1 Introduction and results

Quenched invariance principles and heat kernel bounds for random walks on infinite percolation clusters and among i.i.d. random conductances in \({\mathbb {Z}}^d\) were proved during the last two decades (see [1, 2, 4, 7, 9, 11, 13, 14, 18, 19, 23, 24, 32]). The proofs of these results strongly rely on the i.i.d structure of the models and some stochastic domination with respect to super-critical Bernoulli percolation.

Many important models in probability theory and in statistical mechanics, in particular, models which come from real world phenomena, exhibit long range correlations and offer an incentive to create new tools capable of handling models with dependent structures. In recent years interest arose in understanding such systems, both in specific models such as random interlacements, vacant set of random interlacements and the Gaussian free field, as well as in general systems, see for example [12, 17, 27, 28, 31, 3335]. In the context of invariance principle with long range correlations one should emphasize the results of Biskup [10] and Andres et al. [2], that prove a quenched invariance principle for random walk in ergodic random conductances under some moment assumptions and ellipticity. In this paper we prove a quenched invariance principle for random walks on percolation clusters (i.e., in the non-elliptic situation) in the axiomatic framework of Drewitz et al. [17]. This framework encompasses percolation models with strong correlations, including random interlacements, vacant set of random interlacements, and level sets of the Gaussian free field.

The main novelty of our proof is a new isoperimetric inequality for correlated percolation models, see Theorem 1.2. We should emphasize that existing methods for proving isoperimetric inequalities (see, e.g., [3, 6, 8, 22, 26]) only apply to models which allow for comparison with Bernoulli percolation after a certain coarsening procedure. A common feature of the three examples above is that they cannot be effectively compared with Bernoulli percolation on any scale. Thus, the existing methods for proving isoperimetric inequalities do not apply. Our approach is more combinatorial in nature. It does not rely on any “set counting” arguments and the Liggett–Schonmann–Stacey theorem [21], and can be applied to models which do not dominate supercritical Bernoulli percolation after any coarsening.

1.1 The model

We consider a one parameter family of probability measures \({\mathbb {P}}^u\), \(u\in (a,b)\subseteq {\mathbb {R}}_+\), on the measurable space \((\{0,1\}^{{\mathbb {Z}}^d},{\mathcal {F}})\), \(d\ge 2\), where the sigma-algebra \({\mathcal {F}}\) is generated by the canonical coordinate maps \(\{\omega \mapsto \omega (x)\}_{x\in {\mathbb {Z}}^d}\). The numbers \(0\le a<b\) as well as the dimension \(d\ge 2\) are going to be fixed throughout the paper, and we omit the dependence of various constants on a, b, and d.

For \(x=(x(1),\dots ,x(d))\in {\mathbb {R}}^d\), the \(\ell ^1\) and \(\ell ^\infty \) norms of x are defined in the usual way by \(|x|_1 = \sum _{i=1}^d|x(i)|\) and \(|x|_\infty = \max \{|x(1)|,\ldots |x(d)|\}\), respectively.

For any \(\omega \in \{0,1\}^{{\mathbb {Z}}^d}\), we define

$$\begin{aligned} {\mathcal {S}}= {\mathcal {S}}(\omega ) = \{x\in {\mathbb {Z}}^d~:~\omega (x) = 1\} \subseteq {\mathbb {Z}}^d. \end{aligned}$$

We view \({\mathcal {S}}\) as a subgraph of \({\mathbb {Z}}^d\) in which the edges are drawn between any two vertices of \({\mathcal {S}}\) within \(\ell ^1\)-distance 1 from each other. For \(r\in [0,\infty ]\), we denote by \({\mathcal {S}}_r\), the set of vertices of \({\mathcal {S}}\) which are in connected components of \({\mathcal {S}}\) of \(\ell ^1\)-diameter \(\ge r\). In particular, \({\mathcal {S}}_\infty \) is the subset of vertices of \({\mathcal {S}}\) which are in infinite connected components of \({\mathcal {S}}\).

An event \(G \in {\mathcal {F}}\) is called increasing (respectively, decreasing), if for all \(\omega \in G\) and \(\omega ' \in \{0,1\}^{{\mathbb {Z}}^d}\) with \(\omega (y) \le \omega '(y)\) (respectively, \(\omega (y) \ge \omega '(y)\)) for all \(y\in {\mathbb {Z}}^d\), one has \(\omega ' \in G\).

For \(x \in {\mathbb {Z}}^d\) and \(r \in {\mathbb {R}}_+\), we denote by \({\mathrm {B}}(x,r) = \{y\in {\mathbb {Z}}^d~:~|x-y|_\infty \le \lfloor r \rfloor \}\) the closed \(l^{\infty }\)-ball in \({\mathbb {Z}}^d\) with radius \(\lfloor r \rfloor \) and center x.

We assume that the measures \({\mathbb {P}}^u\), \(u\in (a,b)\), satisfy the axioms from [17], which we now briefly list. The reader is referred to the original paper [17] for a discussion about this setup.

P1 :

(Ergodicity) For each \(u\in (a,b)\), every lattice shift is measure preserving and ergodic on \((\{0,1\}^{{\mathbb {Z}}^d},{\mathcal {F}},{\mathbb {P}}^u)\).

P2 :

(Monotonicity) For any \(u,u'\in (a,b)\) with \(u<u'\), and any increasing event \(G\in {\mathcal {F}}\), \({\mathbb {P}}^u[G] \le {\mathbb {P}}^{u'}[G]\).

P3 :

(Decoupling) Let \(L\ge 1\) be an integer and \(x_1,x_2\in {\mathbb {Z}}^d\). For \(i\in \{1,2\}\), let \(A_i\in \sigma (\{\omega \mapsto \omega (y)\}_{y\in {\mathrm {B}}(x_i,10L)})\) be decreasing events, and \(B_i\in \sigma (\{\omega \mapsto \omega (y)\}_{y\in {\mathrm {B}}(x_i,10L)})\) increasing events. There exist \(R_{\scriptscriptstyle {\mathrm {P}}},L_{\scriptscriptstyle {\mathrm {P}}}<\infty \) and \({\varepsilon _{\scriptscriptstyle {\mathrm {P}}}},{\chi _{\scriptscriptstyle {\mathrm {P}}}}>0\) such that for any integer \(R\ge R_{\scriptscriptstyle {\mathrm {P}}}\) and \(a<\widehat{u}<u<b\) satisfying

$$\begin{aligned} u\ge \left( 1 + R^{-{\chi _{\scriptscriptstyle {\mathrm {P}}}}}\right) \cdot \widehat{u},\ \end{aligned}$$

if \(|x_1 - x_2|_\infty \ge R\cdot L\), then

$$\begin{aligned} {\mathbb {P}}^u\left[ A_1\cap A_2\right] \le {\mathbb {P}}^{\widehat{u}}\left[ A_1\right] \cdot {\mathbb {P}}^{\widehat{u}}\left[ A_2\right] + e^{-f_{\scriptscriptstyle {\mathrm {P}}}(L)},\ \end{aligned}$$

and

$$\begin{aligned} {\mathbb {P}}^{\widehat{u}}\left[ B_1\cap B_2\right] \le {\mathbb {P}}^u\left[ B_1\right] \cdot {\mathbb {P}}^u\left[ B_2\right] + e^{-f_{\scriptscriptstyle {\mathrm {P}}}(L)}, \end{aligned}$$

where \(f_{\scriptscriptstyle {\mathrm {P}}}\) is a real valued function satisfying \(f_{\scriptscriptstyle {\mathrm {P}}}(L) \ge e^{(\log L)^{\varepsilon _{\scriptscriptstyle {\mathrm {P}}}}}\) for all \(L\ge L_{\scriptscriptstyle {\mathrm {P}}}\).

S1 :

(Local uniqueness) There exists a function \(f_{\scriptscriptstyle {\mathrm {S}}}:(a,b)\times {\mathbb {Z}}_+\rightarrow {\mathbb {R}}\) such that for each \(u\in (a,b)\),

$$\begin{aligned} \begin{array}{c} \text {there exist } {\Delta _{\scriptscriptstyle {\mathrm {S}}}}= {\Delta _{\scriptscriptstyle {\mathrm {S}}}}(u)>0 \text { and }R_{\scriptscriptstyle {\mathrm {S}}}= R_{\scriptscriptstyle {\mathrm {S}}}(u)<\infty \\ \text {such that } f_{\scriptscriptstyle {\mathrm {S}}}(u,R) \ge (\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}} \text { for all } R\ge R_{\scriptscriptstyle {\mathrm {S}}}, \end{array} \end{aligned}$$
(1.1)

and for all \(u\in (a,b)\) and \(R\ge 1\), the following inequalities are satisfied:

$$\begin{aligned} {\mathbb {P}}^u\left[ \, {\mathcal {S}}_R\cap {\mathrm {B}}(0,R) \ne \emptyset \, \right] \ge 1 - e^{-f_{\scriptscriptstyle {\mathrm {S}}}(u,R)}, \end{aligned}$$

and

$$\begin{aligned} {\mathbb {P}}^u\left[ \begin{array}{c} \text {for all }x,y\in {\mathcal {S}}_{\scriptscriptstyle {R/10}}\cap {\mathrm {B}}(0,R),\\ x \text { is connected to } y \text { in } {\mathcal {S}}\cap {\mathrm {B}}(0,2R) \end{array} \right] \ge 1 - e^{-f_{\scriptscriptstyle {\mathrm {S}}}(u,R)}. \end{aligned}$$
S2 :

(Continuity) Let \(\eta (u) = {\mathbb {P}}^u\left[ 0\in {\mathcal {S}}_\infty \right] \). The function \(\eta (\cdot )\) is positive and continuous on (ab).

Note that if the family \({\mathbb {P}}^u\), \(u\in (a,b)\), satisfies S1, then a union bound argument gives that for any \(u\in (a,b)\), \({\mathbb {P}}^u\)-a.s., the set \({\mathcal {S}}_\infty \) is non-empty and connected, and there exist \(c_{\scriptscriptstyle 1}=c_{\scriptscriptstyle 1}(u)>0\) and \(C_1 = C_1(u)<\infty \) such that for all \(R\ge 1\),

$$\begin{aligned} {\mathbb {P}}^u\left[ \, {\mathcal {S}}_\infty \cap {\mathrm {B}}(0,R) \ne \emptyset \, \right] \ge 1 - C_1 e^{-c_{\scriptscriptstyle 1}(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}. \end{aligned}$$
(1.2)

We will comment on the use of conditions P2, P3, and S2 in Remark 2.5.

1.2 Results

For \(\omega \in \{0,1\}^{{\mathbb {Z}}^d}\) and \(x\in {\mathcal {S}}\), let \(\deg _\omega (x) =\left| \{y\in {\mathcal {S}}~:~ |y-x|_1=1 \}\right| \) be the degree of x in \({\mathcal {S}}\), and let \(\mathbf {P}_{\omega ,x}\) be the distribution of the random walk \(\{X_n\}_{n \ge 0}\) on \({\mathcal {S}}\) defined by the transition kernel

$$\begin{aligned} \mathbf {P}_{\omega ,x}[X_{n+1} = z|X_{n}=y] = \left\{ \begin{array}{ll} \frac{1}{2d}&{}\quad |z-y|_1=1,~z\in {\mathcal {S}};\\ 1-\frac{\deg _\omega (y)}{2d}&{}\quad z=y;\\ 0 &{} \quad \text {otherwise,} \end{array} \right. \end{aligned}$$

and initial position \(\mathbf {P}_{\omega ,x}[X_0=x]=1\). For \(n\in {\mathbb {N}}\), and \(t\ge 0\), define

$$\begin{aligned} \widetilde{B}_n(t) = \frac{1}{\sqrt{n}}\left( X_{\lfloor tn\rfloor } + (tn - \lfloor tn\rfloor )\cdot (X_{\lfloor tn\rfloor + 1} - X_{\lfloor tn\rfloor })\right) . \end{aligned}$$

Denote by C[0, T] the space of continuous functions from [0, T] to \({\mathbb {R}}^d\) equipped with supremum norm, and by \({\mathcal {W}}_T\) the Borel sigma-algebra on C[0, T]. Our main result is the following theorem.

Theorem 1.1

Let \(d\ge 2\), and assume that the family of measures \({\mathbb {P}}^u\), \(u\in (a,b)\), satisfies assumptions P1P3 and S1S2. Then for all \(u\in (a,b)\), \(T>0\), and for \({\mathbb {P}}^u[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost every \(\omega \), the law of \((\widetilde{B}_n(t))_{0\le t\le T}\) on \((C[0,T],{\mathcal {W}}_T)\) converges weakly to the law of a Brownian motion with zero drift and non-degenerate covariance matrix. In addition, if reflections and rotations of \({\mathbb {Z}}^d\) by \(\frac{\pi }{2}\) preserve \({\mathbb {P}}^u\), then the limiting Brownian motion is isotropic (with positive diffusion constant).

The proof of Theorem 1.1 is based on the well-known construction of the corrector. Moreover, it closely follows the proofs of the main results in [7, 11] using [17, Theorem 1.3] about chemical distance in \({\mathcal {S}}\), and Theorem 1.2 below, which is the main novelty of this paper.

Theorem 1.2

Let \(d\ge 2\) and \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}>0\). For \(A\subset {\mathcal {S}}\), let \(\partial _{\mathcal {S}}A\) be the edge boundary of A in \({\mathcal {S}}\), i.e., the set of edges from \({\mathbb {Z}}^d\) with one end-vertex in A and the other in \({\mathcal {S}}{\setminus } A\). For \(R\ge 1\), let \({\mathcal {C}}_R\) be a largest connected component (in volume, with ties broken arbitrarily) of \({\mathcal {S}}\cap {\mathrm {B}}(0,R)\).

If the family of measures \({\mathbb {P}}^u\), \(u\in (a,b)\), satisfies assumptions P1P3 and S1S2, then for each \(u\in (a,b)\), there exist \(\gamma _{\scriptscriptstyle 1.2} = \gamma _{\scriptscriptstyle 1.2}(u)>0\), \(c=c(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}})>0\), and \(C=C(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}})<\infty \) such that for all \(R\ge 1\),

$$\begin{aligned} {\mathbb {P}}^u\left[ \begin{array}{c} \text {for any }A\subset {\mathcal {C}}_R \text { with }|A|\ge R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}},\\ |\partial _{\mathcal {S}}A| \ge \gamma _{\scriptscriptstyle 1.2}\cdot |A|^{\frac{d-1}{d}} \end{array} \right] \ge 1 - Ce^{-c(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}. \end{aligned}$$
(1.3)

Remark 1.3

  1. 1.

    As we will see in the proof of Theorem 1.2, under assumptions P1P3 and S1S2, for each \(u\in (a,b)\), with \({\mathbb {P}}^u\)-probability \(\ge 1 - Ce^{-c(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}\), there is a unique cluster of largest volume in \({\mathcal {S}}\cap {\mathrm {B}}(0,R)\).

  2. 2.

    Note that we consider here the boundary of A in \({\mathcal {S}}\), and not in \({\mathcal {C}}_R\). This is enough for our purposes. The first proofs of the quenched invariance principle for simple random walk on the infinite cluster of Bernoulli percolation [7, 24, 32] crucially relied on the Gaussian upper bound on \(\mathbf {P}_{\omega ,0}[X_n = x]\) obtained in [3]. To prove the desired bound (as well as the corresponding Gaussian lower bound) one needs to show that with \({\mathbb {P}}^u\)-probability \(\ge 1 - Ce^{-c(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}\), the boundary in \({\mathcal {C}}_R\) of any \(A\subset {\mathcal {C}}_R\) such that \(|A|\le \frac{1}{2}\cdot |{\mathcal {C}}_R|\) has size \(\ge c\cdot R^{-1}\cdot |A|\), see, e.g., [3, Proposition 2.11]. Thanks to simplifications obtained in [11], we do not need to prove such a statement in order to deduce Theorem 1.1. Showing that the Gaussian bounds on the transition density hold under assumptions P1P3 and S1S2 remains an open problem.

  3. 3.

    In fact, we do not need the full strength of Theorem 1.2 to prove Theorem 1.1, see assumption A5 in Sect. 4.

  4. 4.

    Theorem 1.2 implies that under the assumptions P1P3 and S1S2, for any \(u\in (a,b)\) and \({\mathbb {P}}^u[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost every \(\omega \), there exists \(K_u = K_u(\omega )<\infty \) such that for all \(n\ge 1\) and \(x\in {\mathbb {Z}}^d\), \(\mathbf {P}_{\omega ,0}[X_n = x] \le K_u\cdot n^{-d/2}\), see (6.6). This is also a new result, even for the specific models such as random interlacements, vacant set of random interlacements, and the level sets of the Gaussian free field. In the context of random interlacements, a bound close to optimal (with a correcting factor of a multiple logarithm) was obtained in [29].

  5. 5.

    Theorem 1.1 implies that for any \(u\in (a,b)\) and \({\mathbb {P}}^u[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost every \(\omega \), there exists \(k_u = k_u(\omega )>0\) such that for all \(n\ge 1\), \(\mathbf {P}_{\omega ,0}[X_{2n} = 0] \ge k_u\cdot n^{-d/2}\), see [11, Remark 2.2].

Analogue of Theorem 1.2 has only been known before for independent Bernoulli percolation, see, e.g., [3, 6, 8, 22, 26]. All these proofs rely crucially on a “set counting argument” and thus require exponential decay of probabilities of certain events. This is achieved by using Liggett–Schonmann–Stacey theorem [21]. Such approach is quite restrictive and does not apply to models which cannot be compared with Bernoulli percolation on any scale, such as, for example, random interlacements. Our method is more robust and requires only minimal assumptions on the decay of probabilities of some events.

We will now comment on the proof of Theorem 1.2. As in all the proofs of isoperimetric inequalities for subsets of the infinite cluster of Bernoulli percolation, we set up a proper coarsening of \({\mathcal {S}}\) and then translate the given isoperimetric problem for large subsets of \({\mathcal {S}}\) into an isoperimetric problem on the coarsened lattice. Nevertheless, both the coarsening and the analysis of the coarsened lattice are very different from the ones used in existing approaches. The major difficulties, as already discussed, come from the presence of long-range correlations and the fact that the models cannot in general be compared with Bernoulli percolation, which rules out possibilities of using any Peierls-type argument.

