Lower bounds on fluctuations for internal DLA

Abstract

We consider internal diffusion limited aggregation in dimension two or more. This is a random cluster growth model, where random walks start at the origin of the lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. When $n$ random walks are sent from the origin, we establish a lower bound for the inner and outer errors fluctuations of order square root of the logarithm of $n$. When dimension is three or more, this lower bound matches the upper bound recently obtained in independent works of Asselah and Gaudillière (Ann Prob arXiv:1011.4592, 2010) and Jerison et al. (Internal DLA and the Gaussian free field arXiv:1012.3453, 2010). Also, we produce as a corollary of our proof of Asselah and Gaudillière (Ann Prob arXiv:1011.4592, 2010), an upper bound for the fluctuation of the inner error in a specified direction.

1 Introduction

We establish lower bounds for the inner and outer errors in internal diffusion limited aggregation (internal DLA). Internal DLA models a discrete cluster growth, and is defined as follows. Let $\Lambda$ be a subset of $\mathbb{Z }^d$ which represents the explored region at time 0. Let $N$ be an integer, and $\xi =(\xi _1,\ldots ,\xi _N)$ be the initial positions of $N$ independent simple random walks $S_1,\ldots ,S_N$ on $\mathbb{Z }^d$. The cluster that we denote $A(\Lambda , \xi )$ is obtained inductively. First, set $\tilde{A}(0)=\Lambda$, and assume $\tilde{A}(k-1)$ is obtained. Define

$$\begin{aligned} \tau _k=\inf \{ {t\ge 0:\ S_k(t)\not \in \tilde{A}(k-1)} \},\quad \mathrm{and} \quad \tilde{A}(k)=\tilde{A}(k-1)\cup \{S_k(\tau _k)\}.\qquad \end{aligned}$$

(1.1)

The internal DLA cluster is $A(\Lambda ,\xi )=\tilde{A}(N)$. We call explorers the random walks obeying the aggregation rule (1.1). We say that the explorer $k$ settles at time $\tau _k$. If at time 0, the $N$ walks start at the origin, with an empty explored region, we denote $A(\emptyset ,\xi )$ by $A(N)$.

In dimension two or more, Lawler et al. [7] consider $A(N)$, and prove that in order that $A(N)$ covers, without hole, a sphere of radius $n$, we need $N$ of order of the number of sites of $\mathbb{Z }^d$ in this sphere. In other words, the asymptotic shape of the cluster is a sphere. Then, Lawler in [8] shows a subdiffusive upper bound for the worse fluctuation to the spherical shape. More precisely, the latter result is formulated in terms of inner and outer errors, which we now introduce with some notation. We denote with $\Vert \cdot \Vert$ the euclidean norm on $\mathbb{R }^d$. For any $x$ in $\mathbb{R }^d$ and $r$ in $\mathbb{R }^+$, set

$$\begin{aligned} B(x,r) = \{ y\in \mathbb{R }^d :\, \Vert y-x\Vert < r\} \quad \text{ and}\quad \mathbb{B }(x,r) = B(x,r) \cap \mathbb{Z }^d. \end{aligned}$$

(1.2)

For $\Lambda \subset \mathbb{Z }^d,\,|\Lambda |$ denotes the number of sites in $\Lambda$. The inner error $\delta _I(n)$ is such that

$$\begin{aligned} n-\delta _I(n)=\sup \{ {r \ge 0:\, \mathbb{B }(0,r)\subset A(|\mathbb{B }(0,n)|)} \}. \end{aligned}$$

(1.3)

Also, the outer error $\delta _O(n)$ is such that

$$\begin{aligned} n+\delta _O(n)=\inf \{ {r \ge 0:\, A(|\mathbb{B }(0,n)|)\subset \mathbb{B }(0,r)} \}. \end{aligned}$$

(1.4)

Note that $\delta _I$ and $\delta _O$ are associated with the worse fluctuations to the spherical shape.

In 2010, Asselah–Gaudillière in [1, 2], and Jerison et al. in [4, 5], independently showed the following upper bound.

Theorem 1.1

When dimension $d\ge 3$, there are constants $\{\beta _d,\ d\ge 3\}$ such that with probability 1,

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\delta _I(n)}{\sqrt{\log (n)}}\le \beta _d, \quad and \quad \limsup _{n\rightarrow \infty } \frac{\delta _O(n)}{\sqrt{\log (n)}}\le \beta _d. \end{aligned}$$

(1.5)

When dimension is $2$, there is a constant $\{\beta _2\}$ such that with probability 1,

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{\delta _I(n)}{\log (n)}\le \beta _2, \quad and \quad \limsup _{n\rightarrow \infty } \frac{\delta _O(n)}{\log (n)}\le \beta _2. \end{aligned}$$

(1.6)

All these studies are concerned with upper bounds on $\delta _I$ and $\delta _O$, and the matter of showing that they were indeed realized remained untouched. We present now two results on lower bounds. The first one is independent from previous results. Morally, it says that if there are no deep hole, then long tentacles form. The result is optimal in $d\ge 3$.

Proposition 1.2

When dimension is three or more, and $h(k)= \sqrt{\log (k)}$, there is $\alpha _d>0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty } P( { \exists k>n,\ \delta _O(k)\ge \alpha _d h(k)\quad |\quad \delta _I(n)< \alpha _dh(n)} )=1. \end{aligned}$$

(1.7)

When dimension is two, set $h(k)= \sqrt{\log (k)\log (\log (k))}$. There is $\alpha _2>0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty } P( { \exists k>n,\ \delta _O(k)\ge \alpha _2 h(k)\quad |\quad \delta _I(n)< \alpha _2h(n)} )=1. \end{aligned}$$

(1.8)

Remark 1.3

Note that as a consequence of (1.7), we have $\alpha _d$ positive such that

$$\begin{aligned} P( { \delta _O(n)\ge \alpha _d\sqrt{\log (n)}\, \mathrm{or} \, \delta _I(n)\ge \alpha _d\sqrt{\log (n)},\ \mathrm{i.o.}} )=1, \end{aligned}$$

(1.9)

The second result uses the upper bound of Theorem 1.1. It says that deep holes do form.

