On Diffusion Limited Deposition

We propose a simple model of columnar growth through {\it diffusion limited aggregation} (DLA). Consider a graph $G_N\times\N$, where the basis has $N$ vertices $G_N:=\{1,\dots,N\}$, and two vertices $(x,h)$ and $(x',h')$ are adjacent if $|h-h'|\le 1$. Consider there a simple random walk {\it coming from infinity} which {\it deposits} on a growing cluster as follows: the cluster is a collection of columns, and the height of the column first hit by the walk immediately grows by one unit. Thus, columns do not grow laterally. We prove that there is a critical time scale $N/\log(N)$ for the maximal height of the piles, i.e., there exist constants $\alpha<\beta$ such that the maximal pile height at time $\alpha N/\log(N)$ is of order $\log(N)$, while at time $\beta N/\log(N)$ is larger than $N^\chi$. This suggests that a \emph{monopolistic regime} starts at such a time and only the highest pile goes on growing. If we rather consider a walk whose height-component goes down deterministically, the resulting \emph{ballistic deposition} has maximal height of order $\log(N)$ at time $N$. These two deposition models, diffusive and ballistic, are also compared with uniform random allocation and Polya's urn.


Introduction
Motivation. A celebrated model of deposition via diffusion is proposed in the early 80's by Witten and Sanders [31]. The aggregate, denoted A(K), made of K sites of Z d is built inductively as follows. Choose A(1) = {0} and assume A(K). Let ∂A(K) denote its outer boundary. Informally, launch a simple random walk, n → S(n), far away from the origin, and stop it when it reaches ∂A(K), say on random site Y . We set A(K + 1) = A(K) ∪ {Y }. In other words, if τ ∂A(K) is the time at which the walk hits ∂A(K), then for y ∈ ∂A(K), Simulations show that the cluster looks like a ramified tree with long branches. Heuristically, the origin of reinforcement is clear. Think of the walk in terms of its radial component, which performs an almost symmetric one-dimensional walk, and its transverse component. Either the random walk sticks soon after reaching the outer radius of the cluster, and it has to settle on a tip, or it takes time before settling and its radial component diffuses, and has more chances to visit the extremal shells, hence increasing the probability of attaching a tip rather than an inside site. This explains reinforcement, but does not explain why this reinforcement is enough to produce a ramified tree structure. It is clear also, at the heuristic level, that we face two problems: controlling the number of tips in the growing cluster, and controlling in a quantitative way the reinforcement of these tips.
One natural way to measure the dimension of the cluster is to find the scaling of the radius of A(K), and look ford such that Radius(A(K)) ∼ K 1/d . (1.1) If A(K) were a ball, thend = d, and the conjecture is thatd < d. Now, physicists have a much sharper conjectured In dimension 2,d c = 5/3, and simulations gived = 1.7. Kesten in [14,15,16] considers the problem, and shows that the arms of the cluster are not too long. More precisely, his result reads  By reversing time, (see [21] and assume d ≥ 3) one writes the probability of adding Y = y to the cluster as P A(K + 1) = A(K) ∪ {y} A(K) = P y τ ∂A(K) = ∞) z∈∂A P z τ ∂A(K) = ∞) . (1.4) The difficulty is to have estimate on the escape probability when the set A is not a sphere, or some simple geometric shape. Let us mention an interesting result about holes in the DLA cluster, where a hole is a finite maximal connected subset of the complement of A(K). Erbez-Wagner [12] shows that in dimension two, almost surely the number of holes tends to infinity with K. Barlow, Pemantle and Perkins in [7] study DLA on a regular d-ary tree where the conductance between edges joining generation n and n + 1 is α −n for α < 1. These authors show that the infinite cluster has a unique infinite line of descent. Even though there is an explicit formula for the harmonic measure, the proof that r(A(K)) scales like K with normal fluctuations is non-trivial.
Benjamini and Yadin in [9] propose another toy model for DLA. They consider a cylinder G N ×N, where the graph G N has constant degree, N vertices, and is fast mixing: the mixingtime should be less than log 2− (|G N |) for some positive (the class of d-regular random graphs works). They show that if we send H × |G N | simple walks from infinity, then the height of the aggregate is larger than H log(log(|G N |)) for any H and N large enough.