We partition the lattice \({\mathbb {Z}}^d\) into boxes \((x+ [0,L_0)^d)\), \(x\in L_0\cdot {\mathbb {Z}}^d\), and subdivide the set of all boxes into good (very likely as \(L_0\rightarrow \infty \)) and bad. In the restriction of \({\mathcal {S}}\) to each of the good boxes, it is possible to identify uniquely a connected component of largest volume, which we call special, see Lemma 2.6(a). Moreover, for any pair of adjacent good boxes, their special connected components are connected locally, see Lemma 2.6(b). We emphasize that a good box may contain several connected components of large diameter, and in principle a connected component with the largest diameter may be different from the special component. This is the key difference of the coarsening procedure that we use from the ones used in the study of Bernoulli percolation. The main reason for doing this is that a box is good if two local events occur, one of which is increasing (there exists a large in volume connected component of \({\mathcal {S}}\) in the box) and the other decreasing (the cardinality of \({\mathcal {S}}\) in the box is not so big), see Definitions 2.1 and 2.2. Using P3 to control correlations between such monotone events, we set up two multi-scale renormalizations with scales \(L_n\) (one for increasing and one for decreasing events) to identify with high probability a well structured subset of good boxes. We call a \(L_n\)-box n-good if all the \(L_{n-1}\)-subboxes of this box, except for the ones contained in the union of at most two boxes of side length \(r_{n-1}L_{n-1}\), are \((n-1)\)-good. (Here \(r_i\) is a sequence of positive integers growing to infinity, but much slower than \(\frac{L_{i+1}}{L_i}\).) Every \(L_n\)-box is n-good with overwhelming probability. We are interested in the set of (0-)good boxes which are contained in n-good boxes for all \(n\ge 1\). This set is obtained by a perforation of \({\mathbb {Z}}^d\) on multiple levels, and therefore has a well described structure. Indeed, from each n-good box, we delete two boxes of size length \(r_{n-1}L_{n-1}\), from each of the remaining \(L_{n-1}\)-boxes (all of which are \((n-1)\)-good), we delete two boxes of side length \(r_{n-2}L_{n-2}\), and so on until we reach the level 0. Moreover, if \(r_i\ll \frac{L_{i+1}}{L_i}\), then the set of all deleted boxes (the complement of the good set) has a small volume. We should mention that such coarsening and renormalization have already been used before in [17, 30] to study models with long-range correlations satisfying assumption P3.

Fig. 1
figure 1

The colored region on the middle picture corresponds to \({\mathcal {G}}_{z_i}'\). Its restriction to a two dimensional hyperplane is not generally connected, as illustrated here. The colored region on the last picture corresponds to \({\mathcal {G}}_{z_i}''\). The restriction of \({\mathcal {G}}_{z_i}''\) to any two dimensional hyperplane is connected. The three small boxes on these figures are not drawn to the actual scale. (Mind that we assume that \(4r_{i-1}<l_{i-1}\).)

We need to further sparsen the obtained set of good boxes to make sure that it has good connectivity properties. This is done by a “deterministic” multi-level perforation of \({\mathbb {Z}}^d\), where from each n-good box, we delete yet at most one box of side length \(r_{n-1}L_{n-1}\) depending on the location of two already deleted boxes of side length \(r_{n-1}L_{n-1}\). For example, if two boxes of side length \(r_{n-1}L_{n-1}\) are deleted near an edge of an \(L_n\)-box, then we delete another box of side length \(r_{n-1}L_{n-1}\) at the edge, see Figs. 1 and 2. During this discussion, we call the resulting (connected) set of good boxes fat. The fat set is not only connected in the lattice of \(L_0\)-boxes, but also its restriction to any lower dimensional sublattice \((x + \sum _{i=1}^j{\mathbb {Z}}\cdot e_i)\) is connected, where \(x\in L_0\cdot {\mathbb {Z}}^d\), \(2\le j\le d\), and \(e_i\in {\mathbb {Z}}^d\) are pairwise orthogonal unit vectors, see Proposition 3.4. This property is crucially used in the proof of an isoperimetric inequality for subsets of fat set, but we will come to that.

Fig. 2
figure 2

This is an illustration of \({\mathcal {G}}_1\) and \({\mathcal {G}}_0 = \mathbf {G}\) in two dimensions in a box \((z_2 + [0,L_2)^2)\), for some \(z_2\in {\mathcal {G}}_2\). Here \(l_1 = 12\), \(r_1 = 3\), \(l_0 = 9\), and \(r_0 = 2\). The box \((z_2 + [0,L_2)^2)\) consists of \(12\times 12\) boxes of size \(L_1\). The left-bottom corners of those boxes (of size \(L_1\)) which are not colored belong to \({\mathcal {G}}_1\cap (z_2 + [0,L_2)^2)\), and the left-bottom corners of small white boxes (of size \(L_0\)) belong to \({\mathcal {G}}_0\cap (z_2 + [0,L_2)^2)\). Since \(4r_i<l_i\) for \(i\in \{0,1\}\), the resulting set is connected in \({\mathbb {G}}_0\)

If the renormalization scales \(L_n\) are growing fast enough, then the restriction of the fat set to the box \({\mathrm {B}}(0,R)\) serves as a coarsening of the largest connected component \({\mathcal {C}}_R\), which also ensures uniqueness of \({\mathcal {C}}_R\). We would like to reduce the isoperimetric problem for large subsets of \({\mathcal {C}}_R\) to an isoperimetric inequality for large subsets of good boxes of the fat set. The main obstruction here is that our coarsening allows to identify a subset of \({\mathcal {C}}_R\) of large volume (union of special components of good boxes), but the remaining parts of \({\mathcal {C}}_R\) may contain long dangling ends with bad isoperimetric properties. We resolve this issue by two requirements on the set of configurations that we consider. First of all, since we do not have any control of how \({\mathcal {S}}\) looks like in the “deleted” boxes of side length \(r_{n-1}L_{n-1}\), we should at least make sure that each deleted region is not too big in comparison with \(R^{\theta _{\scriptscriptstyle {\mathrm {iso}}}}\), the minimal size of sets which we consider. We require that on a level s of renormalization such that \(L_s^{3d^2}\le R^{\theta _{\scriptscriptstyle {\mathrm {iso}}}}\) [see (3.2)], all the \(L_s\)-boxes intersecting \({\mathrm {B}}(0,2R)\) are s-good, i.e., the biggest box that we “delete” has side length at most \(r_{s-1}L_{s-1}\). Second, to get a partial control of connectivities in the dangling ends, we require that any \(x,y\in {\mathcal {S}}_{L_s}\cap {\mathrm {B}}(0,2R)\) such that \(|x-y|_\infty <2L_s\) are connected in \({\mathrm {B}}(x,4L_s)\). Configurations satisfying these assumptions form an event of high probability, and next we consider only configurations from this event.

Given \(A\subset {\mathcal {C}}_R\) such that \(|A|\ge R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\), we identify a subset \({\mathbf {M}}_A\) of good boxes in the fat set for which A intersects the special connected component. If \(|{\mathbf {M}}_A|\) is small, then we show that the boundary of A is very large (\(\ge c\cdot \frac{|A|}{L_s^d}\)). The reason for this is that while most of the vertices x of A are not in special connected components of good boxes in the fat set, each of them is within distance at most \(L_s\) from the fat set (all \(L_s\)-boxes in \({\mathrm {B}}(0,2R)\) are s-good), and thus from \({\mathcal {C}}_R{\setminus } A\). The weak connectivity assumption then makes sure that there is an edge of \(\partial _{\mathcal {S}}A\) in \({\mathrm {B}}(x,4L_s)\), see Lemma 3.6. On the other hand, if \(|{\mathbf {M}}_A|\) is large, then we prove that it satisfies an isoperimetric inequality in the graph of good boxes, see Lemma 3.7. Noting that for any pair of good boxes from the boundary of \({\mathbf {M}}_A\), their special connected components are locally connected, one of the special components intersects A and the other does not, we obtain a lower bound on \(|\partial _{\mathcal {S}}A|\) in terms of the size of the boundary of \({\mathbf {M}}_A\), see (3.9).

It remains to prove the isoperimetric inequality for large subsets \(\mathbf {A}\) of good boxes of the fat set. We consider the set \({\mathbb {A}}_s\) of disjoint \(L_s\)-boxes such that at least half of \(L_0\)-boxes contained in it are from \(\mathbf {A}\), see (3.13). Again, if \(|{\mathbb {A}}_s|<C\cdot \frac{|\mathbf {A}|}{L_s^d}\), then the boundary of \(\mathbf {A}\) is at least \(c\cdot \frac{|\mathbf {A}|}{L_s^d}\). The interesting case is when \(|{\mathbb {A}}_s|\ge C\cdot \frac{|\mathbf {A}|}{L_s^d}\). By the isoperimetric inequality in the lattice of \(L_s\)-boxes, the boundary of \({\mathbb {A}}_s\) has size \(\ge c\cdot |{\mathbb {A}}_s|^{\frac{d-1}{d}} \ge c\cdot \frac{|\mathbf {A}|^{\frac{d-1}{d}}}{L_s^{d-1}}\). We, roughly speaking, estimate the boundary of \(\mathbf {A}\) from below by the part of its boundary restricted to (disjoint) \(L_s\)-boxes from the boundary of \({\mathbb {A}}_s\) and show that the restrictions to all the boxes are of size \(\ge c\cdot L_s^{d-1}\). Thus the boundary of \(\mathbf {A}\) contains \(c\cdot \frac{|\mathbf {A}|^{\frac{d-1}{d}}}{L_s^{d-1}}\) disjoint pieces of size \(c\cdot L_s^{d-1}\), and we are done.

To be precise, for any pair of adjacent \(L_s\)-boxes from the boundary of \({\mathbb {A}}_s\), one box has large intersection with \(\mathbf {A}\), and the other small. Therefore, the intersection of \(\mathbf {A}\) with a box of side length \(3L_s\) containing both \(L_s\)-boxes is comparable in size with its complement in this \(3L_s\)-box. We show that the boundary of \(\mathbf {A}\) in any such \(3L_s\)-box is at least \(c\cdot L_s^{d-1}\), see Lemma 3.8. For this we prove a stronger statement that the restriction of \(\mathbf {A}\) to j-dimensional subboxes (\(2\le j\le d\)) of a given \(3L_s\)-box which contain a non-trivial (bounded away from 0 and 1) density of vertices from \(\mathbf {A}\) satisfies an isoperimetric inequality in those subboxes, i.e., its boundary in the graph of good boxes in the j-dimensional subbox has size \(\ge c_j\cdot L_s^{j-1}\), see Lemma 3.9. The last statement is proved by induction on j.

In the case \(j=2\), we first reduce the problem to connected sets with complement consisting of large connected components, see (3.15). Then by using the precise construction of the fat set (from each n-good box we delete at most 3 boxes of side length \(r_{n-1}L_{n-1}\)), we show that the boundary of the set in the graph of good boxes has almost the same size as the boundary of the set in \(L_0\cdot {\mathbb {Z}}^2\), i.e., the part of the boundary of the set which touches some “deleted” boxes is small, see (3.17). In the case \(j\ge 3\), we use a dimension reduction argument. We first consider \((j-1)\)-dimensional subcubes (slices) of a given j-dimensional subcube which are stacked along one particular coordinate direction. If there is a positive fraction of slices which have large intersections with \(\mathbf {A}\) and its complement, then we use induction assumption for these slices. Otherwise, we conclude that there are two large disjoint subsets of slices, those that contain many vertices from \(\mathbf {A}\) and very few from its complement, and those that contain few vertices from \(\mathbf {A}\) and many from its complement (overcrowded and undercrowded slices). We then consider two-dimensional slices that intersect all these \((j-1)\)-dimensional slices, see Fig. 5. Most of them will have large intersection with \(\mathbf {A}\) as well as with its complement. We conclude by using the isoperimetric inequality in each of these two dimensional slices (case \(j=2\)).

1.3 Examples

It is well known that classical supercritical Bernoulli percolation satisfies all the requirements P1P3 and S1S2. The main focus of this paper is on models with long range correlations, especially the ones that cannot be studied by comparison with Bernoulli percolation on any scale. The following models with polynomial decay of correlations are known to satisfy all the requirements P1P3 and S1S2, see [17, Section 2]:

  1. (a)

    random interlacements at any level \(u>0\) (see [34]);

  2. (b)

    vacant set of random interlacements at level u (see [33, 34]) in the (non-empty) regime of the so-called local uniqueness;

  3. (c)

    level sets of the Gaussian free field (see [20, 31]) also in the (non-empty) regime of local uniqueness;

The regime of local uniqueness is basically described by those values of u for which S1 is fulfilled. It was shown that the regime of local uniqueness is non-empty for the vacant set of random interlacements in [16, Theorem 1.1], and for the level sets of the Gaussian free field in [17, Theorem 2.6]. In the case of Bernoulli percolation, it is well known that the regime of local uniqueness coincides with the whole supercritical phase, and, based on this, it is believed that the same is true for the models in (b) and (c). It was proved in [17] that both models satisfy all the requirements except for S1 in the whole supercritical regime. Thus, in order to extend the results of this paper to the whole supercritical phase in models (b) and (c), it suffices to check that S1 is satisfied for all supercritical values of u. Currently, this remains an open problem.

1.4 Structure of the paper

In Sect. 2, we recall the renormalization scheme of [17]. Theorem 1.2 is proved in Sect. 3 (see the beginning of the section for a detailed description of its content). In Sect. 4 we state a quenched invariance principle for random walk on the infinite percolation cluster of a random subset of \({\mathbb {Z}}^d\) satisfying a list of general conditions. We show that these conditions are implied by P1P3 and S1S2 in Sect. 5. We discuss possible weakenings of assumption P1 in Sect. 6. Last, in Sect. 1 we give a sketch proof of the general quenched invariance principle stated in Sect. 4; this is a routine adaptation of techniques present in the literature.

Finally, let us make a convention about constants. As already mentioned, we omit the dependence of various constants on a, b, and d from the notation. Dependence on other parameters is reflected in the notation, for example, as \(c(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}})\).

2 Renormalization

In this section we recall the renormalization scheme from [17, Sections 3–5]. (Some ideas are already present in [30] in the context of random interlacements and its vacant set.) The goal is to define a coarsening of \({\mathcal {S}}\) using monotone events from Definitions 2.1 and 2.2, and identify its connectivity patterns using a multi-scale renormalization with scales \(L_n\), see (2.1), (2.2), (2.3), and Lemma 2.4. The key notion is of k-good vertices (boxes), see Definition 2.3 and Lemma 2.4. The main property of a 0-good box is that it contains a unique connected component of \({\mathcal {S}}\) with largest volume, see Lemma 2.6(a), and for any pair of adjacent good boxes, their unique largest connected components are connected locally, see Lemma 2.6(b). For \(k\ge 1\), the k-good box is defined recursively such that all its \((k-1)\)-bad subboxes are contained in at most two subboxes of side length \(r_{k-1}L_{k-1}\), where \(r_{k-1}L_{k-1}\ll L_k\), see Definition 2.3.

Let \({\theta _{\scriptscriptstyle {\mathrm sc}}}= \lceil 1/{\varepsilon _{\scriptscriptstyle {\mathrm {P}}}}\rceil \), where \({\varepsilon _{\scriptscriptstyle {\mathrm {P}}}}\) is defined in P3. Let \(r_0\), \(l_0\), and \(L_0\) be positive integers. (Later in the proofs, we will assume that these integers are sufficiently large, and that the ratio \(\frac{r_0}{l_0}\) is sufficiently small, see the discussion before Sect. 3.1.) Consider the sequences of positive integers

$$\begin{aligned} l_k = l_0\cdot 4^{k^{\theta _{\scriptscriptstyle {\mathrm sc}}}},\qquad r_k = r_0\cdot 2^{k^{\theta _{\scriptscriptstyle {\mathrm sc}}}},\qquad L_k = l_{k-1}\cdot L_{k-1}, \quad k\ge 1. \end{aligned}$$
(2.1)

For \(k\ge 0\), we introduce the renormalized lattice graph \({\mathbb {G}}_k\) by

$$\begin{aligned} {\mathbb {G}}_k = L_k {\mathbb {Z}}^d = \{L_kx ~:~ x\in {\mathbb {Z}}^d\},\ \end{aligned}$$

with edges between any pair of \(\ell ^1\)-nearest neighbor vertices of \({\mathbb {G}}_k\).

Definition 2.1

For \(x\in {\mathbb {G}}_0\) and \(u\in (a,b)\), let \(A^u_x\in {\mathcal {F}}\) be the event that

  1. (a)

    for each \(e\in \{0,1\}^d\), the set \({\mathcal {S}}_{L_0}\cap (x+eL_0 + [0,L_0)^d)\) contains a connected component with at least \(\frac{3}{4} \eta (u) L_0^d\) vertices,

  2. (b)

    all of these \(2^d\) components are connected in \({\mathcal {S}}\cap (x+[0,2L_0)^d)\).