Theorem 1.4

There are positive constants $\{\alpha _d,d\ge 2\}$ such that when $d\ge 2$,

$$\begin{aligned} P(\delta _I(n)> \sqrt{ \alpha _d \log (n)}\quad i.o.)=1 \end{aligned}$$

(1.10)

Remark 1.5

We believe that in dimension 2, for some $\alpha _2>0$, almost surely both $\delta _I(n)$ and $\delta _O(n)$ are larger than $\alpha _2 \log (n)$ infinitely often. However, the way of realizing this event is probably different from what we describe in $d\ge 3$.

We present now a side result dealing with fluctuation in a given direction. The results on fluctuations for internal DLA [1, 2, 4, 5, 7, 8] all focus on inner and outer errors. The paper [6] is still of a different nature: it addresses averaged error, in a sense where inner and outer errors can cancel each other out.

Though interesting, fluctuations in a given direction remained untreated. This is presented as an open problem in Section 3 of [5], and we note that the proof of [2] yields the following bound for the inner error in a given direction.

Proposition 1.6

There are positive constants $\{\kappa _d,d\ge 2\}$, such that for $z\in \mathbb{Z }^d$ with $\Vert z\Vert <n$, we have

$$\begin{aligned} P( { z\not \in A(|\mathbb{B }(0,n)|)} )\le \left\{ \begin{array}{ll} \exp \left({-\kappa _2 \frac{(n-\Vert z\Vert )^2}{\log (n)}}\right)&\quad for \quad d=2, \\ \exp ( {-\kappa _d (n-\Vert z\Vert )^2} )&\quad for \quad d\ge 3. \end{array} \right. \end{aligned}$$

(1.11)

Let us explain briefly why Proposition 1.6 follows from [2]. In [7], the authors consider the number of explorers exiting $\mathbb{B }(0,r)$ from a site $z$ in the boundary of $\mathbb{B }(0,R)$ (actually they consider $z$ inside $\mathbb{B }(0,R)$). This quantity is denoted $W_R(z)$ when the explorers are initially at the origin. Lawler et al. in [7] express $W_R(z)$ as a difference of two sums of independent Bernoulli random variables, say for simplicity $M_R(z)$ and $M_R^{\prime }(z)$. The structure of the relation is that $W_R(z)+M_R^{\prime }(z)\ge M_R(z)$. $M_R(z)$ (resp. $M_R^{\prime }(z)$) counts the number of random walks starting at the origin (resp. starting on each site of $\mathbb{B }(0,r)$), and which exit $\mathbb{B }(0,R)$ on site $z$. Now, if $\mu _R(z)$ is the expectation of $W_R(\eta ,z)$, we have obtained in [1] that for a positive constant $\kappa$

$$\begin{aligned} P(W_{R}(z)=0)\le \exp \left(-\kappa \frac{\mu ^2_R(z)}{\mu _R(z)+\sum _{y\in \mathbb{B }(0,R)}h^2_z(y)}\right), \end{aligned}$$

(1.12)

where $h_z(y)$ is the probability to exit from $\mathbb{B }(0,R)$ on site $z$, when the initial position of the walk is $y$. Now, $\mu _R(z)$ and $y\mapsto h_z(y)$ are estimated in [2].

The rest of the paper is organized as follows. In Sect. 2, we set notation and recall useful results. In Sect. 3, we deal with the outer error and prove Proposition 1.2. In Sect. 4, we deal with the inner error, and prove Theorem 1.4. Finally, in Sect. 5, we explain how to read Proposition 1.6 as a corollary of our previous work [2].

2 Notation and prerequisite

Let $S:\mathbb{N }\rightarrow \mathbb{Z }^d$ denotes a simple random walk on $\mathbb{Z }^d$. When the initial condition is $S(0)=z\in \mathbb{Z }^d$, its law is denoted $\mathbb{P }_z$. The first time $S$ hits a domain $\Lambda \subset \mathbb{Z }^d$, is denoted $H(\Lambda )$, or $H(S;\Lambda )$ to emphasize that $S$ is the walk.

For a positive $\gamma$, we denote by $\rho (\gamma )$ the radius of the largest ball centered at 0 whose volume is less than $\gamma$. In other words,

$$\begin{aligned} \rho (\gamma )=\sup \{ {n\ge 0: |\mathbb{B }(0,n)|\le \gamma } \}. \end{aligned}$$

(2.1)

The abbreviation $b(n)=|\mathbb{B }(0,n)|$ is also handy. We denote with $\Vert \cdot \Vert$ the euclidean norm on $\mathbb{R }^d$. For a subset $\Lambda$ of $\mathbb{Z }^d$, we denote by $\partial \Lambda =\{z\not \in \Lambda :\ \exists y\in \Lambda , \Vert y-z\Vert =1\}$.

We now consider a configuration $\eta \in \mathbb{N }^{\mathbb{Z }^d}$ with a finite number of particles

$$\begin{aligned} |\eta |:=\sum _{z\in \mathbb{Z }^d} \eta (z)<\infty . \end{aligned}$$

(2.2)

For $R>0,\,\Lambda \subset \mathbb{Z }^d$, and $z\in \partial \mathbb{B }(0,R)$, we denote by $W_R(\Lambda ;\eta ,z)$ the number of explorers visiting $z$ at the moment they reach $\partial \mathbb{B }(0,R)$, when the explorers start on $\eta$ with an explored region $\Lambda$. The final positions of explorers which settle before they exit $\mathbb{B }(0,R)$ is denoted $A_R(\Lambda ;\eta )$, and is a subset of $\mathbb{B }(0,R)$.

When we build the internal DLA cluster from $n$ independent random walks, we also consider the unrestricted trajectories. This gives a natural coupling between explorers and independent random walks. Note that if $\mathbb{B }(0,R)\subset \Lambda$ and $z\in \partial \mathbb{B }(0,R)$, then

$$\begin{aligned} W_R(\Lambda ;\eta ,z)=W_R(\mathbb{B }(0,R);\eta ,z), \end{aligned}$$

and this corresponds to the number of independent random walks which exit $\mathbb{B }(0,R)$ at site $z$, and this latter number is denoted $M_R(\eta ,z)$.

