma 2.2 of [23] and Lemma A.1 of [9]. Namely, one has (withč 2 (J) from (A.19)): Lemma A.5.
Adding the two bounds and taking the supremum over I-families, the claim (A.35) follows. ◻