For \(u\in (a,b)\) and \(x\in {\mathbb {G}}_0\), let \(\overline{A}^u_{x,0}\) be the compelement of \(A^u_{x}\), and for \(u\in (a,b)\), \(k\ge 1\), and \(x\in {\mathbb {G}}_k\) define inductively

$$\begin{aligned} \overline{A}^u_{x,k} = \bigcup _{\begin{array}{c}\scriptscriptstyle {x_1,x_2\in {\mathbb {G}}_{k-1}\cap (x + [0,L_k)^d)} \\ \scriptscriptstyle {|x_1-x_2|_\infty \ge r_{k-1} \cdot L_{k-1}}\end{array}} \overline{A}^u_{x_1,k-1} \cap \overline{A}^u_{x_2,k-1}. \end{aligned}$$
(2.2)

Definition 2.2

For \(x\in {\mathbb {G}}_0\) and \(u\in (a,b)\), let \(B^u_x\in {\mathcal {F}}\) be the event that for all \(e\in \{0,1\}^d\),

$$\begin{aligned} \left| {\mathcal {S}}_{L_0}\cap (x+eL_0 + [0,L_0)^d)\right| \le \frac{5}{4}\eta (u) L_0^d. \end{aligned}$$

For \(u\in (a,b)\) and \(x\in {\mathbb {G}}_0\), let \(\overline{B}^u_{x,0}\) be the complement of \(B^u_{x}\), and for \(u\in (a,b)\), \(k\ge 1\), and \(x\in {\mathbb {G}}_k\) define inductively

$$\begin{aligned} \overline{B}^u_{x,k} = \bigcup _{\begin{array}{c}\scriptscriptstyle {x_1,x_2\in {\mathbb {G}}_{k-1}\cap (x + [0,L_k)^d)} \\ \scriptscriptstyle {|x_1-x_2|_\infty \ge r_{k-1} \cdot L_{k-1}}\end{array}} \overline{B}^u_{x_1,k-1} \cap \overline{B}^u_{x_2,k-1}. \end{aligned}$$
(2.3)

Definition 2.3

Let \(u\in (a,b)\). For \(k\ge 0\), we say that \(x\in {\mathbb {G}}_k\) is k-bad if the event \(\overline{A}^u_{x,k}\cup \overline{B}^u_{x,k}\) occurs. Otherwise, we say that x is k-good. Note that \(x\in {\mathbb {G}}_0\) is 0-good, if the event \(A^u_x\cap B^u_x\) occurs.

The following result is [17, Lemmas 4.2 and 4.4].

Lemma 2.4

Assume that the measures \({\mathbb {P}}^u\), \(u\in (a,b)\), satisfy conditions P1P3 and S1S2. Let \(l_k\), \(r_k\), and \(L_k\) be defined as in (2.1). For each \(u\in (a,b)\), there exist \(C = C(u)<\infty \) and \(C' = C'(u,l_0)<\infty \) such that for all \(l_0,r_0\ge C\), \(L_0\ge C'\), and \(k\ge 0\),

$$\begin{aligned} {\mathbb {P}}^u\left[ 0 \text{ is } k\text{-bad }\right] \le 2\cdot 2^{-2^k}. \end{aligned}$$

Remark 2.5

The proof of Lemma 2.4 crucially relies on conditions P2, P3, and S2, see [17]. This is the only place in the proof of Theorem 1.2 where we use these conditions. In the proof of Theorem 1.1 we use these conditions also to prove (5.2), which is a slightly stronger version of Lemma 2.4 (and its proof is essentially the same as the proof of Lemma 2.4).

The next result is [17, Lemma 5.2].

Lemma 2.6

Let \(x,y\in {\mathbb {G}}_0\) be nearest neighbors in \({\mathbb {G}}_0\) such that both are 0-good. Then

  1. (a)

    each of the graphs \({\mathcal {S}}_{L_0}\cap (z + [0,L_0)^d)\), with \(z\in \{x,y\}\), contains the unique connected component \({\mathcal {C}}_z\) with at least \(\frac{3}{4} \eta (u) L_0^d\) vertices,

  2. (b)

    \({\mathcal {C}}_x\) and \({\mathcal {C}}_y\) are connected in the graph \({\mathcal {S}}\cap ((x+[0,2L_0)^d)\cup (y+[0,2L_0)^d))\).

3 Proof of Theorem 1.2

The proof of Theorem 1.2 consists of a probabilistic part, in which we impose some restrictions on the set of allowed configurations (see Defintion 3.2) and estimate the probability of the resulting event \({\mathcal {H}}\) [see (3.7)], and a deterministic part, in which we prove the isoperimetric inequality for subsets of the largest connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,R)\) for each configuration satisfying the a priori restrictions.

We identify two special levels of the renormalization, s defined in (3.2), and \(r = \lfloor \frac{s}{2}\rfloor \). They are defined so that on the one hand \(L_s^d\ll R^{\theta _{\scriptscriptstyle {\mathrm {iso}}}}\) (which means that “deleted” subboxes are rather small), and on the other hand, the probability that a vertex is r-bad is still very small in R. We use these scales to define the event \({\mathcal {H}}\), which consists of configurations for which all the vertices from \({\mathrm {B}}(0,3R)\cap (L_r\cdot {\mathbb {Z}}^d)\) are r-good, and any \(x,y\in {\mathcal {S}}_{L_s}\cap {\mathrm {B}}(0,2R)\) such that \(|x-y|_\infty \le 2L_s\) are connected in \({\mathrm {B}}(x,4L_s)\), see Defintion 3.2. Using Lemma 2.4 and assumption S1 we show that the probability of \({\mathcal {H}}\) is close to 1, see (3.7). (The event \({\mathcal {H}}\) depends on d, u, \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}\), and R, but we do not reflect this in the notation.)

Next, using combinatorics we show that any configuration from \({\mathcal {H}}\) belongs to the event in (1.3). This is done in several steps. First, using the notion of k-good vertices from Definition 2.3, we identify for each configuration in \({\mathcal {H}}\) a well structured connected (in \({\mathbb {G}}_0 = L_0\cdot {\mathbb {Z}}^d\)) subset \(\mathbf {G}\) of 0-good vertices in \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\), obtained from \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\) by a certain multi-scale perforation procedure. The set \(\mathbf {G}\) consists roughly of those 0-good vertices in \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\) which are contained only in j-good boxes for all \(1\le j\le r\). (We need to sparsen this set a bit more in order to obtain the actual set \(\mathbf {G}\) with the desired connectivity properties.) This set is well connected, ubiquitous in \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\), and has almost the same volume as \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\), see Proposition 3.4. A crucial step in the proof is a reduction of the initial isoperimetric problem for subsets of the largest cluster of \({\mathcal {S}}\) in \({\mathrm {B}}(0,R)\) to an isoperimetric problem for large enough subsets of \(\mathbf {G}\), see Lemmas 3.6 and 3.7. The rest of the proof is then about isoperimetric properties of large subsets \(\mathbf {A}\) of \(\mathbf {G}\), see Lemma 3.7. If \(\mathbf {A}\) is sparse, then its boundary is almost comparable with the volume of \(\mathbf {A}\). The most delicate case is when \(\mathbf {A}\) is localized, since in this case its boundary may be much smaller than its volume. In this case, we estimate the boundary of \(\mathbf {A}\) locally in each of the boxes of side length \(3L_s\) which are densely occupied by \(\mathbf {A}\) and by its complement, see Lemma 3.8. We show that in each of such boxes, the boundary of \(\mathbf {A}\) is at least \(c\cdot L_s^{d-1}\). For that we prove a stronger statement that the restriction of \(\mathbf {A}\) to (many) j-dimensional hyperplanes (\(2\le j\le d\)) intersecting the given box of side length \(3L_s\) has boundary \(\ge c_j\cdot L_s^{j-1}\), see Lemma 3.9. This proof is by induction on j.

In the proof of Theorem 1.2 we will work with the scales \(L_k\), \(l_k\), and \(r_k\) defined in (2.1). Throughout the proof we take \(r_0\), \(l_0\), and \(L_0\) satisfying Lemma 2.4. We need to adjust these parameters further in the proof as follows:

  • In the construction of \(\mathbf {G}\), we assume that \(4r_0<l_0\), which is essential for the connectedness of \(\mathbf {G}\).

  • In showing that the largest (in volume) connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,2R)\) is uniquely defined, we assume that \(L_0\) is large enough and \(\frac{r_0}{l_0}\) is small enough (both depending on u) to satisfy (3.8).

  • We choose \(\frac{r_0}{l_0}\) small enough to satisfy Lemmas 3.6 and 3.7.

Most of the conditions on the smallness of \(\frac{r_0}{l_0}\) are formulated in terms of the closeness to 1 of

$$\begin{aligned} f_j\left( \frac{r_0}{l_0}\right) = \prod _{i=0}^\infty \left( 1 - 3\cdot \left( \frac{r_i}{l_i}\right) ^j\right) . \end{aligned}$$
(3.1)

[See, (3.12), (3.14), and (3.22).] The only exception is (3.18).

The reader may notice that in Lemma 3.9 we choose the ratio \(\frac{r_0}{l_0}\) small enough depending on a parameter \(\epsilon \), see (3.14) and (3.22). This is fine, since in the end we only use Lemma 3.9 for a specific choice of \(\epsilon = \frac{1}{2\cdot 3^d}\).

3.1 The event \({\mathcal {H}}\) and its probability

In this section we define the event \({\mathcal {H}}\) containing all the restrictions on the set of allowed configurations (see Definition 3.2), and show that it has probability close to 1 [see (3.7)]. The event \({\mathcal {H}}\) depends on d, u, \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}\), and R, but we do not reflect this in the notation.

Recall the definition of \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}\) from the statement of Theorem 1.2, and note that it suffices to assume that \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}\in (0,1)\).

Let s be the largest integer such that \(L_s^{3d^2} \le R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\), i.e.,

$$\begin{aligned} s=\max \left\{ \, s' \; : \; L_{s'}^{3d^2} \le R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}} \right\} .\ \end{aligned}$$
(3.2)

We assume that \(R\ge L_0^{3d^2/{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\), so that s is well-defined.

Let \(r = \lfloor \frac{s}{2}\rfloor \). By (2.1),

$$\begin{aligned} L_r^2\le L_0\cdot L_s. \end{aligned}$$
(3.3)

Remark 3.1

  1. 1.

    We need to choose the power of \(L_{s'}\) in (3.2) large enough, only in order to deduce Theorem 1.2 from Lemmas 3.6 and 3.7. In fact, any exponent bigger than \(3d^2\) would also do, with a suitable change of constants in (3.4) and (3.5).

  2. 2.

    Property (3.3) will be crucial in the proof of the isoperimetric inequality for two dimensional slices, see Lemma 3.9 and the proof of (3.15).

By (2.1) and (3.2), for all \(R\ge L_0^{3d^2/{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\),

$$\begin{aligned} R^{\frac{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}{3d^2}} \le L_{s+1} = l_s\cdot L_s \le l_0\cdot 4 \cdot (L_s)^{1+2^{\theta _{\scriptscriptstyle {\mathrm sc}}}}, \end{aligned}$$

which implies that

$$\begin{aligned} L_s \ge \frac{1}{4l_0} R^{\frac{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}{3d^2(1+2^{\theta _{\scriptscriptstyle {\mathrm sc}}})}}. \end{aligned}$$
(3.4)

From (2.1) and (3.4) we deduce that there exists \(c = c({\theta _{\scriptscriptstyle {\mathrm {iso}}}}, {\theta _{\scriptscriptstyle {\mathrm sc}}}, l_0, L_0)>0\) such that for all \(R\ge L_0^{3d^2/{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\),

$$\begin{aligned} s\ge c\cdot (\log R)^{\frac{1}{1+{\theta _{\scriptscriptstyle {\mathrm sc}}}}} - 1. \end{aligned}$$
(3.5)

Next we define the event \({\mathcal {H}}\).

Definition 3.2

Let \(u\in (a,b)\). Consider the event \({\mathcal {H}}\) that

  1. (a)

    each \(z\in {\mathbb {G}}_r\cap {\mathrm {B}}(0,3R)\) is r-good,

  2. (b)

    for any \(z\in {\mathrm {B}}(0,2R)\) and \(x,y\in {\mathcal {S}}_{L_s}\cap (z+[-2L_s,2L_s)^d)\), x is connected to y in \({\mathcal {S}}\cap (z+[-4L_s,4L_s)^d)\).

Remark 3.3

Property (a) in the definition of \({\mathcal {H}}\) implies the weaker version (a’) stating that each \(z\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R)\) is s-good. Most of the arguments in the proof of Theorem 1.2 would go through if we used (a’) instead of (a) in the definition of \({\mathcal {H}}\). The only point where we essentially use (a) is in the proof of the two dimesional case, see Lemma 3.9 and the proof of (3.15).

By Definition 2.3, S1, and Lemma 2.4, there exists \(C = C(u)<\infty \) such that

$$\begin{aligned} {\mathbb {P}}^u\left[ {\mathcal {H}}^c \right] \le |{\mathbb {G}}_r\cap {\mathrm {B}}(0,3R)|\cdot 2\cdot 2^{-2^r} + |{\mathrm {B}}(0,2R)|\cdot C\cdot e^{-f_{\scriptscriptstyle {\mathrm {S}}}(u,2L_s)}. \end{aligned}$$

Using (1.1), (3.4), and (3.5), we deduce that there exist \(c' = c'(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}},{\theta _{\scriptscriptstyle {\mathrm sc}}})>0\) and \(C' = C'(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}}, {\theta _{\scriptscriptstyle {\mathrm sc}}}, l_0, L_0)<\infty \) such that for all \(R\ge C'\),

$$\begin{aligned} 2^r \ge (\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}} \quad \text{ and }\quad f_{\scriptscriptstyle {\mathrm {S}}}(u,2L_s) \ge c'(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}} ~.\ \end{aligned}$$
(3.6)

Note that also \(|{\mathbb {G}}_r\cap {\mathrm {B}}(0,3R)| \le |{\mathrm {B}}(0,3R)|\le (6R+1)^d\). Therefore, there exist \(c=c(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}},{\theta _{\scriptscriptstyle {\mathrm sc}}})>0\) and \(C = C(u,{\theta _{\scriptscriptstyle {\mathrm {iso}}}},{\theta _{\scriptscriptstyle {\mathrm sc}}},l_0,L_0)<\infty \) such that for all \(R\ge C\),

$$\begin{aligned} {\mathbb {P}}^u\left[ {\mathcal {H}} \right] \ge 1 - Ce^{-c(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}. \end{aligned}$$
(3.7)

In the remaining part of the proof, we will show that each configuration from \({\mathcal {H}}\) also belongs to the event in (1.3). Together with (3.7) this will imply Theorem 1.2.

$$\begin{aligned} \text {From now on we assume that } {\mathcal {H}} \text { occurs}. \end{aligned}$$

3.2 Construction of \(\mathbf {G}\)

In this section we construct the subset \(\mathbf {G}\) of 0-good vertices in \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R)\) with the property that for every \(z_0\in \mathbf {G}\) and each of the boxes \((z_j+[0,L_j)^d)\), \(z_j\in {\mathbb {G}}_j\), \(0<j\le s\), containing \(z_0\), the vertex \(z_j\) is j-good, and also that the set \(\mathbf {G}\) exhibits good properties of density and connectedness, see Proposition 3.4. The construction is done recursively by going down through the renormalization levels and using Definition 2.3. We assume throughout the construction that \(4r_0<l_0\) (which implies that \(4r_i<l_i\) for all i). This is essential for the connectedness of the sets below.

Let \({\mathcal {G}}_s={\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-2L_s)\). By the definition of \({\mathcal {H}}\) and Remark 3.3, all \(z_s\in {\mathcal {G}}_s\) are s-good. Also note that \(\cup _{z_{s}\in {\mathcal {G}}_{s}}(z_{s} + [0,L_{s})^d)\subset {\mathrm {B}}(0,2R-L_s)\).

For \(r\le i<s\), let \({\mathcal {G}}_i = \cup _{z_{i+1}\in {\mathcal {G}}_{i+1}}({\mathbb {G}}_i\cap (z_{i+1} + [0,L_{i+1})^d)\). By the definition of \({\mathcal {H}}\), all \(z_r\in {\mathcal {G}}_r\) are r-good.

Next we take \(0<i\le r\) and assume that \({\mathcal {G}}_i\subset {\mathbb {G}}_i\) is defined so that

  • all \(z_{i}\in {\mathcal {G}}_{i}\) are i-good,

  • for any \(z_s\in {\mathcal {G}}_s\), the set \({\mathcal {G}}_i\cap (z_s + [0,L_s)^d)\) is connected in \({\mathbb {G}}_i\) and

    $$\begin{aligned} \left| {\mathcal {G}}_{i}\cap (z_s + [0,L_s)^d)\right| \ge \left| {\mathcal {G}}_{i+1}\cap (z_s + [0,L_s)^d)\right| \cdot l_{i}^d\cdot \left( 1 - 3\left( \frac{r_{i}}{l_{i}}\right) ^d\right) , \end{aligned}$$
  • for any \(z_s,\widetilde{z}_s\in {\mathcal {G}}_s\) with \(|z_s - \widetilde{z}_s|_1 = L_s\), the set \({\mathcal {G}}_i\cap ((z_s + [0,L_s)^d)\cup (\widetilde{z}_s + [0,L_s)^d))\) is connected in \({\mathbb {G}}_i\),

  • for any \(z_s\in {\mathcal {G}}_s\), \(x_i\in {\mathbb {G}}_i\cap (z_s + [0,L_s)^d)\), and two orthogonal \(e,e'\in {\mathbb {Z}}^d\) with \(|e|_1 = |e'|_1 = 1\), the two dimensional slice \({\mathcal {G}}_{i}\cap (x_i + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\cap (z_s + [0,L_s)^d)\) is connected in \({\mathbb {G}}_i\cap (x_i + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\), and

    $$\begin{aligned}&\left| {\mathcal {G}}_i\cap (x_i + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\cap (z_s + [0,L_s)^d)\right| \\&\quad \ge \left| {\mathcal {G}}_{i+1}\cap (x_{i+1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\cap (z_s + [0,L_s)^d)\right| \cdot l_{i}^2\cdot \left( 1 - 3\left( \frac{r_{i}}{l_{i}}\right) ^2\right) , \end{aligned}$$

    where \(x_{i+1}\in {\mathbb {G}}_{i+1}\) satisfies \(x_i\in (x_{i+1} + [0,L_{i+1})^d)\),

  • for any \(z_s,\widetilde{z}_s\in {\mathcal {G}}_s\) with \(|z_s - \widetilde{z}_s|_1 = L_s\), \(x_i\in {\mathbb {G}}_i\cap (z_s + [0,L_s)^d)\), \(e = \frac{z_s - \widetilde{z}_s}{L_s}\), and \(e'\in {\mathbb {Z}}^d\) orthogonal to e such that \(|e'|_1 = 1\), the two dimensional slice \({\mathcal {G}}_{i}\cap (x_i + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\cap ((z_s + [0,L_s)^d)\cup (\widetilde{z}_s + [0,L_s)^d))\) is connected in \({\mathbb {G}}_i\cap (x_i + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\).