2.1 The abelian property

Diaconis and Fulton [3] allow explorers to start on distinct sites, and show that the law of the cluster is invariant under permutation of the order in which explorers are launched. This invariance is named the abelian property. As a consequence, one can realize the cluster by sending many exploration waves. Let us illustrate this observation by building $A(\emptyset ,(n+m)\delta _0)$ in three waves, since we need later this very example. The first wave consists in launching $n$ explorers, and they settle in $A_1=A(\emptyset ,n\delta _0)$, which we call the cluster after the first wave. Then, we launch $m$ explorers that we color green for simplicity, and if they reach $\partial \mathbb{B }(0,R)$ before settling, then we stop them on $\partial \mathbb{B }(0,R)$. The settled green explorers make up the cluster $A_R(A_1; m\delta _0)$. The cluster after the second wave is then

$$\begin{aligned} A_2=A_1\cup A_R(A_1; m\delta _0). \end{aligned}$$

For $z\in \partial \mathbb{B }(0,R)$, we call $\zeta _R(z)=W_R(A_1;m\delta _0,z)$. This is the configuration of the green explorers stopped on $\partial \mathbb{B }(0,R)$. Finally, the cluster after the third wave is obtain as we launch the stopped green explorers, and

$$\begin{aligned} A_3=A_2\cup A(A_2;\zeta _R). \end{aligned}$$

The abelian properties implies that $A(\emptyset ,(n+m)\delta _0)$ equals in law to $A_3$. It is convenient to think of the growing cluster as evolving in discrete time, where time counts the number of exploration waves.

2.2 On the harmonic measure.

We first recall a well known property of Poisson variables.

Lemma 2.1

Let $\{U_n,n\in \mathbb{N }\}$ be an i. i. d.  sequence with values in a set $E$, and $\{E_1,\ldots ,E_n\}$ a partition of $E$. Then, if $X$ is an independent Poisson random variable of parameter $\lambda$, and if

$$\begin{aligned} X_i=\sum _{n\le X} {\small 1}\!\!1_{U_n\in E_i}, \end{aligned}$$

then $\{X_i,\ i=1,\ldots ,n\}$ are independent Poisson variables with $E[X_i]=\lambda \times P(U\in E_i)$.

Now, for $z\in \mathbb{Z }^d\backslash \{0\}$, let

$$\begin{aligned} \Sigma (z)=\partial \mathbb{B }(0,\Vert z\Vert ), \end{aligned}$$

(2.3)

and note that $z$ belongs to $\Sigma (z)$ since there is $z^{\prime }\in \mathbb{B }(0,\Vert z\Vert )$ and $\Vert z-z^{\prime }\Vert =1$ (see Lemma 2.1 of [1]). We start independent random walks with initial configuration $\eta$, and for $z\not = 0,h>0$, and $\Lambda \subset \Sigma (z)$, we denote by $N_z(\eta ,\Lambda ,h)$ the number of these walks which exit $\Sigma (z)$ on $\Lambda$, and visit $z$ before exiting $\mathbb{B }(0,\Vert z\Vert +h)$. As a consequence of Lemma 2.1, we have

Corollary 2.2

Let $X$ be a Poisson variable of parameter $\lambda$, and $\eta =X\delta _0$. For any $z\not = 0,h,h^{\prime }>0$, and $\Lambda ,\Lambda ^{\prime } \in \Sigma (z)$ with $\Lambda \cap \Lambda ^{\prime }=\emptyset$, we have that $N_z(\eta ,\Lambda ,h)$ and $N_z(\eta ,\Lambda ^{\prime },h^{\prime })$ are independent Poisson variable, and

$$\begin{aligned} E[ {N_z(\eta ,\Lambda ,h)} ]&= \lambda \times \mathbb{P }_0(S(H(\Sigma (z)))\in \Lambda , \ H(z)<H(\mathbb{B }^c(0,\Vert z\Vert +h)))\nonumber \\&= \lambda \sum _{y\in \Lambda }\mathbb{P }_0(S(H(\Sigma (z))=y)) \mathbb{P }_y(\ H(z)<H(\mathbb{B }^c(0,\Vert z\Vert +h))).\nonumber \\ \end{aligned}$$

(2.4)

By combining well known asymptotics of the harmonic measure with Corollary 2.2, we obtain the following lemma.

Lemma 2.3

Assume that dimension is two or more. Let $\eta$ be as in Corollary 2.2. For $z\not = 0$, and $R>0$ let $\Lambda =\mathbb{B }(z,R)\cap \Sigma (z)$, and $\Lambda ^{\prime }=\Sigma (z)\backslash \Lambda$. There is $\kappa >0$, independent of $z$, and $R$ such that

$$\begin{aligned} P(N_z(\eta ,\Lambda ,\infty )=0,\ N_z(\eta ,\Lambda ^{\prime },R)=0)\ge \exp \left({-\kappa \frac{\lambda R}{\Vert z\Vert ^{d-1}}}\right). \end{aligned}$$

(2.5)

Proof

Since $N_z(\eta ,\Lambda ,\infty )$ and $N_z(\eta ,\Lambda ^{\prime },R)$ are independent Poisson variables, we have

$$\begin{aligned} P(N_z(\eta ,\Lambda ,\infty )&= 0,\ N_z(\eta ,\Lambda ^{\prime },R)=0)\nonumber \\&= \exp (-E[ {N_z(\eta ,\Lambda ,\infty )} ] -E[ {N_z(\eta ,\Lambda ^{\prime },R)} ]). \end{aligned}$$

(2.6)

It remains to compute expected values. To estimate $E[ {N_z(\eta ,\Lambda ,\infty )} ]$, we recall that there is a constant $c_d$ such that

$$\begin{aligned} \mathbb{P }_y(H(z)<\infty )\le \frac{c_d}{1+\Vert y-z\Vert ^{d-2}}. \end{aligned}$$

(2.7)

Using (2.4), there is a constant $c$

$$\begin{aligned} E[ {N_z(\eta ,\Lambda ,\infty )} ]\le&\sum _{y\in \Lambda } \lambda \mathbb{P }_0(S(H(\Sigma (z)))=y)\times \frac{c_d}{1+\Vert y-z\Vert ^{d-2}}\nonumber \\ \le&\frac{c}{2}\frac{\lambda }{\Vert z\Vert ^{d-1}} \left(1+\sum _{k=1}^{R} \frac{k^{d-2}}{1+k^{d-2}}\right)\le c \frac{\lambda R}{\Vert z\Vert ^{d-1}}. \end{aligned}$$