There is a two-dimensional model, the Hastings-Levitov model, which takes advantage of the conformal invariance of two-dimensional brownian motion, and Riemman's mapping Theorem to map the complement of the cluster into the complement of the unit disk, and then attach on the unit circle a stick at a random uniform angle. Recently Norris and Turner [26] have studied very precisely the limiting cluster obtained by iteration of randomly rotated conformal mappings.
In a series of three recent papers, Amir, Angel, Benjamini and Kozma [1,2,3] study DLA on Z with long-range random walks. The cluster is no longer connected, and they discover many phase transitions in the growth rate of the cluster according to the tail decay of the increment of the walk.
Our model is a further simplification of Benjamini and Yadin's model [9] in two ways: (i) no lateral hairs are produced, and (ii) the basis graph has no geometry. In our toy model of DLA, the radial component does a one-dimensional random walk, and the transverse component samples uniformly the section of our graph. Still we believe that our model is interesting, and one can answer some of the following questions in a quantitative way.
• What is the origin of reinforcement?
• What is the critical height to overcome ?
• What are the different regimes in the cluster's growth?
Models. We shall consider two deposition models, diffusive deposition and ballistic deposition.
We start with defining diffusive deposition. Our graph is a half-cylinder G N × N, where the basis has N vertices G N := {1, . . . , N }, and two vertices (x, h) and (x , h ) are adjacent if |h − h | = 1. The set G N × {0} is called the ground.
Let n → A(n) be the evolution of random subsets of G N × N that we call the cluster. The cluster is built inductively with A(0) = G N × {0}. For an integer k, the cluster A(k) is made of columns, that is, (1.5) We shall write for simplicity A(k) = (σ 1 (k), . . . , σ N (k)). Assume that A(k) is built. We consider a simple random walk n → S n = (X n , Z n ) on our graph. In other words, 3. the initial condition Z 0 is above the maximal height of the cluster A(k). For defininetness we take Z 0 = max i σ i (k) + 1.
The following rule of aggregation, or deposition, makes the cluster grow. The walk S n , roams until it hits the cluster A(k). Let (X * , Z * ) be the hitting site on A(k), and necessarily 0 ≤ Z * ≤ σ X * (k). We build A(k + 1) by increasing the height of column X * by one unit. That is We shall also say that the walk attaches to the column, or pile, at X * . The walk with the aggregation rule is called an explorer. We shall denote by P the probability associated with this process.
In diffusive deposition there are two relevant phenomena: one is diffusion, the other is deposition which happens instantly and this explains the name diffusion limited deposition.
Ballistic deposition is defined similarly, with the same notation, but with a totally asymmetric walk {Z n+1 − Z n = −1}. One could consider a continuum of biased models with a drift parameter.
Given a configuration σ, we denote by ζ(σ) the height occupation of σ, i.e., Note that j≥1 ζ j (σ) = |σ|, and that ζ(σ) = ζ(σ). Given two configurations σ and η such that |σ| = |η|, we say that σ is more monopolistic than η, writing σ η, when Equivalently, one realizesη fromσ by moving particles from the highest columns to the lowest ones. Urn models are paradigms of reinforcement phenomena (see for instance the survey [27]), and our deposition models actually can be stochastically compared with urns with N colors. We briefly recall Polya's urn with N colors: starting with one ball of each color, at each unit time one draws a ball and put it back in the urn with an additional ball of the same color. Calling η i , with i = 1, . . . , N , the number of added balls of color i after |η| draws, the probability of drawing a ball of color i is We consider also a generalized urn by replacing the r.h.s. in (1.8) When f (x) = x 2 + 1, we call the model the quadratic urn. When f ≡ 1, we call the model the uniform random allocation and denote by q U i (η) = 1/N the corresponding probability of drawing a ball of color i at any time. Finally, we say that a process t → σ(t) is more monopolistic than process t → η(t) if there is a coupling of the processes such that for any t > 0 we have σ(t) η(t), if this is the case initially.
Main Results. In this Section we collect our main results. The first Theorem gives an estimate of the number of explorers necessary, in diffusive deposition, to form a cluster with at least one column proportional to a power of N . Theorem 1.1 Consider diffusive deposition. There are constants α < β, such that almost surely, when N is large enough 9) and there exists a positive constant χ such that In ballistic deposition, we prove that the growth of the height of the cluster is much slower. Indeed, it is unlikely that N explorers produce a column of height log(N ).  N )). We obtain that a large number of columns reach this height by comparison with random allocation. We obtain interesting comparison with other urns, with the observation that ballistic deposition looks like Polya's urn with N colors, whereas diffusive deposition looks like a quadratic urn (see below (1.8) for the definition). Then, one of these subcritical columns reaches the critical height log(N ). Since our estimate requires the configuration to stay in the early regime, one has to bound the number of critical columns. We show that the number of critical columns is less than N 1−2χ for some positive χ, and this implies that the evolution remains in the early regime as long as the highest column has not crossed N χ . We now can state our comparison result.