We now define \({\mathcal {G}}_{i-1}\subset {\mathbb {G}}_{i-1}\) which satisfies the same properties as \({\mathcal {G}}_i\) with i replaced everywhere by \((i-1)\). By Definition 2.3, for each \(z_i\in {\mathcal {G}}_i\), there exist \(a_{z_i},b_{z_i}\in {\mathbb {G}}_{i-1}\cap (z_i + [0,L_i)^d)\) such that all the vertices in \({\mathcal {G}}_{z_i}' = ({\mathbb {G}}_{i-1}\cap (z_i + [0,L_i)^d)){\setminus }((a_{z_i}+[0,r_{i-1}L_{i-1})^d) \cup (b_{z_i}+[0,r_{i-1}L_{i-1})^d))\) are \((i-1)\)-good. Note that the set \({\mathcal {G}}_{z_i}'\) is connected in \({\mathbb {G}}_{i-1}\) if \(d\ge 3\) (and \(4r_{i-1}<l_{i-1}\)), but not necessarily connected if \(d=2\). However, for each \(z_i\in {\mathcal {G}}_i\), there exists \(c_{z_i}\in {\mathbb {G}}_{i-1}\cap (z_i + [0,L_i)^d)\) (see Fig. 1) such that the sets \({\mathcal {G}}_{z_i}'' = {\mathcal {G}}_{z_i}'{\setminus }(c_{z_i}+[0,r_{i-1}L_{i-1})^d)\), \(z_i\in {\mathcal {G}}_i\), satisfy the following properties:

  • for any \(z_i\in {\mathcal {G}}_i\), all \(z_{i-1}\in {\mathcal {G}}_{z_i}''\) are \((i-1)\)-good,

  • for any \(z_i\in {\mathcal {G}}_i\), the set \({\mathcal {G}}_{z_i}''\) is connected in \({\mathbb {G}}_{i-1}\) and \(|{\mathcal {G}}_{z_i}''| \ge l_{i-1}^d\cdot \left( 1 - 3\left( \frac{r_{i-1}}{l_{i-1}}\right) ^d\right) \),

  • for any \(z_i,\widetilde{z}_i\in {\mathcal {G}}_i\) with \(|z_i - \widetilde{z}_i|_1 = L_i\), the set \({\mathcal {G}}_{z_i}''\cup {\mathcal {G}}_{\widetilde{z}_i}''\) is connected in \({\mathbb {G}}_{i-1}\),

  • for any \(z_i\in {\mathcal {G}}_i\), \(x_{i-1}\in {\mathbb {G}}_{i-1}\cap (z_i + [0,L_i)^d)\), and two orthogonal \(e,e'\in {\mathbb {Z}}^d\) with \(|e|_1 = |e'|_1 = 1\), the two dimensional slice \({\mathcal {G}}_{z_i}''\cap (x_{i-1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\) is connected in \({\mathbb {G}}_{i-1}\cap (x_{i-1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\), and \(\left| {\mathcal {G}}_{z_i}''\cap (x_{i-1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\right| \ge l_{i-1}^2\cdot \left( 1 - 3\left( \frac{r_{i-1}}{l_{i-1}}\right) ^2\right) \),

  • for any \(z_i,\widetilde{z}_i\in {\mathcal {G}}_i\) with \(|z_i - \widetilde{z}_i|_1 = L_i\), \(x_{i-1}\in {\mathbb {G}}_{i-1}\cap (z_i + [0,L_i)^d)\), \(e = \frac{z_i - \widetilde{z}_i}{L_i}\), and \(e'\in {\mathbb {Z}}^d\) orthogonal to e such that \(|e'|_1 = 1\), the two dimensional slice \(({\mathcal {G}}_{z_i}''\cup {\mathcal {G}}_{\widetilde{z}_i}'')\cap (x_{i-1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\) is connected in \({\mathbb {G}}_{i-1}\cap (x_{i-1} + {\mathbb {Z}}\cdot e + {\mathbb {Z}}\cdot e')\).

We define \({\mathcal {G}}_{i-1} = \cup _{z_i\in {\mathcal {G}}_i}{\mathcal {G}}_{z_i}''\subset {\mathbb {G}}_{i-1}\). From the above properties of \(({\mathcal {G}}_{z_i}'')_{z_i\in {\mathcal {G}}_i}\), one can see that \({\mathcal {G}}_{i-1}\) satisfies the same properties as \({\mathcal {G}}_i\) with i replaced everywhere by \((i-1)\).

The outcome of such a recursive procedure is the set \({\mathcal {G}}_0\subset \cup _{z_s\in {\mathcal {G}}_s}(z_s + [0,L_s)^d)\subset {\mathbb {G}}_0\cap {\mathrm {B}}(0,2R-L_s)\) on the 0-level of the renormalization. We denote it by \(\mathbf {G}\). (See an illustration of part of \(\mathbf {G}\) on Fig. 2.) Note that \(\mathbf {G}\) satisfies all the properties of \({\mathcal {G}}_i\) listed above with i replaced everywhere by 0. For the ease of references, we summarize most of the properties of \(\mathbf {G}\) used in the proof of Theorem 1.2 in Proposition 3.4 (see also Remark 3.5). These properties follow from the construction.

Proposition 3.4

The set \(\mathbf {G}\subset {\mathbb {G}}_0\cap {\mathrm {B}}(0,2R-L_s)\) constructed above satisfies the following properties [with \(f_j\) defined in (3.1)]:

  1. (a)

    any \(z\in \mathbf {G}\) is 0-good,

  2. (b)

    for any \(x_s\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-2L_s)\), the set \(\mathbf {G}\cap (x_s + [0,L_s)^d)\) is connected in \({\mathbb {G}}_0\) and \(\left| \mathbf {G}\cap (x_s + [0,L_s)^d)\right| \ge \left( \frac{L_s}{L_0}\right) ^d\cdot f_d\left( \frac{r_0}{l_0}\right) \),

  3. (c)

    for any \(x_s\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\), \(x_0\in {\mathbb {G}}_0\cap (x_s+[-L_s,2L_s)^d)\), \(2\le j\le d\), and pairwise orthogonal \(e_1,\dots ,e_j\in {\mathbb {Z}}^d\) with \(|e_i|_1 = 1\), the j-dimensional slice \(\mathbf {G}\cap (x_s + [-L_s,2L_s)^d)\cap (x_0 + \sum _{i=1}^j{\mathbb {Z}}\cdot e_i)\) is connected in \({\mathbb {G}}_0\) and \(\left| \mathbf {G}\cap (x_s + [-L_s,2L_s)^d)\cap (x_0 + \sum _{i=1}^j{\mathbb {Z}}\cdot e_i)\right| \ge \left( \frac{3L_s}{L_0}\right) ^j\cdot f_j\left( \frac{r_0}{l_0}\right) \).

Remark 3.5

Most part of the proof of Theorem 1.2 relies on the properties of \(\mathbf {G}\) listed in Proposition 3.4, and not on the specifics of the construction of \(\mathbf {G}\). The only exception is the proof of the isoperimetric inequality for two dimensional slices, where we need to use the definition of all \({\mathcal {G}}_i\)’s, see Lemma 3.9 and especially the proof of (3.17).

In what follows, we will use ordinary font to denote subsets of \({\mathcal {S}}\) (e.g., A and \({D_A}\)), bold font for subsets of \(\mathbf {G}\) (e.g., \(\mathbf {A}\), \({\mathbf {M}}_A\), \(\mathbf {a}\), \(\mathbf {g}\), etc.), and blackboard bold for subsets of \({\mathbb {G}}_s\) (e.g., \({\mathbb {A}}_s\)).

3.3 Reduction of Theorem 1.2 to isoperimetry in \(\mathbf {G}\)

In this section we show how the initial isoperimetric problem for large subsets of \({\mathcal {C}}_R\) can be reduced to an isoperimetric problem for large subsets of \(\mathbf {G}\). We first show that the set \(\mathbf {G}\) can be viewed as a coarsening of the largest connected subset \({\mathcal {C}}_{2R}\) of \({\mathcal {S}}\cap {\mathrm {B}}(0,2R)\), in particular, that \({\mathcal {C}}_{2R}\) and \({\mathcal {C}}_R\) are uniquely defined for any configuration in \({\mathcal {H}}\) under a mild tuning of the renormalization scales, see (3.8). The key reduction step is formalized in Lemmas 3.6 and 3.7. We finish this section with the proof of Theorem 1.2 given the results of the lemmas, and prove the lemmas in later sections.

We will first show that the largest (in volume) connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,2R)\) is uniquely defined, and that the set \(\mathbf {G}\) can be viewed as its coarsening on the scale \(L_0\).

Recall that \(\mathbf {G}\subset {\mathbb {G}}_0\cap \cup _{z_s\in {\mathcal {G}}_s}(z_s + [0,L_s)^d)\subset {\mathbb {G}}_0\cap {\mathrm {B}}(0,2R-L_s)\). By the definition of a 0-good vertex and Lemma 2.6, each of the boxes \(x + [0,L_0)^d\), \(x\in \mathbf {G}\), contains a unique connected subset \({\mathcal {C}}_x\) of \({\mathcal {S}}\) of size \(\ge \frac{3}{4} \eta (u)L_0^d\), and all these sets are connected in \((\mathbf {G} + [0,2L_0)^d)\subset {\mathrm {B}}(0,2R)\). Therefore, all the \({\mathcal {C}}_x\), \(x\in \mathbf {G}\), are part of the same connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,2R)\), which has size

$$\begin{aligned} \ge \left| \bigcup _{x\in \mathbf {G}}{\mathcal {C}}_x\right| \ge \frac{3}{4}\eta (u)L_0^d\cdot |\mathbf {G}|. \end{aligned}$$

On the other hand, by the definition of a 0-good vertex and Lemma 2.6, each of the boxes \(x + [0,L_0)^d\), \(x\in \mathbf {G}\), contains \(\le \frac{5}{4} \eta (u)L_0^d\) vertices from \({\mathcal {S}}_{L_0}\). Since in addition by Proposition 3.4(b),

$$\begin{aligned} |\mathbf {G}|\ge & {} \left| {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-2L_s)\right| \cdot f_d\left( \frac{r_0}{l_0}\right) \cdot \left( \frac{L_s}{L_0}\right) ^d\\\ge & {} \frac{|{\mathrm {B}}(0,2R-2L_s)|}{L_s^d}\cdot f_d\left( \frac{r_0}{l_0}\right) \cdot \left( \frac{L_s}{L_0}\right) ^d, \end{aligned}$$

it follows that

$$\begin{aligned} |{\mathcal {S}}_{L_0}\cap {\mathrm {B}}(0,2R)|\le & {} \frac{5}{4}\eta (u)L_0^d \cdot |\mathbf {G}| + \left( |{\mathrm {B}}(0,2R)| - L_0^d\cdot |\mathbf {G}|\right) \\\le & {} L_0^d\cdot |\mathbf {G}|\cdot \left( \frac{5}{4}\eta (u) + \frac{|{\mathrm {B}}(0,2R)|}{|{\mathrm {B}}(0,2R-2L_s)|}\cdot f_d\left( \frac{r_0}{l_0}\right) ^{-1} - 1 \right) \\\le & {} L_0^d\cdot |\mathbf {G}|\cdot \left( \frac{5}{4}\eta (u) + \left( 1 - \frac{L_s}{R}\right) ^{-d}\cdot f_d\left( \frac{r_0}{l_0}\right) ^{-1} - 1 \right) . \end{aligned}$$

Since we assume that \({\theta _{\scriptscriptstyle {\mathrm {iso}}}}<1\), it follows from (3.2) that \(\frac{L_s}{R} < \frac{1}{L_s}\le \frac{1}{L_0}\). Therefore, there exist \(C = C(u)<\infty \) and \(\rho = \rho (u)>0\) such that for all \(L_0>C(u)\) and for all choices of the ratio \(\frac{r_0}{l_0}<\rho (u)\),

$$\begin{aligned} \left( 1 - \frac{1}{L_0}\right) ^{-d}\cdot f_d\left( \frac{r_0}{l_0}\right) ^{-1} - 1 < \frac{1}{4}\eta (u). \end{aligned}$$
(3.8)

With such a choice of \(L_0\), \(r_0\), and \(l_0\), the largest (in volume) connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,2R)\) is uniquely defined and

$$\begin{aligned} {\mathcal {C}}_{2R}\text { contains }{\mathcal {C}}_x,\text { for all }x\in \mathbf {G}. \end{aligned}$$

Similar reasoning together with the above conclusion imply that \({\mathcal {C}}_{2R}\) contains \({\mathcal {C}}_R\).

For any subset A of \({\mathcal {S}}\), we denote by \(\partial _{\mathcal {S}}A\) the edge boundary of A in \({\mathcal {S}}\), i.e., the set of edges from \({\mathbb {Z}}^d\) with one end-vertex in A and the other in \({\mathcal {S}}{\setminus } A\). Similarly, for any subset \(\mathbf {A}\) of \(\mathbf {G}\), we denote by \(\partial _{\mathbf {G}} \mathbf {A}\), the boundary of \(\mathbf {A}\) in \(\mathbf {G}\), i.e., those pairs of vertices in \({\mathbb {G}}_0\) which are at \(\ell ^1\)-distance \(L_0\) (in \({\mathbb {Z}}^d\)) from each other, one of them is in \(\mathbf {A}\) and the other in \(\mathbf {G}{\setminus } \mathbf {A}\).

The next two lemmas allow to reduce the initial isoperimetric problem for subsets of \({\mathcal {C}}_R\) to an isoperimetric problem for subsets of \(\mathbf {G}\). Recall the definition of \({\mathcal {C}}_x\) from Lemma 2.6(a).

Lemma 3.6

Let A be a subset of \({\mathcal {C}}_R\). Let \({\mathbf {M}}_A\) be the set of all \(x\in \mathbf {G}\) such that \({\mathcal {C}}_x\cap A \ne \emptyset \), and denote by \({D_A}\) the set of \(x\in A\) such that there exists \(y\in {\mathcal {C}}_{2R}{\setminus } A\) with \(|x-y|_\infty \le 2L_s\). Then

$$\begin{aligned} |\partial _{\mathcal {S}}A| \ge \max \left\{ \frac{1}{d\cdot 2^d}\cdot |\partial _{\mathbf {G}}{\mathbf {M}}_A|,\frac{|{D_A}|}{(11\cdot L_s)^d} \right\} ,\ \end{aligned}$$
(3.9)

and there exists \(\rho _{\scriptscriptstyle 3.6}>0\) such that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.6}\) then

$$\begin{aligned} |A| \le 6^d\cdot L_0^d\cdot |{\mathbf {M}}_A| + |{D_A}|. \end{aligned}$$
(3.10)

Lemma 3.7

There exist \(\gamma _{\scriptscriptstyle 3.7} >0\) and \(\rho _{\scriptscriptstyle 3.7}>0\) such that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.7}\), then for any \(\mathbf {A}\subset \mathbf {G}\cap {\mathrm {B}}(0,2R-4L_s)\) such that \(|\mathbf {A}|\ge 7^{-d}\cdot (\frac{L_s}{L_0})^{2d^2}\), we have \(|\partial _{\mathbf {G}} \mathbf {A}|\ge \gamma _{\scriptscriptstyle 3.7}\cdot |\mathbf {A}|^{\frac{d-1}{d}}\).

Now we finish the proof of Theorem 1.2 using the two lemmas. We prove Lemma 3.6 in Sect. 3.4 and Lemma 3.7 in Sect. 3.5.

Proof of Theorem 1.2

We take \(L_0\), \(l_0\), and \(r_0\) as in (2.1) satisfying the statements of Lemmas 2.4, 3.6, and 3.7, and also (3.8). We also assume that \(4r_0<l_0\) and \(5L_s<R\). It suffices to show that the event \({\mathcal {H}}\) implies the event in (1.3).

Fix a subset \(A\subset {\mathcal {C}}_R\) such that \(|A|\ge R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\), and define \({\mathbf {M}}_A\) and \({D_A}\) as in the statement of Lemma 3.6. Note that \({\mathbf {M}}_A\subset \mathbf {G}\cap {\mathrm {B}}(0,R+L_0)\subset \mathbf {G}\cap {\mathrm {B}}(0,2R-4L_s)\) if \(5L_s<R\). First, we claim that

$$\begin{aligned} \max \left\{ |\partial _{\mathbf {G}}{\mathbf {M}}_A|,\frac{|{D_A}|}{L_s^d} \right\} \ge \min (1,\gamma _{\scriptscriptstyle 3.7})\cdot \max \left\{ |{\mathbf {M}}_A|^{\frac{d-1}{d}},\frac{|{D_A}|}{L_s^d} \right\} . \end{aligned}$$
(3.11)

Indeed, if \(|{\mathbf {M}}_A|^{\frac{d-1}{d}} < \frac{|{D_A}|}{L_s^d}\), then (3.11) trivially holds. On the other hand, if \(|{\mathbf {M}}_A|^{\frac{d-1}{d}} \ge \frac{|{D_A}|}{L_s^d}\), then using (3.10) we get

$$\begin{aligned} |A| \le 6^d\cdot L_0^d\cdot |{\mathbf {M}}_A| + L_s^d\cdot |{\mathbf {M}}_A|^{\frac{d-1}{d}} \le 7^d\cdot L_s^d\cdot |{\mathbf {M}}_A|. \end{aligned}$$

By the assumption on |A| and using (3.2), we have that \(|A|\ge R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\ge L_s^{3d^2}\). Hence \(|{\mathbf {M}}_A|\ge 7^{-d}\cdot L_s^{2d^2}\ge 7^{-d}\cdot (\frac{L_s}{L_0})^{2d^2}\), and (3.11) follows from Lemma 3.7 applied to \(\mathbf {A} = {\mathbf {M}}_A\).