(2.8)

Now, to estimate $E[ {N_z(\eta ,\Lambda ^{\prime },R)} ]$, we recall Lemma 5(b) of [4], which states that for some constant $c_d^{\prime }$, for $y\in \Sigma (z)$

$$\begin{aligned} \mathbb{P }_y( H(z)< H( \mathbb{B }^c(0,\Vert z\Vert +R)))\le \frac{ c_d^{\prime } R^2}{\Vert z-y\Vert ^d} \end{aligned}$$

(2.9)

Using (2.4) and (2.9), there is a constant $c^{\prime }$ such that

$$\begin{aligned} E[ {N_z(\eta ,\Lambda ^{\prime },R)} ]&\le \sum _{y\in \Sigma (z)\backslash \mathbb{B }(z,R)} \lambda \mathbb{P }_0(S(H(\Sigma (z)))=y) \frac{ c_d^{\prime } R^2}{\Vert z-y\Vert ^d} \nonumber \\&\le \ \frac{c^{\prime }}{2} \frac{\lambda R^2}{\Vert z\Vert ^{d-1}} \sum _{k=R}^{2\Vert z\Vert +2} \frac{k^{d-2}}{k^d}\le \ c^{\prime } \frac{\lambda R}{\Vert z\Vert ^{d-1}} \end{aligned}$$

(2.10)

Combining (2.8), and (2.10), we obtain the desired result.

3 The outer error

In this section, we prove Proposition 1.2. Let us explain the proof in dimension three or more, and explain, in Remark 3.1 of Step 2, how we adapt the proof to dimension two.

For positive reals $\alpha$ and $\gamma$, to be chosen later, we set $h(n)=\alpha \sqrt{\log (n)}$ and $\underline{L}(n)= \gamma \sqrt{\log (n)}$ for estimates on the outer and inner fluctuations. Even though we eventually take $h(n)=\underline{L}(n)$, it is useful to keep in mind their distinct nature. The limit (1.9) follows if for some small $\gamma =\alpha$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty } P( { \exists k\ge n,\ \delta _O(k)\ge h(k) \ |\ \delta _I(n)<\underline{L}(n)} )=1. \end{aligned}$$

(3.1)

Indeed,

$$\begin{aligned}&P(\exists k\ge n,\ \delta _O(k)\ge h(k)\quad \text{ or} \quad \delta _I(k)\ge \underline{L}(k)) \nonumber \\&\ge P(\delta _I(n)\ge \underline{L}(n), \text{ or} \ \exists k\ge n,\ \delta _O(k)\ge h(k))\nonumber \\&\ge P(\delta _I(n)\ge \underline{L}(n)) +P( { \exists k\ge n,\ \delta _O(k)\ge h(k)\ |\ \delta _I(n)< \underline{L}(n)} )P(\delta _I(n)< \underline{L}(n))\nonumber \\&\ge 1-(1-P(\exists k\ge n,\ \delta _O(k)\ge h(k)\ |\ \delta _I(n)< \underline{L}(n))) P(\delta _I(n)< \underline{L}(n)). \end{aligned}$$

(3.2)

We now prove 3.1. For an integer $n$, assume that $A(b(n)\delta _0)$ is realized. Let $X_n$ be a Poisson random variable with parameter $\lambda _n= |\mathbb{B }(0,n+h(n))\backslash \mathbb{B }(0,n)|$. We realize the cluster $A((b(n)+X_n)\delta _0)$ through three exploration waves, as explained in Sect. 2.1. After the first wave with $b(n)$ explorers, we launch $X_n$ explorers, the green ones, and stop them on $\Sigma :=\partial \mathbb{B }(0,n-\underline{L}(n))$. Recall that $W_{n-\underline{L}(n)}(A(b(n)\delta _0);X_n\delta _0,z)$ corresponds to the number of explorers, out of $X_n$ initially at 0, which exit the explored region $A(b(n)\delta _0)$ at site $z$. Under the event $\{\delta _I(n)<\underline{L}(n)\}$, we have for $z\in \Sigma$

$$\begin{aligned} W_{n-\underline{L}(n)}(A(b(n)\delta _0);X_n\delta _0,z)= M_{n-\underline{L}(n)}(X_n\delta _0,z). \end{aligned}$$

The configuration of the random walks (associated with the green explorers) stopped on $\Sigma$ is denoted $\zeta ^{\prime }$. Note that $\zeta ^{\prime }$ is independent of $A(b(n)\delta _0)$ and that on the event $\{\delta _I(n)<\underline{L}(n)\}$ we have $\zeta ^{\prime }=\zeta _{n-\underline{L}(n)}$ (with the notation of Sect. 2.1). The key observation now is that $\{ \zeta ^{\prime }(z),\ z\in \Sigma \}$ are independent Poisson variables which are also independent of $\delta _I(n)$. Indeed, $\{ \zeta ^{\prime }(z),\ z\in \Sigma \}$ deals with the walks associated with the green explorers, whereas $\delta _I(n)$ depends on the $b(n)$ explorers which we launch first. Moreover, the expected value of $\zeta ^{\prime }(z)$ is easy to estimate. Note that $E[\zeta ^{\prime }(z)]=E[X_n]\times \mathbb{P }_0(S(H(\Sigma ))=z)$, and that $|\Sigma |$ is of order $n^{d-1}$, so that there are two positive constants $\bar{c},\underline{c}$ such that

$$\begin{aligned} \underline{c} h(n)\le E[\zeta ^{\prime }(z)]\le \bar{c} h(n). \end{aligned}$$

(3.3)

Also, since $\zeta ^{\prime }(z)$ is a Poisson variable, for $A\ge 1$, we mention an obvious tail estimate.

$$\begin{aligned} P(\zeta ^{\prime }(z)\ge A E[\zeta ^{\prime }(z)]) \ge \frac{\exp (-\log (A) A E[\zeta ^{\prime }(z)])}{\sqrt{\pi A E[\zeta ^{\prime }(z)]}}. \end{aligned}$$

(3.4)

For $z\in \Sigma$, we call cov$(z)$ the event that $\zeta ^{\prime }(z)$ explorers starting on $z$ produce a cluster $A(\emptyset ,\zeta ^{\prime }(z)\delta _z)$ which satisfies

$$\begin{aligned} A(\emptyset ,\zeta ^{\prime }(z)\delta _z)\cap \mathbb{B }^c(0,n+4h(n))\not = \emptyset . \end{aligned}$$

(3.5)

Note that the explorers contributing to cov$(z)$ start on the positions of the random walks stopped on $z\in \Sigma$, which are associated with the green explorers. Assume, for a moment, that when cov$(z)$ happens, there is a tentacle of $A((b(n)+X_n)\delta _0)$ which protrudes $\mathbb{B }(0,n+4h(n))$. Assume also that under condition on $X_n$, we have

$$\begin{aligned} n+4h(n)\ge R_n+h(R_n),\quad \mathrm{where}\quad R_n=\rho (b(n)+X_n). \end{aligned}$$

(3.6)

We would deduce that $\delta _O(R_n)\ge h(R_n)$.