Theorem 1.4 Both deposition models (diffusive and ballistic) are more monopolistic than Polya's urn, which itself is more monopolistic than random allocation.
The following corollary is a side result interesting on its own right which seems new, to the best of our knowledge.
Proposition 1.5 Polya's urn with N colors is monotone with respect to the order .
Related Models. There are many models of cluster growth similar in definition to DLA. They differ according to the law of Y , the site we add on the boundary of the cluster A. This can also be expressed according to the site, say X, from where the random walks are launched and lead to different phenomenology.
• If X = 0, we rather define a dual model of erosion. The cluster represents the eroded materia, and A(0) = ∅. Each new walk starts at 0, and settles on the first visited site outside the cluster (a site which we interpreted as being eroded). This is internal DLA, and was introduced by Meakin and Deutch in [25]. The cluster is spherical as was first seen Lawler, Bramson and Griffeath in [22]. The fluctuations were studied in [4,5,6] and independently in [18,19,20].
• If X is uniformly drawn in the cluster, then Benjamini, Duminil-Copin, Kozma, Lucas in [8] show that the cluster is spherical.
• If Y is uniform on the boundary of the cluster, then this is the celebrated Eden model [11], which was proposed in the '60, and studied first by Richardson [28].
• If particles do not erode immediately the materia, but do it with an exponential clock, and if they can be activated again when another walk stands on their site, this is Activated Random Walks. This model has been introduced by Spitzer in the 70, and much discussed in the physics literature as an example of self-organized criticality. This has been studied mathematically by Rolla and Sidoravicius [29] (and references therein), and recently by Sidoravicius and Teixera [30] among others. Recent efforts have focused on the case of an initial condition drawn from a product Poisson measure. As one tunes the density there is phase transition between settlement of explorers (in any finite box), and their perpetual activity.
Pictures and simulations. In order to illustrate our main results, we show some numerics. In particular, we emphasize the freezing phenomenon which leads to the monopolistic regime: after a given time the highets pile grows linearly catching all particles. We stress that simulations do not capture quantitative aspects of the problem (scalings or exponents), but serve merely as qualitative illustrations. For both the diffusive and the ballistic model we have simulated the systems for N = 50, 100, 200, . . . , 1000. For the diffusive model we have considered also the cases N = 2000, 4000, 5000, 6000, 8000, 10000. In all the cases averages have been computed over 10 4 independent realizations of the process. We have checked in all the cases that the sample is large enought to get stable averages.  We have also tested numerically our main results in Theorems 1.1 and 1.2. Indeed, we have computed, by averaging over different realizations of the process, the typical height of the highest pile at time N . Rigorous results suggest that this quantity should scale as a power law in the diffusive case and logarithmically in the ballistic one. The related numerical results are shown in the right panels in Figures 1 and 2. The qualitative agreement between simulations and theoretical results is striking. We stress again that the numerical results cannot be interpreted as a quantitave description of the model behavior since, for instance, too small values of the graph size N have been considered.  Finally, in the diffusive case we have also tested numerically our results about the critical character of the time scale N/ log(N ). In Figure 3 we compare the highest column height measured at times N/ log(N ) and 2N/ log(N ). In the first case the numerical (solid circles) data can be perfectly fitted by a logarithmic function. In the latter case, on the other hand, the poor logarithmic fitting is opposed to a perfect power law one of the numerical data (solid squares). This result is in perfect agreement with the one proved in Theorem 1.1 and, in particular, it suggests that 1 < α < β < 2. We stress that our numerics cannot in any case be considered quantitative, indeed, we have no clue to state that, by considering larger sizes of the graph, our numerical results would be confirmed.  Plan. The rest of the paper is organized as follows. In Section 2 we present our main tool, which is the probability an explorer hits the ground, as well as a heuristic explanation for the logarithmic scale of the critical height. In Section 3, we present comparison with urns models. The proof of Proposition 1.5 is given in Section 3.2. The proof of the Theorem 1.4 is given in Section 3.3. In Section 4, we establish our main tool, and related estimates. We study the very early regime in Section 5. Then, we study the growth of cluster in Section 6, and the reason why a large number of columns cannot overcome height log(N ). Finally, we gather all the needed estimates to prove Theorem 1.1 in Section 7.