It follows from (3.9), (3.10), and (3.11) that

$$\begin{aligned} \frac{|\partial _{\mathcal {S}}A|}{|A|^{\frac{d-1}{d}}}\ge & {} \frac{\frac{1}{d\cdot 11^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})\cdot \max \left\{ |{\mathbf {M}}_A|^{\frac{d-1}{d}},\frac{|{D_A}|}{L_s^d} \right\} }{(6^d\cdot L_0^d\cdot |{\mathbf {M}}_A| + |{D_A}|)^{\frac{d-1}{d}}}\\\ge & {} \frac{\frac{1}{d\cdot 11^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})\cdot \max \left\{ |{\mathbf {M}}_A|^{\frac{d-1}{d}},\frac{|{D_A}|}{L_s^d} \right\} }{6^{d-1}\cdot L_0^{d-1}\cdot |{\mathbf {M}}_A|^{\frac{d-1}{d}} + |{D_A}|^{\frac{d-1}{d}}}. \end{aligned}$$

On the one hand, if \(L_0^d\cdot |{\mathbf {M}}_A| \ge |{D_A}|\), then

$$\begin{aligned} \frac{|\partial _{\mathcal {S}}A|}{|A|^{\frac{d-1}{d}}} \ge \frac{\frac{1}{d\cdot 11^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})\cdot |{\mathbf {M}}_A|^{\frac{d-1}{d}}}{6^{d-1}\cdot L_0^{d-1}\cdot |{\mathbf {M}}_A|^{\frac{d-1}{d}} + |{D_A}|^{\frac{d-1}{d}}} \ge \frac{\frac{1}{d\cdot 11^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})}{(7\cdot L_0)^{d-1}}. \end{aligned}$$

On the other hand, if \(L_0^d\cdot |{\mathbf {M}}_A| \le |{D_A}|\), then by (3.10) and (3.2), \(|{D_A}| \ge 7^{-d}\cdot |A| \ge 7^{-d}\cdot R^{{\theta _{\scriptscriptstyle {\mathrm {iso}}}}}\ge 7^{-d}\cdot L_s^{3d^2}\), and we get

$$\begin{aligned} \frac{|\partial _{\mathcal {S}}A|}{|A|^{\frac{d-1}{d}}} \ge \frac{\frac{1}{d\cdot 11^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})\cdot |{D_A}|^{\frac{1}{d}}}{7^{d-1}\cdot L_s^d} \ge \frac{1}{d\cdot 77^d}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7}). \end{aligned}$$

This finishes the proof of Theorem 1.2 with the choice of \(\gamma _{\scriptscriptstyle 1.2} = \frac{1}{d\cdot 77^d\cdot L_0^{d-1}}\cdot \min (1,\gamma _{\scriptscriptstyle 3.7})\). \(\square \)

3.4 Proof of Lemma 3.6

The proof of both (3.9) and (3.10) goes by constructing certain mappings from \(\partial _{\mathbf {G}}{\mathbf {M}}_A\) to \(\partial _{\mathcal {S}}A\) (a \(d\cdot 2^d\) to 1 map), from \({D_A}\) to \(\partial _{\mathcal {S}}A\) (a \((11\cdot L_s)^d\) to 1 map), and from \(A{\setminus }{D_A}\) to \({\mathbf {M}}_A\) (a \((6\cdot L_0)^d\) to 1 map).

Recall the definition of \({\mathcal {C}}_x\) from Lemma 2.6(a).

Proof of (3.9)

Note that for any \(x\in {\mathbf {M}}_A\) and \(y\in \mathbf {G}{\setminus } {\mathbf {M}}_A\) such that \(|x-y|_1 = L_0\), \({\mathcal {C}}_x\cap A\ne \emptyset \) and \({\mathcal {C}}_y \subset {\mathcal {C}}_{2R} {\setminus } A\). By Lemma 2.6, \({\mathcal {C}}_x\) and \({\mathcal {C}}_y\) are connected in \({\mathcal {S}}\cap ((x+[0,2L_0)^d)\cup (y+[0,2L_0)^d))\). Each path in \({\mathcal {S}}\) connecting \({\mathcal {C}}_x\cap A\) and \({\mathcal {C}}_y\) contains an edge from \(\partial _{{\mathcal {S}}} A\). This implies that

$$\begin{aligned} |\partial _{\mathcal {S}}A| \ge \frac{1}{d\cdot 2^d}\cdot |\partial _{\mathbf {G}}{\mathbf {M}}_A|,\ \end{aligned}$$

where the constant \(\frac{1}{d\cdot 2^d}\) takes care for overcounting. We next show that

$$\begin{aligned} |\partial _{\mathcal {S}}A| \ge \frac{|{D_A}|}{(11\cdot L_s)^d}. \end{aligned}$$

Indeed, by the definition of \({D_A}\), for any \(x\in {D_A}\), there exists \(y\in {\mathcal {C}}_{2R}{\setminus } A\) such that \(|x-y|_\infty \le 2L_s\). By the second part of the definition of \({\mathcal {H}}\), we conclude that any such x and y are connected by a path in \({\mathcal {S}}\cap {\mathrm {B}}(x,4L_s)\). Since \(x\in A\) and \(y\notin A\), this path necessarily contains an edge from \(\partial _{\mathcal {S}}A\). This implies that

$$\begin{aligned} |{D_A}| \le |\{(x,e)~:~x\in {D_A}, e\in \partial _{\mathcal {S}}A\cap {\mathrm {B}}(x,4L_s)\}|\le |{\mathrm {B}}(0,4L_s+1)|\cdot |\partial _{\mathcal {S}}A|, \end{aligned}$$

and the claim follows. \(\square \)

Proof of (3.10)

We need to show that \(|A{\setminus }{D_A}| \le 6^d\cdot L_0^d\cdot |{\mathbf {M}}_A|\).

We choose \(\rho _{\scriptscriptstyle 3.6}>0\) such that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.6}\) then

$$\begin{aligned} f_d\left( \frac{r_0}{l_0}\right) >\frac{1}{2}. \end{aligned}$$
(3.12)

Then, by Proposition 3.4(b), for any \(z\in {\mathrm {B}}(0,R)\), the box \((z+[-2L_s,2L_s)^d)\) contains at least \((\frac{L_s}{L_0})^d\) vertices from \(\mathbf {G}\). Since for any \(y\in \mathbf {G}\), \({\mathcal {C}}_y\subset {\mathcal {C}}_{2R}\), we have that for any \(x\in A{\setminus } {D_A}\), the set \(x + [-2L_s,2L_s)^d\) contains at least \((\frac{L_s}{L_0})^d\) vertices from \({\mathbf {M}}_A\). (Mind that every vertex from \({\mathcal {C}}_{2R}\cap (x+[-2L_s,2L_s)^d\) must be in A by the definition of \({D_A}\).) Thus we have a map from \(A{\setminus }{D_A}\) to a subset of \({\mathbf {M}}_A\) of size at least \((\frac{L_s}{L_0})^d\) such that every vertex of \({\mathbf {M}}_A\) is in the image of at most \(6^d\cdot L_s^d\) vertices from \(A{\setminus }{D_A}\). This implies that \(|A{\setminus }{D_A}| \le 6^d\cdot L_0^d\cdot |{\mathbf {M}}_A|\), and (3.10) is proved. \(\square \)

3.5 Proof of Lemma 3.7

The statement of the lemma concerns with sets \(\mathbf {A}\subset \mathbf {G}\cap {\mathrm {B}}(0,2R-4L_s)\) of large enough size, but not necessarily comparable with the size of \({\mathbb {G}}_0\cap {\mathrm {B}}(0,2R-4L_s)\). We distinguish the cases when \(\mathbf {A}\) is sparse, and when it is localized. In the first case, we prove that the boundary of \(\mathbf {A}\) is almost of the same size as the volume of \(\mathbf {A}\). In the second case, we estimate the boundary of \(\mathbf {A}\) locally in each of the boxes of side length \(3L_s\) which has dense intersection with \(\mathbf {A}\) and with its complement, see Lemma 3.8. More precisely, we show that the boundary of \(\mathbf {A}\) in each of these boxes is at least of order \(L_s^{d-1}\). Using the isoperimetric inequality for subsets of the lattice \({\mathbb {G}}_s\) we show that the number of disjoint such boxes is of order \(\frac{|\mathbf {A}|^{\frac{d-1}{d}}}{L_s^{d-1}}\). Thus we show that the boundary of \(\mathbf {A}\) contains an order of \(\frac{|\mathbf {A}|^{\frac{d-1}{d}}}{L_s^{d-1}}\) disjoint pieces of size \(L_s^{d-1}\). Before we proceed with the proof, we state the key ingredient of the proof as Lemma 3.8.

Lemma 3.8

Let \(x\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\). Denote by \(\mathbf {g}\) the graph \(\mathbf {G}\cap (x+[-L_s,2L_s)^d)\). For any subset \(\mathbf {a}\) of \(\mathbf {g}\), let \(\partial _{\mathbf {g}} \mathbf {a}\) be the set of pairs of vertices in \(\mathbf {g}\) at \(\ell ^1\)-distance \(L_0\) (in \({\mathbb {Z}}^d\)) from each other so that one of them is in \(\mathbf {a}\) and the other in \(\mathbf {g}{\setminus } \mathbf {a}\).

There exists \(\gamma _{\scriptscriptstyle 3.8}>0\) and \(\rho _{\scriptscriptstyle 3.8}>0\) such that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.8}\) then for any subset \(\mathbf {a}\) of \(\mathbf {g}\) with \(|\mathbf {a}|\in [\frac{1}{2} (\frac{L_s}{L_0})^d, (3^d-\frac{1}{2})(\frac{L_s}{L_0})^d]\), we have \(|\partial _{\mathbf {g}} \mathbf {a}| \ge \gamma _{\scriptscriptstyle 3.8}\cdot (\frac{L_s}{L_0})^{d-1}\).

We postpone the proof of Lemma 3.8 until Sect. 3.6, and now show how Lemma 3.7 follows from Lemma 3.8.

Take \(\mathbf {A}\subset \mathbf {G}\cap {\mathrm {B}}(0,2R-4L_s)\) such that \(|\mathbf {A}| \ge 7^{-d}\cdot (\frac{L_s}{L_0})^{2d^2}\). Note that

$$\begin{aligned} \left| \{x\in {\mathbb {G}}_s~:~\mathbf {A}\cap (x+[0,L_s)^d)\ne \emptyset \}\right| \ge |\mathbf {A}|\cdot \left( \frac{L_s}{L_0}\right) ^{-d}. \end{aligned}$$

Let \({\mathbb {A}}_s\) be the set of \(x\in {\mathbb {G}}_s\) such that

$$\begin{aligned} |\mathbf {A}\cap (x+[0,L_s)^d)| \ge \frac{1}{2}\cdot \left( \frac{L_s}{L_0}\right) ^d. \end{aligned}$$
(3.13)

Note that \({\mathbb {A}}_s\subset {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\).

By Proposition 3.4(b), for any \(x\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-2L_s)\), \(|\mathbf {G}\cap (x+[0,L_s)^d)|>f_d(\frac{r_0}{l_0})\cdot (\frac{L_s}{L_0})^d\), where \(f_d\) is defined in (3.1). By (3.12), for any choice of the ratio \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.6}\), \(f_d(\frac{r_0}{l_0})>\frac{1}{2}\). This implies that for any \(x\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-2L_s)\), \(|\mathbf {G}\cap (x+[0,L_s)^d)| > \frac{1}{2}\cdot (\frac{L_s}{L_0})^d\). Thus, if \(|{\mathbb {A}}_s|< \frac{1}{2}\cdot |\mathbf {A}|\cdot (\frac{L_s}{L_0})^{-d}\), then the number of \(x\in {\mathbb {G}}_s\) such that \(x + [0,L_s)^d\) intersects both \(\mathbf {A}\) and \(\mathbf {G}{\setminus } \mathbf {A}\) is at least \(\frac{1}{2} \cdot |\mathbf {A}|\cdot (\frac{L_s}{L_0})^{-d}\). Since \(\mathbf {G}\cap (x+[0,L_s)^d)\) is connected for each such x, there is an edge in \(\partial _{\mathbf {G}} \mathbf {A}\) with both end-vertices in \(\mathbf {G}\cap (x+[0,L_s)^d)\). Therefore,

$$\begin{aligned} |\partial _{\mathbf {G}} \mathbf {A}|\ge \frac{1}{2} \cdot |\mathbf {A}|\cdot \left( \frac{L_s}{L_0}\right) ^{-d} \ge \frac{1}{14}\cdot |\mathbf {A}|^{\frac{d-1}{d}}\cdot \left( \frac{L_s}{L_0}\right) ^{2d}\cdot \left( \frac{L_s}{L_0}\right) ^{-d} \ge \frac{1}{14}\cdot |\mathbf {A}|^{\frac{d-1}{d}}. \end{aligned}$$

Assume now that \(|{\mathbb {A}}_s|\ge \frac{1}{2}\cdot |\mathbf {A}|\cdot (\frac{L_s}{L_0})^{-d}\). Let \(\partial _{{\mathbb {G}}_s} {\mathbb {A}}_s\) be the set of edges of \({\mathbb {G}}_s\) with exactly one end-vertex in \({\mathbb {A}}_s\). By the isoperimetric inequality on \({\mathbb {G}}_s\),

$$\begin{aligned} |\partial _{{\mathbb {G}}_s} {\mathbb {A}}_s|\ge c_\star \cdot |{\mathbb {A}}_s|^{\frac{d-1}{d}} \ge \frac{1}{2}\cdot c_\star \cdot |\mathbf {A}|^{\frac{d-1}{d}}\cdot \left( \frac{L_s}{L_0}\right) ^{1-d}, \end{aligned}$$

where \(c_\star >0\) is the isoperimetric constant for \({\mathbb {Z}}^d\). By the definition of \({\mathbb {A}}_s\), for any \(y\in {\mathbb {G}}_s{\setminus }{\mathbb {A}}_s\), \(|\mathbf {A}\cap (y+[0,L_s)^d)| \le \frac{1}{2} \cdot (\frac{L_s}{L_0})^d\). Therefore, for any \(x\in {\mathbb {A}}_s\) such that there exists \(y\in {\mathbb {G}}_s{\setminus }{\mathbb {A}}_s\) with \(|x-y|_1 = L_s\), \(\frac{1}{2} \cdot (\frac{L_s}{L_0})^d\le |\mathbf {A}\cap (x+[-L_s,2L_s)^d)| \le (3^d - \frac{1}{2}) \cdot (\frac{L_s}{L_0})^d\). Since \({\mathbb {A}}_s\subset {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\), we can apply Lemma 3.8 to \(\mathbf {g} = \mathbf {G}\cap (x+[-L_s,2L_s)^d)\) and \(\mathbf {a} = \mathbf {A}\cap (x+[-L_s,2L_s)^d)\), to obtain that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.8}\), then

$$\begin{aligned} |\partial _{\mathbf {G}}\mathbf {A} \cap (x+[-L_s,2L_s)^d)|\ge \gamma _{\scriptscriptstyle 3.8}\cdot \left( \frac{L_s}{L_0}\right) ^{d-1}. \end{aligned}$$

We are essentially done. Let \({\sum }^*\) be the sum over \(x\in {\mathbb {A}}_s\) such that there exists \(y\in {\mathbb {G}}_s{\setminus }{\mathbb {A}}_s\) with \(|x-y|_1 = L_s\), i.e., \(\{x,y\}\in \partial _{{\mathbb {G}}_s} {\mathbb {A}}_s\). Combining the last two estimates we get

$$\begin{aligned} |\partial _{\mathbf {G}} \mathbf {A}|\ge & {} \frac{1}{3^d}\cdot {\sum }^* |\partial _{\mathbf {G}}\mathbf {A} \cap (x+[-L_s,2L_s)^d)|\\\ge & {} \frac{1}{3^d}\cdot \left( \frac{1}{2d}\cdot \frac{1}{2}\cdot c_\star \cdot |\mathbf {A}|^{\frac{d-1}{d}}\cdot \left( \frac{L_s}{L_0}\right) ^{1-d}\right) \cdot \left( \gamma _{\scriptscriptstyle 3.8}\cdot \left( \frac{L_s}{L_0}\right) ^{d-1}\right) \\\ge & {} \left( \frac{1}{4d\cdot 3^d}\cdot c_\star \cdot \gamma _{\scriptscriptstyle 3.8}\right) \cdot |\mathbf {A}|^{\frac{d-1}{d}}. \end{aligned}$$

Our final choice of \(\rho _{\scriptscriptstyle 3.7}\) and \(\gamma _{\scriptscriptstyle 3.7}\) is

$$\begin{aligned} \rho _{\scriptscriptstyle 3.7} = \min \left( \rho _{\scriptscriptstyle 3.6},\rho _{\scriptscriptstyle 3.8}\right) \quad \text{ and }\quad \gamma _{\scriptscriptstyle 3.7} = \min \left( \frac{1}{14},\frac{1}{4d\cdot 3^d}\cdot c_\star \cdot \gamma _{\scriptscriptstyle 3.8}\right) , \end{aligned}$$

where \(c_\star \) is the isoperimetric constant for \({\mathbb {Z}}^d\). The proof of Lemma 3.7 is complete, subject to Lemma 3.8.