We now proceed through four steps. First, we show that (3.5) implies that the final cluster is not inside $\mathbb{B }(0,n+4h(n))$. Secondly, we estimate the cost of producing a tentacle realizing cov$(z)$. Then, we establish conditions ensuring (3.6). Finally, we show that for an appropriate choice of $\alpha$, one event cov$(z)$ realizes for some $z\in \Sigma$.

Step 1: Coupling By coupling, it is easy to see that for any subset $\Lambda$, and $z\in \Sigma$

$$\begin{aligned} A(\emptyset ;\zeta ^{\prime }(z)\delta _z)\subset \Lambda \cup A(\Lambda ;\zeta ^{\prime }(z)\delta _z)\subset \Lambda \cup A(\Lambda ;\zeta ^{\prime }). \end{aligned}$$

(3.7)

If we denote by $A_2$ the cluster after the second exploration wave (see Sect. 2.1), then we have with an equality in law

$$\begin{aligned} A_2\cup A(A_2; \zeta ^{\prime })= A((b(n)+X_n)\delta _0). \end{aligned}$$

(3.8)

Now, using (3.7), (3.8), and (3.5), we conclude that the final cluster is not in $\mathbb{B }(0,n+4h(n))$.

Step 2: Long tentacles To produce cov$(z)$, we first bring a number of explorers at $z$ proportional to $h(n)$, and force them to make a tentacle normal to $\Sigma$ at $z$, with a height $4h(n)+ \underline{L}(n)$. More precisely, draw unit cubes centered on the points of the sequence

$$\begin{aligned} x_n=(\Vert z\Vert +n) \frac{z}{\Vert z\Vert }\in \mathbb{R }^d\quad (\text{ and}\quad \Vert x_n\Vert =\Vert z\Vert +n). \end{aligned}$$

Each such cube contains at least a site of $\mathbb{Z }^d$, say $z_n$. Note that $\Vert z_n-z_{n-1}\Vert \le 2 \sqrt{d}+1$, so that we can exhibit a sequence $\{z=y_1,y_2,\ldots ,y_N\}$ of nearest neighbors in $\mathbb{Z }^d$ such that $\Vert y_N-z\Vert \ge 4h(n)+\underline{L}(n)$, with $N\le c(4h(n)+\underline{L}(n))$ for some constant $c$ independent of $n$. Now, if $\zeta ^{\prime }(z)\ge N$, and if we launch the green explorers stopped on $z$ and force the first $N$ of them them to walk along the sequence $\{y_1,y_2,\ldots ,y_N\}$, with the $k$-th explorer settling on $y_k$, then we realize cov$(z)$, and

$$\begin{aligned}&P(\text{ cov}(z))\ge P(\zeta ^{\prime }(z)\ge N)\times \left({\frac{1}{2d}}\right)^{\sum \nolimits _{k=1}^N k}\ge P(\zeta ^{\prime }(z)\ge N) \nonumber \\&\quad \times \exp (-c (4h(n)+\underline{L}(n))^2). \end{aligned}$$

(3.9)

Remark 3.1

In dimension 2, there is a better strategy to build a tentacle. Since we believe that it yields an estimate which is not optimal, we do not give the full proof, but give enough steps of the construction so that the interested reader can easily fill the details.

We first bring a larger number of green explorers at $z$, about $\frac{h(n)}{\log ^2(h(n))}\times h(n)$ of them. The probability of so doing is larger than

$$\begin{aligned} \exp \left(-C \frac{h(n)}{\log ^2(h(n))}\log \left(\frac{h(n)}{\log ^2(h(n))} \right) \times h(n)\right)\ge \exp \left(-c\frac{h^2(n)}{\log (h(n))}\right),\qquad \qquad \end{aligned}$$

(3.10)

for two positive constants $C,c$. Then, the explorers are forced to fill sequentially cylindrical compartments of a telescope-like domain that we now describe. Let $R$ be the integer part of $4h(n)+\underline{L}(n)$, and divide $B(z,R)$ into $R$ shells of length $h_1,\ldots ,h_R$ with for $i=1,\ldots ,R$

$$\begin{aligned} h_i=\frac{R}{i\times \log (R)}, \quad \text{ and}\quad H_i=\sum _{j=1}^i h_j. \end{aligned}$$

(3.11)

Choose the sequence of $R$ points of $\mathbb{R }^d$

$$\begin{aligned} \forall i\in \{1,\ldots ,R\}\quad x_i=(\Vert z\Vert +H_i) \frac{z}{\Vert z\Vert } \end{aligned}$$

There is $z_i\in \partial \mathbb{B }(z,H_i)$ with $\Vert z_i-x_i\Vert \le 2$. Now, for a constant $A_0$ appearing in Lemma 1.3 of [2] (and which is independent of $R$), we bring $n_i:=A_0\pi h_{i+1}^2$ explorers in $\mathbb{B }(z_i,h_i/4)$. Then, with a probability larger than $1-1/h_i^2$, they cover the ball $\mathbb{B }(z_i,h_{i+1})$ by Lemma 1.3 of [2]. Now, if we bring additional explorers in $\mathbb{B }(z_i,h_i/4)$, they can reach $\mathbb{B }(z_{i+1},h_{i+1}/4)$ with a positive probability, say $\exp (-\kappa )$: indeed, they only need to escape a square-like domain centered on $z_i$, of side-length $h_{i+1}/4$ on the side which separate $z_i$ from $z_{i+1}$. The cost of the scenario, for which we only described the $i$-th step, is therefore of order

$$\begin{aligned} \prod _{i=1}^{R-1}\left({1-\frac{1}{h_i^2}}\right)\;\times\;e^{-\kappa \sum \nolimits _{i=2}^R n_i}\nonumber \\&\times\;e^{-\kappa \sum \nolimits _{i=3}^R n_i}\cdots \times e^{-\kappa n_R}\ge C\exp \left(-\kappa \sum _{i=2}^R (i-1)n_i\right),\qquad \qquad \end{aligned}$$