3.6 Proof of Lemma 3.8

We would like to prove that for any subset \(\mathbf {a}\) of \(\mathbf {g}\) which occupies a non-trivial (bounded away from 0 and 1) fraction of vertices in \({\mathbb {G}}_0\cap (x_s + [-L_s,2L_s)^d)\), \(x_s\in {\mathbb {G}}_s\), its boundary is at least an order of \((\frac{L_s}{L_0})^{d-1}\). For this we prove a much stronger statement that for any j-dimensional (\(2\le j\le d\)) subbox of \({\mathbb {G}}_0\cap (x_s + [-L_s,2L_s)^d)\) containing a non-trivial fraction of vertices of \(\mathbf {a}\), the boundary of \(\mathbf {a}\) in the restriction of \(\mathbf {g}\) to this j-dimensional subbox is at least an order of \((\frac{L_s}{L_0})^{j-1}\), see Lemma 3.9. This statement is proved by induction on j. The case \(j=2\) is the most involved. We first reduce the problem to connected sets with complement consisting of large connected components, see (3.15). The boundary (in \({\mathbb {G}}_0\)) of such sets is large [see (3.19)] and consists of only large \(*\)-connected pieces [see (3.20)]. The key step in the proof is to show that each individual \(*\)-connected piece of the boundary consists mostly of the edges from \(\mathbf {g}\) (see (3.17)). This is done by exploiting further the multi-scale construction of \(\mathbf {G}\). In the case \(j\ge 3\), we use a dimension reduction argument. We partition the j-dimensional box into smaller dimensional subboxes, and estimate the part of the boundary of \(\mathbf {a}\) in each individual subbox where \(\mathbf {a}\) has a non-trivial density.

The main result of this section is the following lemma.

Lemma 3.9

For any \(x\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\), \(y\in {\mathbb {G}}_0\cap (x+[-L_s,2L_s)^d)\), \(2\le j\le d\), and pairwise orthogonal \(e_1,\dots ,e_j\in {\mathbb {Z}}^d\) with \(|e_i|_1=1\), let \(\mathbf {g}'\) be the restriction of \(\mathbf {g}\) to the j-dimensional subcube \((y + \sum _{i=1}^j{\mathbb {Z}}\cdot e_i)\cap (x+[-L_s,2L_s)^d)\) of \((x+[-L_s,2L_s)^d)\). For any \(\epsilon >0\) there exists \(\gamma _{\scriptscriptstyle 3.9} = \gamma _{\scriptscriptstyle 3.9}(\epsilon , j)>0\) and \(\rho _{\scriptscriptstyle 3.9} = \rho _{\scriptscriptstyle 3.9}(\epsilon ,j)>0\) such that if \(\frac{r_0}{l_0}<\rho _{\scriptscriptstyle 3.9}\) then for any subset \(\mathbf {a}'\) of \(\mathbf {g}'\) with \(|\mathbf {a}'|\in [\epsilon (\frac{3L_s}{L_0})^j, (1-\epsilon )(\frac{3L_s}{L_0})^j]\), we have \(|\partial _{\mathbf {g}'} \mathbf {a}'| \ge \gamma _{\scriptscriptstyle 3.9}(\epsilon ,j) (\frac{L_s}{L_0})^{j-1}\).

Note that Lemma 3.8 is a special case of Lemma 3.9 corresponding to the choice of \(j=d\) and \(\epsilon = \frac{1}{2\cdot 3^{d}}\). In particular, Lemma 3.9 implies Lemma 3.8 with the choice of \(\rho _{\scriptscriptstyle 3.8} = \rho _{\scriptscriptstyle 3.9}(\frac{1}{2\cdot 3^d},d)\) and \(\gamma _{\scriptscriptstyle 3.8} = \gamma _{\scriptscriptstyle 3.9}(\frac{1}{2\cdot 3^d},d)\). Thus it only remains to prove Lemma 3.9. We first prove Lemma 3.9 in the case \(j=2\), and then use induction on j to prove Lemma 3.9 in the case \(j\ge 3\).

Proof of Lemma 3.9 (\(j=2\)). Fix any pair of orthogonal \(e_1,e_2\in {\mathbb {Z}}^d\) with \(|e_i|_1 = 1\), \(y\in {\mathbb {G}}_0\cap (x+[-L_s,2L_s)^d)\), and let

$$\begin{aligned} Q = {\mathbb {G}}_0\cap (x+[-L_s,2L_s)^d)\cap (y + {\mathbb {Z}}\cdot e_1 + {\mathbb {Z}}\cdot e_2). \end{aligned}$$

Denote by \(\mathbf {g}'\) the restriction of \(\mathbf {g}\) to Q. By Proposition 3.4(c), \(\mathbf {g}'\) is connected and \(|\mathbf {g}'|> f_2(\frac{r_0}{l_0})\cdot (\frac{3L_s}{L_0})^2\). We choose \(\rho _1 = \rho _1(\epsilon )>0\) so that if \(\frac{r_0}{l_0}<\rho _1\) then

$$\begin{aligned} f_2\left( \frac{r_0}{l_0}\right) > 1 - \frac{\epsilon }{4}, \end{aligned}$$
(3.14)

which implies that \(|\mathbf {g}'| \ge (1 - \frac{\epsilon }{4})\cdot (\frac{3L_s}{L_0})^2\).

Next, we claim that it suffices to show that there exists \(c=c(\epsilon )>0\) and \(\rho _2>0\) such that if \(\frac{r_0}{l_0}<\rho _2\), then

$$\begin{aligned} \begin{array}{c} \text {for any } connected \text { subset } \mathbf {a}''\text { of } \mathbf {g}' \text { satisfying } |\mathbf {a}''|\in \left[ \left( \frac{3L_s}{L_0}\right) , \left( 1-\frac{3\epsilon }{8}\right) \left( \frac{3L_s}{L_0}\right) ^2\right] \\ \text {and such that each connected component of } \mathbf {g}'{\setminus } \mathbf {a}''\text { has size }\ge \left( \frac{3L_s}{L_0}\right) ,\\ \text {we have } |\partial _{\mathbf {g}'} \mathbf {a}''| \ge c(\epsilon )\cdot |\mathbf {a}''|^{1/2}. \end{array} \end{aligned}$$
(3.15)

Indeed, assume that \(\mathbf {a}' = \mathbf {a}_1'\cup \mathbf {a}_2'\), where \(\mathbf {a}_1'\) is the subset of \(\mathbf {a}'\) consisting of connected components of \(\mathbf {a}'\) of size \(\ge \frac{3L_s}{L_0}\), and \(\mathbf {a}_2'\) is the rest of \(\mathbf {a}'\). Let N be the number of connected components in \(\mathbf {a}_2'\). If \(|\mathbf {a}_1'| \le |\mathbf {a}_2'| (\le \frac{3L_s}{L_0}\cdot N)\), then \(N\ge \frac{1}{2}\cdot \frac{L_0}{3L_s}\cdot |\mathbf {a}'|\ge \frac{\epsilon }{2}\cdot (\frac{3L_s}{L_0})\), and

$$\begin{aligned} \frac{|\partial _{\mathbf {g}'} \mathbf {a}'|}{|\mathbf {a}'|^{1/2}} \ge \frac{N}{\left( \frac{6L_s}{L_0}\cdot N\right) ^{1/2}} \ge \frac{1}{2}\cdot \epsilon ^{1/2}. \end{aligned}$$

On the other hand, if \(|\mathbf {a}_1'| > |\mathbf {a}_2'|\), then

$$\begin{aligned} \frac{|\partial _{\mathbf {g}'} \mathbf {a}'|}{|\mathbf {a}'|^{1/2}} \ge \frac{1}{\sqrt{2}}\cdot \frac{|\partial _{\mathbf {g}'} \mathbf {a}_1'|}{|\mathbf {a}_1'|^{1/2}}. \end{aligned}$$

The same reasoning applied to \(\mathbf {g}'{\setminus }\mathbf {a}_1'\) implies that we may assume that the total volume of connected components of \(\mathbf {g}'{\setminus } \mathbf {a}_1'\) with size \(< (\frac{3L_s}{L_0})\) is at most \(\frac{1}{2} |\mathbf {g}'{\setminus }\mathbf {a}_1'|\). By merging all these small connected components of \(\mathbf {g}'{\setminus } \mathbf {a}_1'\) into \(\mathbf {a}_1'\), we obtain the set \(\mathbf {a}'''\) such that \(|\partial _{\mathbf {g}'} \mathbf {a}'''| \le |\partial _{\mathbf {g}'} \mathbf {a}_1'|\), all connected components of \(\mathbf {a}'''\) and \(\mathbf {g}'{\setminus } \mathbf {a}'''\) have size \(\ge \frac{3L_s}{L_0}\), and \(|\mathbf {g}'{\setminus } \mathbf {a}'''| \ge \frac{1}{2} |\mathbf {g}'{\setminus }\mathbf {a}_1'| \ge \frac{1}{2} |\mathbf {g}'{\setminus }\mathbf {a}'| \ge \frac{3\epsilon }{8}\cdot (\frac{3L_s}{L_0})^2\). Moreover, using the same ideas as, e.g., in [22, Section 3.1], we get that for some \(c>0\)

$$\begin{aligned} \frac{|\partial _{\mathbf {g}'} \mathbf {a}_1'|}{|\mathbf {a}_1'|^{1/2}} \ge \frac{|\partial _{\mathbf {g}'} \mathbf {a}'''|}{|\mathbf {a}'''|^{1/2}} \ge c\cdot \inf _{\mathbf {a}''}\frac{|\partial _{\mathbf {g}'} \mathbf {a}''|}{|\mathbf {a}''|^{1/2}},\ \end{aligned}$$

where the infimum is over all connected subsets \(\mathbf {a}''\) of \(\mathbf {g}'\) with \(|\mathbf {a}''|\in [(\frac{3L_s}{L_0}), (1-\frac{3\epsilon }{8})(\frac{3L_s}{L_0})^2]\) and such that each connected component of \(\mathbf {g}'{\setminus } \mathbf {a}''\) has size \(\ge \frac{3L_s}{L_0}\). Thus, if (3.15) holds, then Lemma 3.9 follows in the case \(j=2\) with the choice of

$$\begin{aligned} \rho _{\scriptscriptstyle 3.9}(\epsilon ,2) = \min (\rho _1(\epsilon ),\rho _2). \end{aligned}$$

We proceed with the proof of (3.15). Here we will need the full strength of property (a) in the definition of the event \({\mathcal {H}}\) (see Remark 3.3). We will also use the definition of sets \(({\mathcal {G}}_i)_{0\le i\le r}\) from the construction of \(\mathbf {G}\) (see Remark 3.5).

Recall that \(r = \lfloor \frac{s}{2} \rfloor \). It follows from (3.3) that

$$\begin{aligned} \frac{3L_s}{L_0} \ge 3 \left( \frac{L_r}{L_0}\right) ^2. \end{aligned}$$
(3.16)

Let \(\mathbf {b}''\) be the connected components (in \({\mathbb {G}}_0\)) of \(Q{\setminus } \mathbf {a}''\) which do not intersect \(\mathbf {g}'{\setminus } \mathbf {a}''\), and let \(\overline{\mathbf {a}} = \mathbf {a}''\cup \mathbf {b}''\). (In other words, \(\overline{\mathbf {a}}\) is obtained from \(\mathbf {a}''\) by “filling in holes” in \(\mathbf {a}''\), see Fig. 3.)

Fig. 3
figure 3

This is an illustration of the set \(\overline{\mathbf {a}}\). The black region on the left picture corresponds to \(Q{\setminus }\mathbf {g}'\), but the black boxes are not drawn to the actual scale. The light grey region corresponds to \(\mathbf {a}''\), the white to \(\mathbf {g}'{\setminus }\mathbf {a}''\), and the grey on the right picture to \(\mathbf {b}''\). Thus, the union of light grey and grey regions corresponds to \(\overline{\mathbf {a}}\). Note that a black box turns grey only if it does not have a white neighbor

Note that \(\overline{\mathbf {a}}\) is connected, each connected component of \(Q{\setminus } \overline{\mathbf {a}}\) has size \(\ge \frac{3L_s}{L_0}\), \(|\overline{\mathbf {a}}| \ge |\mathbf {a}''| \ge \frac{3L_s}{L_0}\), and \(|Q{\setminus }\overline{\mathbf {a}}| \ge |Q| - |\mathbf {a}''| - |Q{\setminus } \mathbf {g}'| \ge \frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^2\).

Let \((\overline{\mathbf {b}}_i)_{i\ge 1}\) be connected components of \(Q{\setminus }\overline{\mathbf {a}}\). Let \(\Delta _i\) be the set of edges \(\{x,y\}\) with \(x\in \overline{\mathbf {a}}\) and \(y\in \overline{\mathbf {b}}_i\) (note that necessarily \(x\in \mathbf {a}''\) by the definition of \(\overline{\mathbf {a}}\) and \(\overline{\mathbf {b}}_i\)), and denote by \(\delta _i\) the set of edges \(\{x,y\}\in \Delta _i\) such that \(x\in \overline{\mathbf {a}}\) and \(y\in \mathbf {g}'{\setminus }\overline{\mathbf {a}}\). Note that

$$\begin{aligned} |\partial _{\mathbf {g}'}\mathbf {a}''| = \sum _{i\ge 1}|\delta _i| \qquad \text{ and }\qquad |\partial _Q\overline{\mathbf {a}}| = \sum _{i\ge 1}|\Delta _i|. \end{aligned}$$

We will show that there exists \(C<\infty \) such that for all \(i\ge 1\),

$$\begin{aligned} |\delta _i| \ge |\Delta _i|\cdot \left( 1 - C\sum _{k=0}^{r-1}\frac{r_k}{l_k}\right) . \end{aligned}$$
(3.17)

Once (3.17) is proved, we choose \(\rho _2>0\) so that for \(\frac{r_0}{l_0}<\rho _2\),

$$\begin{aligned} 1 - C\sum _{k=0}^{r-1}\frac{r_k}{l_k}>\frac{1}{2}. \end{aligned}$$
(3.18)

Then using the fact that \(|\overline{\mathbf {a}}|\in [(\frac{3L_s}{L_0}),(1-\frac{\epsilon }{8})(\frac{3L_s}{L_0})^2]\), we apply the isoperimetric inequality for \(\overline{\mathbf {a}}\) in Q (see [15, Proposition 2.2]) and get

$$\begin{aligned} |\mathbf {a}''|\le & {} |\overline{\mathbf {a}}| \le C(\epsilon )\cdot \sum _{i\ge 1}|\Delta _i|^2 \le C(\epsilon )\cdot \left( \sum _{i\ge 1}|\Delta _i|\right) ^2 \le 4\cdot C(\epsilon )\cdot \left( \sum _{i\ge 1}|\delta _i|\right) ^2\nonumber \\= & {} 4\cdot C(\epsilon )\cdot |\partial _{\mathbf {g}'}\mathbf {a}''|^2, \end{aligned}$$
(3.19)

and (3.15) follows. Before we prove (3.17), we show that there exists \(c>0\) such that for each \(i\ge 1\),

$$\begin{aligned} |\Delta _i| \ge c\cdot \frac{L_r}{L_0}. \end{aligned}$$
(3.20)

Indeed, if \(|\overline{\mathbf {b}}_i|<\frac{1}{2}(\frac{3L_s}{L_0})^2\), then by the isoperimetric inequality in Q (see [15, Proposition 2.2]), \(|\Delta _i| \ge c|\overline{\mathbf {b}}_i|^{1/2} \ge c(\frac{3L_s}{L_0})^{1/2}\). On the other hand, if \(|\overline{\mathbf {b}}_i|\ge \frac{1}{2}(\frac{3L_s}{L_0})^2\), then \(Q{\setminus } \overline{\mathbf {b}}_i\) is connected (since \(\overline{\mathbf {a}}\) is connected), and \(|Q{\setminus } \overline{\mathbf {b}}_i|\in [(\frac{3L_s}{L_0}), \frac{1}{2}(\frac{3L_s}{L_0})^2]\). Thus, again by the isoperimetric inequality in Q (see [15, Proposition 2.2]), \(|\Delta _i| \ge c|Q{\setminus } \overline{\mathbf {b}}_i|^{1/2} \ge c(\frac{3L_s}{L_0})^{1/2}\). Using (3.16), we get (3.20).

We now prove (3.17). For this we recall the construction of \(\mathbf {G}\), namely the definition of \({\mathcal {G}}_k\). In particular, note that by part (a) of the definition of \({\mathcal {H}}\), \({\mathcal {G}}_r \cap {\mathrm {B}}(0,R+L_r) = {\mathbb {G}}_r\cap {\mathrm {B}}(0,R+L_r)\), and for \(0\le k\le r-1\), \({\mathcal {G}}_k\) is obtained by deleting at most 3 boxes of side length \(r_kL_k\) from each of the boxes \((z + [0,L_{k+1})^2)\), \(z\in {\mathcal {G}}_{k+1}\). A useful implication of this construction is that for each such deleted box of side length \(r_kL_k\), there exist at most \(26 (= 3\cdot 3^2 - 1)\) other deleted boxes of side length \(r_kL_k\) which are within \(\ell ^\infty \)-distance \(L_{k+1}\) from the specified box.

Fix \(i\ge 1\). We write the set of “bad” edges \(\Delta _i{\setminus }\delta _i\) as the union \(\cup _{k=0}^{r-1}E_k\), where \(E_k\) consists of edges \(\{z,z'\}\) in \({\mathbb {G}}_0\) such that \(z\in \overline{\mathbf {a}}\) and \(z'\in \overline{\mathbf {b}}_i \cap (({\mathcal {G}}_{k+1} + [0,L_{k+1})^2){\setminus } ({\mathcal {G}}_k + [0,L_k)^2))\). This is the part of \(\Delta _i\) which “touches” the boxes of side length \(r_kL_k\) deleted from \({\mathrm {B}}(0,2R)\) (more specifically, from \(({\mathcal {G}}_{k+1} + [0,L_{k+1})^2)\)) in the definition of \(\mathbf {G}\), i.e., when defining \({\mathcal {G}}_k\). Let \(N_k\) be the total number of such “touched” boxes of side length \(r_kL_k\). Since each of these boxes has boundary \(\le 4r_k\frac{L_k}{L_0}\), it follows that \(|E_k| \le N_k\cdot 16 r_k\frac{L_k}{L_0}\). Consider separately the cases \(N_k> 27\) and \(N_k\le 27\). If \(N_k\le 27\), then

$$\begin{aligned} |E_k| \le N_k\cdot 16 r_k\frac{L_k}{L_0}\le 16\cdot 27\cdot \frac{r_k}{l_k}\cdot \frac{L_r}{L_0} \le C\cdot \frac{r_k}{l_k}\cdot |\Delta _i|, \end{aligned}$$

where the last inequality follows from (3.20). Assume now that \(N_k>27\). From all these boxes we can choose \(\ge \lceil \frac{N_k}{27}\rceil (\ge 2)\) boxes so that each pair of them is at \(\ell ^\infty \)-distance \(\ge L_{k+1}\) from each other. By [15, Lemma 2.1(ii)], the set \(\{x\in \overline{\mathbf {a}}~:~\{x,y\}\in \Delta _i \text{ for } \text{ some } y\}\) is \(*\)-connected. Thus, we can choose disjoint simple \(*\)-paths in \(\{x\in \overline{\mathbf {a}}~:~\{x,y\}\in \Delta _i \text{ for } \text{ some } y\}\) of \(\frac{L_{k+1}}{3L_0}\) vertices each, originating near each of such boxes. (See Fig. 4.) Therefore,

$$\begin{aligned} |\Delta _i| \ge |\{x\in \overline{\mathbf {a}}~:~\{x,y\}\in \Delta _i \text{ for } \text{ some } y\}| \ge \frac{1}{3}\cdot \frac{L_{k+1}}{L_0}\cdot \frac{N_k}{27},\ \end{aligned}$$
(3.21)

and we conclude that

$$\begin{aligned} |E_k| \le N_k\cdot 16 r_k\frac{L_k}{L_0}\le 16\cdot 81\cdot \frac{r_k}{l_k}\cdot |\Delta _i|. \end{aligned}$$

Combining the bounds of \(|E_k|\) for all k gives

$$\begin{aligned} |\delta _i| = |\Delta _i| - |\Delta _i{\setminus }\delta _i| \ge |\Delta _i| - \sum _{k=0}^{r-1}|E_k| \ge |\Delta _i|\cdot \left( 1 - C\cdot \sum _{k=0}^{r-1}\frac{r_k}{l_k}\right) . \end{aligned}$$

This is precisely (3.17). Thus, the proof of Lemma 3.9 is complete in the case \(j=2\). \(\square \)

Fig. 4
figure 4

The shaded region is \(\overline{\mathbf {b}}_i\). Its boundary is the \(*\)-connected set \(\{x\in \overline{\mathbf {a}}~:~\{x,y\}\in \Delta _i \text{ for } \text{ some } y\}\). The colored boxes correspond to \(({\mathcal {G}}_{k+1} + [0,L_{k+1})^2){\setminus } ({\mathcal {G}}_k + [0,L_k)^2)\). The total number of boxes at pairwise distance \(\ge L_{k+1}\) from each other is \(\ge \lceil \frac{N_k}{27}\rceil (\ge 2)\). Each piece of the boundary of \(\overline{\mathbf {b}}_i\) (illustrated with solid lines) touching one of the well-separated boxes consist of \(\frac{L_{k+1}}{3L_0}\) vertices. Since these paths are disjoint, the total number of vertices in these paths is \(\ge \lceil \frac{N_k}{27}\rceil \cdot \frac{L_{k+1}}{3L_0}\), which implies (3.21)

We proceed with the proof of Lemma 3.9 in the case \(j\ge 3\).

Proof of Lemma 3.9 (\(j\ge 3\)). The proof is by induction on j and using the result of Lemma 3.9 for \(j=2\) proved before. Given \(j\ge 3\), we assume that the statement of Lemma 3.9 holds for all \(j'<j\) and prove that it also holds for j.

Fix \(x\in {\mathbb {G}}_s\cap {\mathrm {B}}(0,2R-3L_s)\), \(y\in {\mathbb {G}}_0\cap (x+[-L_s,2L_s)^d)\), \(2\le j\le d\), and pairwise orthogonal \(e_1,\dots ,e_j\in {\mathbb {Z}}^d\) with \(|e_i|_1=1\), and let \(\mathbf {g}'\) be the restriction of \(\mathbf {g}\) to \((y + \sum _{i=1}^j{\mathbb {Z}}\cdot e_i)\). Fix \(\epsilon >0\) and a subset \(\mathbf {a}'\) of \(\mathbf {g}'\) with \(|\mathbf {a}'|\in [\epsilon (\frac{3L_s}{L_0})^j, (1-\epsilon )(\frac{3L_s}{L_0})^j]\).

By Proposition 3.4(c), \(\mathbf {g}'\) is connected and \(|\mathbf {g}'| \ge f_j(\frac{r_0}{l_0})\cdot (\frac{3L_s}{L_0})^j\), where \(f_j\) is defined in (3.1). There exists \(\rho _3 = \rho _3(\epsilon ,j)>0\) so that if \(\frac{r_0}{l_0}<\rho _3\), then

$$\begin{aligned} f_j\left( \frac{r_0}{l_0}\right) \ge f_{j-1}\left( \frac{r_0}{l_0}\right) > 1 - \frac{\epsilon }{2},\ \end{aligned}$$
(3.22)

which implies that \(|\mathbf {g}'| \ge (1 - \frac{\epsilon }{2})\cdot (\frac{3L_s}{L_0})^j\). (The inequality for \(f_{j-1}(\frac{r_0}{l_0})\) is used later in the proof.)

Consider the \((j-1)\)-dimensional slices \(M_k = k\cdot L_0\cdot e_1 + (y + \sum _{i=2}^j{\mathbb {Z}}\cdot e_i)\cap (x + [-L_s,2L_s)^d)\cap {\mathbb {G}}_0\), \(k\in {\mathbb {Z}}\). Since \(|\mathbf {a}'| \ge \epsilon (\frac{3L_s}{L_0})^j\), there exist at least \(\frac{\epsilon }{2}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge \frac{\epsilon }{2}\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {a}'\). Since \(|\mathbf {g}'{\setminus } \mathbf {a}'| \ge \frac{\epsilon }{2}\cdot (\frac{3L_s}{L_0})^j\), there exists at least \(\frac{\epsilon }{4}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge \frac{\epsilon }{4}\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {g}'{\setminus } \mathbf {a}'\).

If there exists at least \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge \frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from each of the sets \(\mathbf {a}'\) and \(\mathbf {g}'{\setminus } \mathbf {a}'\), then the restriction of \(\mathbf {g}'\) to any such slice satisfies the induction hypothesis. Therefore, by applying Lemma 3.9 to the restriction of \(\mathbf {g}'\) in each of these slices, we conclude that

$$\begin{aligned} |\partial _{\mathbf {g}'} \mathbf {a}'|\ge & {} \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{8},j-1\right) \cdot \left( \frac{3L_s}{L_0}\right) ^{j-2}\cdot \frac{\epsilon }{8}\cdot \left( \frac{3L_s}{L_0}\right) \\= & {} \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{8},j-1\right) \cdot \frac{\epsilon }{8}\cdot \left( \frac{3L_s}{L_0}\right) ^{j-1}. \end{aligned}$$

If there exists \(<\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge \frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from each of the sets \(\mathbf {a}'\) and \(\mathbf {g}'{\setminus } \mathbf {a}'\), then by earlier conclusion, there exist at least \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(<\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {g}'{\setminus } \mathbf {a}'\). By Proposition 3.4(c) and (3.22), each such slice contains at least \(f_{j-1}(\frac{r_0}{l_0})\cdot (\frac{3L_s}{L_0})^{j-1}\ge (1 - \frac{\epsilon }{2})\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {g}'\). Therefore, there exist at least \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge (1-\frac{5\epsilon }{8})\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {a}'\). We choose \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) of them and call these slices overcrowded.

Similarly one shows that there exist at least \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) slices \(M_k\) containing \(\ge (1-\frac{5\epsilon }{8})\cdot (\frac{3L_s}{L_0})^{j-1}\) vertices from \(\mathbf {g}'{\setminus } \mathbf {a}'\). We choose \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})\) of them and call these slices undercrowded.

Fig. 5
figure 5

An illustration of the overcrowded and undercrowded slices \(M_k\). Every two-dimensional slice \(N_t\) intersects each of the overcrowded and undercrowded slices in \(\frac{3L_s}{L_0}\) vertices, so it intersects the union of all the overcrowded slices and also the union of all the undercrowded slices in \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^2\) vertices

Consider now the two-dimensional slices \(N_t = \sum _{i=3}^jt_i\cdot L_0\cdot e_i + (y + {\mathbb {Z}}\cdot e_1 + {\mathbb {Z}}\cdot e_2)\cap (x + [-L_s,2L_s)^d)\cap {\mathbb {G}}_0\), \(t = (t_3,\dots ,t_j)\in {\mathbb {Z}}^{j-2}\), see Fig. 5. Note that every non-empty two-dimensional slice \(N_t\) intersects the union of all the overcrowded \((j-1)\)-dimensional slices and also the union of all the undercrowded \((j-1)\)-dimensional slices in \(\frac{\epsilon }{8}\cdot (\frac{3L_s}{L_0})^2\) vertices. Therefore, there exist at least \(\frac{2}{3}\cdot (\frac{3L_s}{L_0})^{j-2}\) two-dimensional slices \(N_t\) containing at least \(\frac{\epsilon }{16}\cdot (\frac{3L_s}{L_0})^2\) vertices from \(\mathbf {a}'\), and at least \(\frac{2}{3}\cdot (\frac{3L_s}{L_0})^{j-2}\) slices \(N_t\) containing at least \(\frac{\epsilon }{16}\cdot (\frac{3L_s}{L_0})^2\) vertices from \(\mathbf {g}'{\setminus } \mathbf {a}'\). Indeed, assume that there exist \({\mathrm N}< \frac{2}{3}\cdot (\frac{3L_s}{L_0})^{j-2}\) slices containing at least \(\frac{\epsilon }{16}\cdot (\frac{3L_s}{L_0})^2\) vertices from \(\mathbf {a}'\). The other case is considered similarly. Then the total number of vertices of \(\mathbf {a}'\) in overcrowded \((j-1)\)-dimensional slices is

$$\begin{aligned}\le & {} {\mathrm N}\cdot \left( \frac{\epsilon }{8}\cdot \left( \frac{3L_s}{L_0}\right) ^{2}\right) + \left( \left( \frac{3L_s}{L_0}\right) ^{j-2} - {\mathrm N}\right) \cdot \left( \frac{\epsilon }{16}\cdot \left( \frac{3L_s}{L_0}\right) ^{2}\right) \\< & {} \frac{5}{3}\cdot \frac{\epsilon }{16}\cdot \left( \frac{3L_s}{L_0}\right) ^j < \left( 1-\frac{5\epsilon }{8}\right) \cdot \frac{\epsilon }{8}\cdot \left( \frac{3L_s}{L_0}\right) ^j,\ \end{aligned}$$

for \(\epsilon <\frac{4}{15}\), which contradicts with the definition of overcrowded slices.

Therefore, there exist at least \(\frac{1}{3}\cdot (\frac{3L_s}{L_0})^{j-2}\) slices \(N_t\) containing at least \(\frac{\epsilon }{16}\cdot (\frac{3L_s}{L_0})^2\) vertices from \(\mathbf {a}'\) and at least \(\frac{\epsilon }{16}\cdot (\frac{3L_s}{L_0})^2\) vertices from \(\mathbf {g}'{\setminus } \mathbf {a}'\). We can now apply Lemma 3.9 to each of such slices to obtain that the boundary of \(\mathbf {a}'\) in the restriction of \(\mathbf {g}'\) to each of such slices is at least \(\gamma _{\scriptscriptstyle 3.9}(\frac{\epsilon }{16},2)\cdot \frac{3L_s}{L_0}\). Since the total number of slices is at least \(\frac{1}{3}\cdot (\frac{3L_s}{L_0})^{j-2}\), we conclude that

$$\begin{aligned} |\partial _{\mathbf {g}'} \mathbf {a}'| \ge \frac{1}{3}\cdot \left( \frac{3L_s}{L_0}\right) ^{j-2}\cdot \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{16},2\right) \cdot \frac{3L_s}{L_0} = \frac{1}{3}\cdot \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{16},2\right) \cdot \left( \frac{3L_s}{L_0}\right) ^{j-1}. \end{aligned}$$

Thus the result of Lemma 3.9 for the given j follows with the choice of

$$\begin{aligned} \gamma _{\scriptscriptstyle 3.9}(\epsilon ,j) = \min \left( \frac{\epsilon }{8}\cdot \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{8},j-1\right) , \frac{1}{3}\cdot \gamma _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{16},2\right) \right) \end{aligned}$$

and

$$\begin{aligned} \rho _{\scriptscriptstyle 3.9}(\epsilon ,j) = \min \left( \rho _3(\epsilon ,j),\rho _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{8},j-1\right) ,\rho _{\scriptscriptstyle 3.9}\left( \frac{\epsilon }{16},2\right) \right) . \end{aligned}$$

The proof of Lemma 3.9 in the case \(j\ge 3\) is complete. \(\square \)

4 Quenched invariance principle

In this section we state the quenched invariance principle for simple random walk on percolation clusters satisfying some general conditions. Later, in Sect. 5, we show that these conditions are satisfied by any probability measure \({\mathbb {P}}^u\), for \(u\in (a,b)\), given that the family \(\{{\mathbb {P}}^u\}_{u\in (a,b)}\) satisfies the axioms P1P3 and S1S2.

Consider a probability measure \({\mathbb {P}}\) on the measurable space \((\Omega ,{\mathcal {F}})\), where \(\Omega = \{0,1\}^{{\mathbb {Z}}^d}\), \(d\ge 2\), and \({\mathcal {F}}\) is the sigma-algebra generated by the canonical coordinate maps \(\{\omega \mapsto \omega (x)\}_{x\in {\mathbb {Z}}^d}\). For \(x\in {\mathbb {Z}}^d\), denote by \(\tau _x~:~\Omega \rightarrow \Omega \) the shift in direction x, i.e., \((\tau _x\omega )(y) = \omega (x+y)\). For each \(\omega \in \Omega \), let

$$\begin{aligned} {\mathcal {S}}= {\mathcal {S}}(\omega ) = \{x\in {\mathbb {Z}}^d~:~\omega (x) = 1\}. \end{aligned}$$

We think about \({\mathcal {S}}\) as a subgraph of \({\mathbb {Z}}^d\) in which edges are added between any two vertices of \({\mathcal {S}}\) of \(\ell ^1\)-distance 1. As before, we denote by \({\mathcal {S}}_\infty \) the subset of vertices of \({\mathcal {S}}\) which belong to infinite connected components of \({\mathcal {S}}\). We assume that \({\mathbb {P}}\) satisfies the following axioms.

A1 :

For all \(e\in {\mathbb {Z}}^d\) with \(|e|_1 = 1\), the shift \(\tau _e\) is measure preserving and ergodic on \((\Omega ,{\mathcal {F}}, {\mathbb {P}})\).

A2 :

The subgraph \({\mathcal {S}}_\infty \) is non-empty and connected, \({\mathbb {P}}\)-a.s. (In particular, \({\mathbb {P}}[0\in {\mathcal {S}}_\infty ] > 0\).)

A3 :

There exist constants \(c>0\), \(C<\infty \), and \(\Delta _3>0\) such that for all \(R\ge 1\) and for all \(e\in {\mathbb {Z}}^d\) with \(|e|_1 = 1\),

$$\begin{aligned} {\mathbb {P}}\left[ k\cdot e\in {\mathcal {S}}_\infty \text{ for } \text{ some } 0\le k\le R \right] \ge 1-C\cdot e^{-c(\log R)^{1+\Delta _3}}. \end{aligned}$$

Our next axioms on \({\mathbb {P}}\) concern intrinsic geometry of \({\mathcal {S}}_\infty \). For \(x,y\in {\mathcal {S}}\), let \(\rho _{\mathcal {S}}(x,y)\in {\mathbb {N}} \cup \{\infty \}\) denote the distance between x and y in \({\mathcal {S}}\), i.e.,

$$\begin{aligned} \rho _{\mathcal {S}}(x,y)=\inf \left\{ n\ge 0 ~:~ \begin{array}{c} \text {there exist } x_0,\dots ,x_n\in {\mathcal {S}}\text { such that}\\ x_0=x, x_n=y,\text { and}\\ |x_k-x_{k-1}|_1=1\text { for all }k=1,\dots ,n \end{array} \right\} , \end{aligned}$$

where we use the convention \(\inf \emptyset =\infty \), and let \({{\mathrm {B}}_{\scriptscriptstyle {\mathcal {S}}}}(x,R) = \{y\in {\mathcal {S}}~:~\rho _{\mathcal {S}}(x,y)\le R\}\).

A4 :

There exist constants \(c>0\), \(C<\infty \), and \(\Delta _4>0\) such that for all \(R\ge 1\),

$$\begin{aligned} {\mathbb {P}}\left[ \text {for all }x,y\in {\mathcal {S}}_\infty \cap {\mathrm {B}}(0,R), ~\rho _{\mathcal {S}}(x,y)\le C\cdot R \right] \ge 1-C\cdot e^{-c(\log R)^{1+\Delta _4}}. \end{aligned}$$
A5 :

\({\mathbb {P}}[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost surely,

$$\begin{aligned} \inf _{k\ge 1}\inf \left\{ \frac{|\partial _{\mathcal {S}}A|}{|A|^{\frac{d-1}{d}}}~:~A\subset {\mathcal {S}}_\infty \cap {{\mathrm {B}}_{\scriptscriptstyle {\mathcal {S}}}}(0,2k), |A|\ge k^{1/3} \right\} >0. \end{aligned}$$

Next we describe the random walk on \({\mathcal {S}}\). For \(\omega \in \Omega \) and \(x\in {\mathcal {S}}\), let \(\deg _\omega (x)=\left| \{y\in {\mathcal {S}}~:~ |y-x|_1=1 \}\right| \) be the degree of x in \({\mathcal {S}}\). For a configuration \(\omega \in \Omega \) and \(x\in {\mathcal {S}}\), let \(\mathbf {P}_{\omega ,x}\) be the distribution of the random walk \(\{X_n\}_{n \ge 0}\) on \({\mathcal {S}}\) defined by the transition kernel

$$\begin{aligned} \mathbf {P}_{\omega ,x}[X_{n+1} = z|X_{n}=y] = \left\{ \begin{array}{ll} \frac{1}{2d}&{}\quad |z-y|_1=1,~z\in {\mathcal {S}};\\ 1-\frac{\deg _\omega (y)}{2d}&{}\quad z=y;\\ 0 &{} \quad \text {otherwise,} \end{array} \right. \end{aligned}$$
(4.1)

and initial position \(\mathbf {P}_{\omega ,x}[X_0=x]=1\). The corresponding expectation is denoted by \(\mathbf {E}_{\omega ,x}\).