(3.12)

for positive constants $\kappa ,C$. Note that with the choice of $h_i$ in (3.11), and $n_i=A_0\pi h_{i+1}^2$, we have with $\underline{L}(n)=h(n)$ and for a positive constant $\kappa ^{\prime }$

$$\begin{aligned} \sum _{i=2}^R (i-1)n_i\ge \kappa ^{\prime } \frac{h^2(n)}{\log (h(n))}. \end{aligned}$$

Thus, the probability of bringing $h^2(n)/\log ^2(n)$ explorers in $z$, and the probability of building a tentacles of height $5h(n)$ is larger than

$$\begin{aligned} \exp \left(-\kappa _2\frac{h^2(n)}{\log (h(n))}\right)\!, \end{aligned}$$

(3.13)

for some positive constant $\kappa _2$.

Step 3: Bounding $X_n$ We impose that $X_n\le 2\lambda _n$.

$$\begin{aligned} X_n&\le 2|\mathbb{B }(0,n+h(n))\backslash \mathbb{B }(0,n)| \nonumber \\&\le |\mathbb{B }(0,n+2h(n))\backslash \mathbb{B }(0,n)| \Longrightarrow X_n+b(n) \nonumber \\&\le |\mathbb{B }(0,n+2h(n))| \Longrightarrow R_n\le n+2h(n). \end{aligned}$$

(3.14)

The conclusion of (3.14) implies also, for $n$ large enough, that $h(R_n)\le 2h(n)$, and this implies (3.6). Note that since $X_n$ is a Poisson variable of mean $\lambda _n$, there is a constant $c$, such that

$$\begin{aligned} P(X_n> 2\lambda _n)\le \exp (-ch(n)n^{d-1}). \end{aligned}$$

(3.15)

Step 4: Many possible tentacles We summarize Step 1 to Step 3, as establishing that

$$\begin{aligned}&\{ { X_n\le 2 \lambda _n,\ \exists z\in \Sigma \ \text{ cov}(z),\ \delta _I(n)<\underline{L}(n)} \}\subset \{ { \delta _O(R_n)\ge h(R_n),\ \delta _I(n)<\underline{L}(n)} \} \nonumber \\&\quad \subset \{ {\exists k>n, \delta _O(k)\ge h(k),\ \delta _I(n)<\underline{L}(n)} \}. \end{aligned}$$

(3.16)

Taking probability on both sides of (3.16), and dividing by $P(\delta _I(n)<\underline{L}(n))$, we obtain

$$\begin{aligned} P(\exists k>n, \delta _O(k)\ge h(k)| \delta _I(n)<\underline{L}(n))\ge P( X_n\le 2 \lambda _n,\ \exists z\in \Sigma \quad \mathrm{cov}(z)).\qquad \quad \end{aligned}$$

(3.17)

Thus,

$$\begin{aligned} P(\exists k>n, \delta _O(k)\ge h(k)| \delta _I(n)<\underline{L}(n))\ge P\left(\bigcup _{z\in \Sigma } \mathrm{cov} (z)\right)- P( X_n> 2 \lambda _n).\nonumber \\ \end{aligned}$$

(3.18)

Using now (3.9), (3.3), the lower bound (3.4), and that $|\Sigma |$ is of order $n^{d-1}$, there are some positive constants $\kappa ,\kappa ^{\prime }$ such that

$$\begin{aligned} P\left({\bigcup _{z\in \Sigma } \text{ cov}(z)}\right)\ge&1- \prod _{z\in \Sigma } (1-P(\text{ cov}(z) \cap \ \{\zeta ^{\prime }(z)>c(4h(n)+\underline{L}(n))\})) \nonumber \\ \ge&1-\exp ( {-\kappa ^{\prime } n^{d-1} \exp ( {-\kappa (4h(n)+\underline{L}(n))^2} )} ). \end{aligned}$$

(3.19)

Now, for $\alpha >0$ small enough, and $\underline{L}(n)=h(n)= \alpha \sqrt{\log (n)}$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }n^{d-1}\exp ( {-\kappa (h^2(n)+h(n))} )=\infty . \end{aligned}$$

(3.20)

The proof is now completed. In dimension 2, the estimate (3.19) has to be replaced with (3.13).

4 The inner error

In this section, we prove Theorem 1.4. We show that in the process of going from a cluster of volume $b(n)$ to one of volume $b(2n)$, chances tend to one as $n$ tends to infinity, that there appears a cluster $A$ whose inner error is larger than $\alpha \sqrt{\log (\rho (|A|))}$ for some positive $\alpha$ independent of $n$. To do so, we launch many exploration waves, each one is made up of a Poisson number of explorers.

We proceed inductively. For positive reals $\alpha ,\beta$, to be chosen later, we set $h(n)=\alpha \sqrt{\log (n)}$, and $\bar{L}(n) =\beta \sqrt{\log (n)}$. $\bar{L}(n)$ will refer to an outer radius. Let $\{\mathcal{{G}}_n,\ n\ge 0\}$ denotes the natural filtration associated with the evolution by waves.

First, we launch $b(n)$ explorers. Assume that explorers of wave $k-1$ have been launched, and are settled. Knowing $\mathcal{{G}}_{k-1}$, the size of the $k$-th wave, denoted $X_k$, is a Poisson variable of parameter

$$\begin{aligned} \lambda (k)&= |\mathbb{B }(0,R_{k-1}+2h(R_{k-1}))\backslash \mathbb{B }(0,R_{k-1}))|, \quad \mathrm{where}\nonumber \\&R_{k-1}=\rho (b(n)+X_1+\cdots +X_{k-1}). \end{aligned}$$

(4.1)

Since $R_{k-1}$ is of order $n$, each wave fills approximately a peel of width $2h(n)$, and $n/2h(n)$ waves fill approximately $\mathbb{B }(0,2n)$. We prove in this section that for an appropriate $\alpha$

$$\begin{aligned} \lim _{n\rightarrow \infty } P\left({\bigcup _{1\le k<n/2h(n)} \{ {\delta _I(R_k)> \alpha \sqrt{\log (R_k)}} \}}\right)=1. \end{aligned}$$

(4.2)

We now proceed with estimating the probability of observing a deep hole after each exploration waves. We set $\mathcal{{A}}_k=\{ {\delta _I(R_k)> \alpha \sqrt{\log (R_k)}} \}$.

On the holes left after wave $k-1$ Observe that by definition, on $\mathcal{{A}}^c_{k-1}$

$$\begin{aligned} \mathbb{B }(0,R_{k-1}-h(R_{k-1}))\subset A(b(n)+X_1+\cdots +X_{k-1}), \end{aligned}$$

which implies that

$$\begin{aligned}&(\mathbb{B }(0,R_{k-1})\backslash \mathbb{B }(0,R_{k-1}-h(R_{k-1}))\cup \partial \mathbb{B }(0,R_{k-1})) \nonumber \\&\quad \cap A(b(n)+X_1+\cdots +X_{k-1})\not = \emptyset . \end{aligned}$$

(4.3)

Choose any $Z_k$ in the intersection of the non-empty set of (4.3), and note that

$$\begin{aligned} R_{k-1}-h(R_{k-1})\le \Vert Z_k\Vert \le R_{k-1}+1. \end{aligned}$$

(4.4)

Recall that we have defined $\Sigma (Z_k):=\partial \mathbb{B }(0,\Vert Z_k\Vert )$. We launch the $X_k$ explorers, that we name the green explorers, and we stop them as they reach $\Sigma (Z_k)$. The green explorers which settle before reaching $\Sigma (Z_k)$ play no role here, and we bound the number of green explorers stopped on some region $\Lambda \in \Sigma (Z_k)$, by the number of corresponding random walks exiting $\Sigma (Z_k)$ on $\Lambda$. Thus, if we choose

$$\begin{aligned} \Lambda _k=\mathbb{B }(Z_k,\bar{L}(R_k))\cap \Sigma (Z_k), \quad \text{ and} \quad \Lambda ^{\prime }_k=\Sigma (Z_k)\backslash \Lambda _k, \end{aligned}$$

(4.5)

and if we denote

$$\begin{aligned} I_k=\{ {N_{Z_k}(X_k\delta _0,\Lambda _k,\infty )=0, \quad N_{Z_k}(X_k\delta _0,\Lambda ^{\prime }_k,7\bar{L}(R_{k-1}))=0} \} \end{aligned}$$

(4.6)

then, on the event $I_k$, green explorers either exit a ball of radius $\Vert Z_k\Vert +7\bar{L}(R_{k-1})$, or do not visit $Z_k$. In other words,

$$\begin{aligned} I_k\subset \{ {R_k+\delta _O(R_k)\ge \Vert Z_k\Vert +7\bar{L}(R_{k-1})} \}\cup \{ {\delta _I(R_k)> R_k-\Vert Z_k\Vert } \}. \end{aligned}$$

In order to conclude that $\{\delta _I(R_k)\ge h(R_k)\}$ or $\{\delta _O(R_k)\ge \bar{L}(R_{k})\}$, we need to find conditions on $X_k$ that guarantee that

$$\begin{aligned} R_k-\Vert Z_k\Vert \ge h(R_k),\quad \text{ and}\quad \Vert Z_k\Vert -R_k+7\bar{L}(R_{k-1})\ge \bar{L}(R_{k}). \end{aligned}$$

(4.7)

Conditions on $X_k$ fulfilling (4.7) We call

$$\begin{aligned} \mathcal{{C}}_k=\{ {2\lambda (k)\ge X_k} \}\cap \left\{ X_k\ge \frac{2}{3}\lambda (k)\right\} . \end{aligned}$$

On the one hand, if $X_k\ge \frac{2}{3}\lambda (k)$, then

$$\begin{aligned} X_k&\ge \frac{2}{3}\big |\mathbb{B }(0,R_{k-1}+2h(R_{k-1}))\backslash \mathbb{B }(0,R_{k-1})\big | \nonumber \\&\ge \big |\mathbb{B }(0,R_{k-1}+\frac{4}{3}h(R_{k-1}))\backslash \mathbb{B }(0,R_{k-1})\big | \Longrightarrow R_k\ge R_{k-1}+\frac{4}{3} h(R_{k-1}).\qquad \qquad \end{aligned}$$

(4.8)

On the other hand, if $X_k\le 2 \lambda _k$, then

$$\begin{aligned} X_k&\le 2\big |\mathbb{B }(0,R_{k-1}+2h(R_{k-1}))\backslash \mathbb{B }(0,R_{k-1})\big |\nonumber \\&\le \big |\mathbb{B }(0,R_{k-1}+4h(R_{k-1}))\backslash \mathbb{B }(0,R_{k-1})\big | \Longrightarrow R_k\le R_{k-1}+4h(R_{k-1}).\qquad \qquad \end{aligned}$$

(4.9)

Now, for $x$ large enough, the following implication is obvious

$$\begin{aligned} x\le y+4h(y)\Longrightarrow h(x)\le \frac{4}{3}h(y)-1 \quad \mathrm{and}\quad \bar{L}(x)\le 2 \bar{L}(y). \end{aligned}$$

(4.10)

If $n$ is large enough, (4.10) and (4.9) imply that $h(R_k)\le \frac{4}{3}h(R_{k-1})-1$, which in turn, with (4.8), yields

$$\begin{aligned} R_k\ge R_{k-1}+1+h(R_k). \end{aligned}$$

(4.11)

Also, $\bar{L}(R_k)\le 2 \bar{L}(R_{k-1})$, and when $\alpha$ is small enough, then $\bar{L}(R_{k-1})\ge h(R_{k-1})$. Recalling (4.9) we have

$$\begin{aligned} R_{k-1}-h(R_{k-1})+7\bar{L}(R_{k-1})-\bar{L}(R_{k})\ge R_{k-1}+4h(R_{k-1})\ge R_k. \end{aligned}$$

(4.12)

Thus, if $\mathcal{{C}}_k\cap \mathcal{{A}}^c_{k-1}$ holds, then (4.4), (4.11) and (4.12) imply that conditions (4.7) holds.