Let \(\Omega _0=\left\{ \omega \in \Omega ~:~ 0\in {\mathcal {S}}_\infty \right\} \), and define the measure \({\mathbb {P}}_0\) by \({\mathbb {P}}_0[A]={\mathbb {P}}[A~|~\Omega _0]\). We denote by \({\mathbb {E}}_0\) the expectation with respect to \({\mathbb {P}}_0\).

For \(\omega \in \Omega _0\), \(n\in {\mathbb {N}}\), and \(t\ge 0\), let

$$\begin{aligned} \widetilde{B}_n(t) = \frac{1}{\sqrt{n}}\left( X_{\lfloor tn\rfloor } + \left( tn - \lfloor tn\rfloor \right) \cdot \left( X_{\lfloor tn\rfloor + 1} - X_{\lfloor tn\rfloor }\right) \right) ,\ \end{aligned}$$

where \((X_k)_{k\ge 0}\) is the random walk on \({\mathcal {S}}\) (actually on \({\mathcal {S}}_\infty \)) with distribution \(\mathbf {P}_{\omega ,0}\). Theorem 1.1 follows from Theorem 4.1, as we demonstrate in Sect. 5.

Theorem 4.1

Let \(d\ge 2\), and assume that the measure \({\mathbb {P}}\) satisfies assumptions A1A5. Then for all \(T>0\) and for \({\mathbb {P}}_0\)-almost every \(\omega \), the law of \((\widetilde{B}_n(t))_{0\le t\le T}\) on \((C[0,T],{\mathcal {W}}_T)\) converges weakly to the law of a Brownian motion with zero drift and non-degenerate covariance matrix. In addition, if reflections and rotations of \({\mathbb {Z}}^d\) by \(\frac{\pi }{2}\) preserve \({\mathbb {P}}\), then the limiting Brownian motion is isotropic (with positive diffusion constant).

The proof of Theorem 4.1 is a routine adaptation of the proof of [7, Theorem 1.1]. Instead of proving [7, Theorem 6.3], which relies on the upper bound on heat kernel obtained in [3, Theorem 1], we follow the proof of [11, Theorem 2.1], which uses softer arguments (still relying very much on observations from [5, 25] exploited in [3], but not using the full strenth of the upper bound in [3, Theorem 1]). We give a sketch proof of Theorem 4.1 in Sect. 1.

5 Proof of Theorem 1.1

In this section we derive Theorem 1.1 from Theorem 4.1. Namely, we prove that for a family of probability measures \(\{{\mathbb {P}}^u\}_{u\in (a,b)}\) satisfying P1P3 and S1S2, every probability measure \({\mathbb {P}}^u\) in the family satisfies the conditions A1A5 of Sect. 4. Our proof can mostly be read independently of Sects. 2 and 3, except for the proof of A3, where we need to use and generalize some results from Sect. 2. Fix \(u\in (a,b)\). We prove that \({\mathbb {P}}^u\) satisfies A1A5.

  • Condition A1 follows from P1.

  • Condition A2 follows from S1S2.

  • The fact that A4 follows from P1P3 and S1S2 is proved in [17, Theorem 1.3].

  • Condition A5 follows from Theorem 1.2, P1 (only translation invariance part), and S1. It suffices to show that for \({\mathbb {P}}^u[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost every realization \(\omega \) and all R sufficiently large, the connected component of 0 in \({\mathcal {S}}_\infty \cap {\mathrm {B}}(0,R)\) is the unique largest in volume connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,R)\), i.e., using the notation of Theorem 1.2, \(0\in {\mathcal {C}}_R\). Indeed, as soon as \(0\in {\mathcal {C}}_R\) for all large R, the inclusion \({{\mathrm {B}}_{\scriptscriptstyle {\mathcal {S}}}}(0,R)\subset {\mathcal {C}}_R\) holds for all large R, and A5 follows from Theorem 1.2 and the Borel-Cantelli lemma.

To prove the remaining claim, we apply S1 to all the boxes \({\mathrm {B}}(x,R^{1/2d})\), \(x\in {\mathrm {B}}(0,R-4R^{1/2d})\), (this is possible by P1) and use the Borel-Cantelli lemma to conclude that \({\mathbb {P}}^u\)-almost surely for all large R,

  1. (a)

    each box \({\mathrm {B}}(x,R^{1/2d})\), \(x\in {\mathrm {B}}(0,R-4R^{1/2d})\), intersects \({\mathcal {S}}_{R^{1/2d}}\),

  2. (b)

    for any \(x,x'\in {\mathrm {B}}(0,R-4R^{1/2d})\) such that \(|x-x'|_1 = 1\), there exists a unique connected component of \({\mathcal {S}}_{R^{1/2d}}\cap {\mathrm {B}}(x,4R^{1/2d})\) which intersects \({\mathrm {B}}(x,R^{1/2d})\cup {\mathrm {B}}(x',R^{1/2d})\).

Statements (a) and (b) together imply that \({\mathbb {P}}^u\)-almost surely for all large R, there exists a connected component of \({\mathcal {S}}_{R^{1/2d}}\cap {\mathrm {B}}(0,R)\) which intersects every box \({\mathrm {B}}(x,R^{1/2d})\), \(x\in {\mathrm {B}}(0,R-4R^{1/2d})\), and it is the unique connected component of \({\mathcal {S}}_{R^{1/2d}}\cap {\mathrm {B}}(0,R)\) which intersects \({\mathrm {B}}(0,R - 3R^{1/2d})\). Therefore, \({\mathbb {P}}^u[\cdot ~|~0\in {\mathcal {S}}_\infty ]\)-almost surely for all large R, (1) the connected component of 0 in \({\mathcal {S}}_\infty \cap {\mathrm {B}}(0,R)\) intersects each box \({\mathrm {B}}(x,R^{1/2d})\), \(x\in {\mathrm {B}}(0,R-4R^{1/2d})\), and (2) it is the unique connected component of \({\mathcal {S}}_{R^{1/3}}\cap {\mathrm {B}}(0,R)\) which intersects \({\mathrm {B}}(0,R - R^{\frac{1}{3}})\). By (1), the connected component of 0 in \({\mathcal {S}}_\infty \cap {\mathrm {B}}(0,R)\) has volume \(\ge \frac{1}{2} R^{d-\frac{1}{2}}\). Note that for all large R, any connected component of \({\mathcal {S}}\cap {\mathrm {B}}(0,R)\) with volume \(\ge \frac{1}{2} R^{d-\frac{1}{2}}\) has diameter \(\ge R^{\frac{1}{3}}\) and intersects \({\mathrm {B}}(0,R - R^{\frac{1}{3}})\). By (2), such connected component must be unique. This implies the claim, and A5 follows.

  • It remains to show that A3 follows from P1P3 and S1S2. This is done by exploiting the renormalization structure of [17] and adding an additional increasing event to the structure. More precisely, we modify Definition 2.1 of event \(A^u_x\). Let \(\{e_i\}_{i=1}^d\) be the unit coordinate vectors in \({\mathbb {Z}}^d\). For \(x\in {\mathbb {G}}_0\) and \(u\in (a,b)\), let \({\mathcal {A}}^u_x\in {\mathcal {F}}\) be the event that

  1. (a)

    for each \(e\in \{0,1\}^d\), the set \({\mathcal {S}}_{L_0}\cap (x+eL_0 + [0,L_0)^d)\) contains a connected component with at least \(\frac{3}{4} \eta (u) L_0^d\) vertices,

  2. (b)

    all of these \(2^d\) components are connected in \({\mathcal {S}}\cap (x+[0,2L_0)^d)\),

  3. (c)

    for each \(1\le i\le d\), the “special” connected component of \({\mathcal {S}}_{L_0}\) in \((x + [0,L_0)^d)\) contains a vertex in each of the d line segments \(I_{x,i} = (x+(\lfloor \frac{L_0}{2}\rfloor , \dots , \lfloor \frac{L_0}{2}\rfloor ) + {\mathbb {Z}}\cdot e_i)\cap (x + [\lfloor \frac{L_0}{3}\rfloor ,\lfloor \frac{2L_0}{3}\rfloor )^d)\).

For \(u\in (a,b)\) and \(x\in {\mathbb {G}}_0\), let \(\overline{{\mathcal {A}}}^u_{x,0}\) be the complement of \({\mathcal {A}}^u_{x}\), and for \(u\in (a,b)\), \(k\ge 1\), and \(x\in {\mathbb {G}}_k\) define inductively

$$\begin{aligned} \overline{{\mathcal {A}}}^u_{x,k} = \bigcup _{\begin{array}{c}\scriptscriptstyle {x_1,x_2\in {\mathbb {G}}_{k-1}\cap (x + [0,L_k)^d)} \\ \scriptscriptstyle {|x_1-x_2|_\infty \ge r_{k-1} \cdot L_{k-1}}\end{array}} \overline{{\mathcal {A}}}^u_{x_1,k-1} \cap \overline{{\mathcal {A}}}^u_{x_2,k-1}. \end{aligned}$$

By P1 and Birkhoff’s ergodic theorem, for any \(u\in (a,b)\), \(x\in {\mathbb {G}}_0\), and \(1\le i\le d\),

$$\begin{aligned} \lim _{L_0 \rightarrow \infty } \frac{3}{L_0} \sum _{ y \in I_{x,i}} \mathbbm {1}_{\{y \in {\mathcal {S}}_{L_0}\}} \; \mathop {=}\limits ^{\mathbb {P}^u\text {-a.s.}} \; \lim _{L_0 \rightarrow \infty } \frac{3}{L_0} \sum _{ y \in I_{x,i}} \mathbbm {1}_{\{y \in {\mathcal {S}}_\infty \}} \; \mathop {=}\limits ^{\mathbb {P}^u\text {-a.s.}} \; \eta (u). \end{aligned}$$

We conclude from S1, S2, and [17, (4.3)] that for any \(u\in (a,b)\) there exists \(\delta = \delta (u)>0\) such that \((1-\delta )u>a\) and

$$\begin{aligned} {\mathbb {P}}^{(1-\delta )u}\left[ {\mathcal {A}}_0^u\right] \rightarrow 1 ,\quad \text{ as } \quad L_0\rightarrow \infty . \end{aligned}$$

As in the proof of [17, Lemma 4.2], this implies that for each \(u\in (a,b)\), there exist \(C = C(u)<\infty \) and \(C' = C'(u,l_0)<\infty \) such that for all \(l_0,r_0\ge C\), \(L_0\ge C'\), and \(k\ge 0\),

$$\begin{aligned} {\mathbb {P}}^u\left[ \overline{{\mathcal {A}}}^u_{0,k}\right] \le 2^{-2^k}. \end{aligned}$$
(5.1)

We modify Definition 2.3 by replacing the events \(A^u_x\) by \({\mathcal {A}}^u_x\). Let \(u\in (a,b)\). For \(k\ge 0\), we say that \(x\in {\mathbb {G}}_k\) is k-bad if the event \(\overline{{\mathcal {A}}}^u_{x,k}\cup \overline{B}^u_{x,k}\) occurs, where \(B^u_{x,k}\) is defined in (2.3). Otherwise, we say that x is k-good. It follows from (5.1) and [17, Lemma 4.4] that for each \(u\in (a,b)\), there exist \(C = C(u)<\infty \) and \(C' = C'(u,l_0)<\infty \) such that for all \(l_0,r_0\ge C\), \(L_0\ge C'\), and \(k\ge 0\),

$$\begin{aligned} {\mathbb {P}}^u\left[ 0 \text{ is } k\text{-bad }\right] \le 2\cdot 2^{-2^k}. \end{aligned}$$
(5.2)

Note that if 0 is k-good, then \({\mathbb {G}}_0\cap [0,L_k)^d\) contains a connected component \({\mathcal {G}}\) of 0-good vertices of diameter \(\frac{L_k}{L_0}\) (in \({\mathbb {G}}_0\)) which intersects every line segment \({\mathbb {Z}}\cdot e_i\cap [0,L_k)^d\), \(1\le i\le d\). This is easily proved by induction from the definition of k-good vertex. By Lemma 2.6 and noting that any 0-good vertex in the new sense is also 0-good in the sense of Definition 2.3, the set \(\cup _{x\in {\mathcal {G}}}{\mathcal {C}}_x\) is contained in the same connected component of \({\mathcal {S}}\) with diameter at least \(\frac{L_k}{2}\). (\({\mathcal {C}}_x\) is the “special” component of \({\mathcal {S}}\cap (x + [0,L_0)^d)\) defined in Lemma 2.6(a).) By S1 and (1.2), with probability \(\ge 1 - Ce^{-c(\log L_k)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}} - 2\cdot 2^{-2^k}\), \(\cup _{x\in {\mathcal {G}}}{\mathcal {C}}_x\subset {\mathcal {S}}_\infty \). By the definition of 0-good vertex, namely using part (c) in the definition of \({\mathcal {A}}^u_x\), we obtain that

$$\begin{aligned}&{\mathbb {P}}^u\left[ {\mathcal {S}}_\infty \cap \left( \left( \left\lfloor \frac{L_0}{2}\right\rfloor , \dots , \left\lfloor \frac{L_0}{2}\right\rfloor \right) + {\mathbb {Z}}\cdot e_i\right) \cap [0,L_k)^d\ne \emptyset \right] \\&\quad \ge 1 - Ce^{-c(\log L_k)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}} - 2\cdot 2^{-2^k}. \end{aligned}$$

For \(R\ge 1\), choose the largest k such that \(L_k\le R\). Then as in (3.4) and (3.6), we obtain that \(\log L_k \ge c\log R\) and \(2^k \ge (\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}\) for all R large enough. This implies that

$$\begin{aligned} {\mathbb {P}}^u\left[ {\mathcal {S}}_\infty \cap \left( \left( \left\lfloor \frac{L_0}{2}\right\rfloor , \dots , \left\lfloor \frac{L_0}{2}\right\rfloor \right) + {\mathbb {Z}}\cdot e_i\right) \cap {\mathrm {B}}(0,R)\ne \emptyset \right] \ge 1 - Ce^{-c(\log R)^{1+{\Delta _{\scriptscriptstyle {\mathrm {S}}}}}}. \end{aligned}$$
(5.3)

Assumption A3 now follows from P1 and (5.3).

We have checked that for any \(u\in (a,b)\), the probability measure \({\mathbb {P}}^u\) satisfies the assumptions A1A5, given that the family \(\{{\mathbb {P}}^u\}_{u\in (a,b)}\) satisfies P1P3 and S1S2. Thus, Theorem 1.1 follows from Theorem 4.1.

6 Remarks on ergodicity assumption

In this section we discuss possible weakenings of assumption P1, more precisely, its part concerning with ergodicity of \({\mathbb {P}}^u\). Condition P1 requires ergodicity of \({\mathbb {P}}^u\) with respect to every shift of \({\mathbb {Z}}^d\), i.e., \({\mathbb {P}}^u[E]\in \{0,1\}\) for every \(E\in {\mathcal {F}}\) such that \(\tau _x(E) = E\) for some \(x\in {\mathbb {Z}}^d\). This is crucially used in the proof of the shape theorem in [17]. However, the proof of [17, Theorem 1.3] goes through under the milder assumption of ergodicity of \({\mathbb {P}}^u\) with respect to the group \({\mathbb {Z}}^d\), i.e., \({\mathbb {P}}^u[E]\in \{0,1\}\) for every \(E\in {\mathcal {F}}\) such that \(\tau _x(E) = E\) for all \(x\in {\mathbb {Z}}^d\). Indeed, the only place where ergodicity is used in the proof of [17, Theorem 1.3] is [17, (4.1)], which still holds under the weaker assumption. Since [17, (4.1)] is used in the proof of Lemma 2.4 (Lemmas 4.2 and 4.4 in [17]), and since we do not use any form of ergodicity of \({\mathbb {P}}^u\) elsewhere in the proof of Theorem 1.2, we conclude that the result of Theorem 1.2 holds even if we replace the ergodicity of \({\mathbb {P}}^u\) with respect to every shift of \({\mathbb {Z}}^d\) in P1 by the ergodicity of \({\mathbb {P}}^u\) with respect to the group \({\mathbb {Z}}^d\).

Similarly, in the proof of the quenched invariance principle we do not need the full strength of assumption P1. Apart from the proof of Theorem 1.2, we use ergodicity of \({\mathbb {P}}^u\) to check assumptions A1, A3, and A4. Assumptions A1 and A3 hold under the milder assumption of ergodicity of \({\mathbb {P}}^u\) with respect to each shift along a coordinate direction, i.e., \({\mathbb {P}}^u[E]\in \{0,1\}\) for every \(E\in {\mathcal {F}}\) such that \(\tau _e(E) = E\) for some \(e\in {\mathbb {Z}}^d\) with \(|e|_1 = 1\). Assumption A4 holds under assumption of ergodicity of \({\mathbb {P}}^u\) with respect to the group \({\mathbb {Z}}^d\), as discussed just above. Therefore, the result of Theorem 1.1 holds when the ergodicity of \({\mathbb {P}}^u\) with respect to every shift of \({\mathbb {Z}}^d\) in P1 is replaced by the ergodicity of \({\mathbb {P}}^u\) with respect to each shift along a coordinate direction of \({\mathbb {Z}}^d\). We remark that in the case of the random conductance model with elliptic coefficients, the quenched invariance principle holds under the ergodicity of random coefficients with respect to the group \({\mathbb {Z}}^d\) and some moment assumptions, see [2, 10]. The tricky part is discussed at the end of the proof of [10, Lemma 4.8]. It crucially relies on the positivity of all the coefficients (every vertex of \({\mathbb {Z}}^d\) can be visited by the random walk) and does not generally apply if some coefficients are 0.