On a deep hole in one shell We choose an integer $k<n/2h(n)$. We have seen that

$$\begin{aligned} \mathcal{{A}}^c_{k-1}\cap I_k\cap \mathcal{{C}}_k\subset \mathcal{{A}}^c_{k-1}\cap (\mathcal{{A}}_k \cup \{ {\delta _0(R_k)\ge \bar{L}(R_k)} \}). \end{aligned}$$

(4.13)

Taking conditional probabilities on both sides of (4.13), we obtain,

$$\begin{aligned}&{\small 1}\!\!1_{\mathcal{{A}}^c_{k-1}} P(\mathcal{{A}}^c_k\cap \mathcal{{C}}_k\ |\mathcal{{G}}_{k-1})\nonumber \\&\quad = {\small 1}\!\!1_{\mathcal{{A}}^c_{k-1}}(P(\mathcal{{C}}_k)-P(\mathcal{{A}}_k\cap \mathcal{{C}}_k|\mathcal{{G}}_{k-1})) \nonumber \\&\quad \le {\small 1}\!\!1_{\mathcal{{A}}^c_{k-1}}( P(\mathcal{{C}}_k)-P(I_k\cap \mathcal{{C}}_k|\mathcal{{G}}_{k-1})+ P(\{ {\delta _0(R_k)\ge \bar{L}(R_k)} \}|\mathcal{{G}}_{k-1}))\nonumber \\&\quad \le P(\{ {\delta _0(R_k)\ge \bar{L}(R_k)} \}|\mathcal{{G}}_{k-1}) +{\small 1}\!\!1_{\mathcal{{A}}^c_{k-1}}(1-P(I_k|\mathcal{{G}}_{k-1})). \end{aligned}$$

(4.14)

Now, we invoke Lemma 2.3 with $\Vert z\Vert ,R$ and $\lambda$ respectively of order $n,\bar{L}(n)$, and $h(n) n^{d-1}$. As a consequence, we have on $\mathcal{{C}}_k\cap \mathcal{{A}}^c_{k-1}$ for a constant $\kappa$,

$$\begin{aligned} \inf _{k\le n/2h(n)} P(I_k\ |\ \mathcal{{G}}_{k-1})\ge \exp (-\kappa h(n) \bar{L}(n)). \end{aligned}$$

(4.15)

If we denote by $N$ the integer part of $n/2h(n)$, and proceed inductively, we obtain

$$\begin{aligned}&P\left(\cup _{k\le N} \mathcal{{A}}_k\right) -P\left(\cap _{k\le N} \mathcal{{C}}_k\right)\\&\quad \ge - P(\forall k\le N,\ \mathcal{{A}}^c_k\cap \mathcal{{C}}_k) \\&\quad \ge -E[ {{\small 1}\!\!1} ]_{\forall k<N,\ \mathcal{{A}}^c_k\cap \mathcal{{C}}_k} P(\mathcal{{A}}^c_N\cap \mathcal{{C}}_N\ |\mathcal{{G}}_{N-1}) \\&\quad \ge -E[ {{\small 1}\!\!1_{\forall k<N,\ \mathcal{{A}}^c_k\cap \mathcal{{C}}_k}} ] (P(\{ {\delta _0(R_N)\ge \bar{L}(R_N)} \}|\mathcal{{G}}_{N-1}) +1-P(I_N|\mathcal{{G}}_{N-1}))\\&\quad \ge -P(\{ {\delta _0(R_N)\ge \bar{L}(R_N)} \})-(1- \exp (-\kappa h(n) \bar{L}(n)))P( {\forall k<N,\ \mathcal{{A}}^c_k\cap \mathcal{{C}}_k} )\\&\quad \ge -\sum _{k\le N} P(\{ {\delta _0(R_k)\ge \bar{L}(R_k)} \})- (1-\exp (-\kappa h(n) \bar{L}(n)))^{N}. \end{aligned}$$

Thus,

$$\begin{aligned} P(\cup _{k\le N} \mathcal{{A}}_k)&\ge 1- \sum _{k\le N} (P(\{ {\delta _0(R_k)\ge \bar{L}(R_k)} \})\nonumber \\&\quad + P(\mathcal{{C}}_k^c)) -(1-\exp (-\kappa h(n) \bar{L}(n)))^{N}. \end{aligned}$$

(4.16)

Now, we have established in [2], that for $\beta$ large enough, the probability of $\{\delta _0(R_k)\ge \bar{L}(R_k)\}$ decays faster than any power in $n$, whereas the fact that $X_k$ is Poisson implies that for some constant $c$, we have $P(\mathcal{{C}}_k^c)\le \exp (-c h(n)n^{d-1})$. The last term on the last display of (4.16) tends to 0 if

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{n}{2h(n)}\exp (-\kappa h(n) \bar{L}(n))= \infty . \end{aligned}$$

(4.17)

In dimension 2 or more, (4.17) holds for $\alpha$ small enough.

5 Proof of Proposition 1.6

The proof is a direct corollary of formula (3.11) of [2]. We consider actually tiles of size 1, that is site of $\mathbb{Z }^d$. Inequality (3.8) of [2] shows that for some constant $c_d$ (depending only on dimension) and $R=\Vert z\Vert <n$, we have

$$\begin{aligned} E[W_R(\emptyset , b(n)\delta _0,z)]\ge c_d (n-\Vert z\Vert ). \end{aligned}$$

(5.1)

Inequality (3.10) of [2] is written a little differently as

$$\begin{aligned} P( { W_{R}(\emptyset , b(n)\delta _0,z)=0} )\le \left\{ \begin{array}{ll} \exp ( {-\lambda \kappa _2 (n-\Vert z\Vert ) + \lambda ^2 c^{\prime }_2 \log (n)} )&\quad \text{ for} \quad d=2, \\ \exp ( {-\lambda \kappa _d (n-\Vert z\Vert ) + \lambda ^2 c^{\prime }_d } )&\quad \text{ for} \quad d\ge 3. \end{array} \right.\nonumber \\ \end{aligned}$$

(5.2)

As we optimize (5.2) in $\lambda >0$, we obtain (1.11).