2 Key Tools and Sketch.
The key to Theorem 1.1 is an estimate of the probability of attaching to a given column. Before, we need a lower bound on the probability of hitting first the ground.
Lemma 2.1 Consider diffusive deposition with a configuration σ such that |σ| < N/2 explorers, The time spent on the slab G N × {0, . . . , σ i } before touching the ground is typically σ 2 i for a SRW, but only if the walk has good chances to cross the whole slab. Our key attachement estimate follows.
Lemma 2.2 Consider diffusive deposition. Let σ be a configuration such that |σ| < N/2. Then there exists a positive constant κ D such that In a sense (2.2) and (2.1) are saying opposite things: the former inequality tells how easy it is to get trapped, whereas the latter tells how easy it is to reach the ground. Imagine a regime where i σ 2 i N . In view of (2.1) the probability of hitting the ground would be small, and very likely the walk would not go below H, where H is such that In other words, H of (2.3) would play the role of an effective ground. We then replace (2.2) by the following estimate.

Corollary 2.3
In the diffusive case, and for any positive H, Sketch. We wish to sketch heuristically the reason why log(N ) is the critical height at which a monopole forms. We fix a given column, say column 1, and we estimate the number of explorers needed to produce a given height. Lemma 2.2 allows us to bound this number by a sum of independent geometric variables, for which we know everything. Indeed, introduce τ 1 the number of explorers needed so that the height of our distinguished site reaches height 1, that is τ 1 := inf{n > 0 : σ 1 (n) = 1}. By induction, for any integer h knowing τ h we define τ h+1 := inf{n > 0 : . Assume now that we are in a regime where hitting the ground is likely. The estimate (2.2) means that for any integers h, n Then, we will show that for any height H means that X explorers produce a column of height H at site 1. Now, we want to find X such that a given height H is likely to be reached. This would be the case if the probability that any distinguished site reaches height H is above 1/N . Thus, we look for X such that Let us write X as N/f (N ), and try to guess the size of f (N ) which produces a monopole. Note also that for any H > f (N ), .
} imposes a constraint only on the first f (N ) variables in (2.7). We then have to estimate f (N ) such that We use now that {τ i } are independent geometric variables (2.8) Now, using Stirling's formula, we obtain Thus, if f (N ) is of order log(N ), it is likely that one monopole forms.

Comparison with Urns
In this section, we establish a coupling between our deposition processes and simpler ones which preserves a natural order on ordered configurations, to be defined below.
We consider growth evolution on N N such that at each unit time we add a unit height to a configuration, say η, at a given site, say i, with a probability p i (η) which depends only on the value η i , and on the unordered set {η j , j = i}. In this case, it is useful to reorder the indices through a permutation of the indices to obtain configurations whose heights are in decreasing order. We call p = {(p 1 (η), . . . , p N (η)), η ∈ N N } the law of the growth process.

Comparing evolutions
It will be important to compare configurations with the same number of explorers. Our main results are the following. Proposition 3.1 Consider two processes t → η(t) and t → σ(t) on N N evolving, respectively, according to the laws p = p 1 (·), . . . , p N (·) and q = q 1 (·), . . . , q N (·) . Assume that for any η, σ ∈ O N such that η ≺ σ we have that Then, the process σ(t) is more monopolistic than η(t),i.e. there is a coupling between the two processes such that σ(t) η(t) for any t.
Then, for any k = 1, . . . , N , we have that Proof. The proof is by induction on N . Assume the Lemma is true with N − 1 sets of positive numbers, and define the renormalized N − 1 numbers The induction hypothesis states that for k = 1 to N − 1 In other words, The question is whether As a corollary of Proposition 3.1 and Lemma 3.2 we have the following result.
Corollary 3.3 With the notation of Proposition 3.1, assume that for any η, σ ∈ O N such that η ≺ σ we have that Then, there is an order preserving coupling between η(t) and σ(t).
In order to prove Proposition 3.1 we need some notation and some simple observations. We define the action A j : N N → N N of adding one explorer to site j: Assume that η ∈ O N and define In other words, d(η, i) is the last position of a height decrease up to position i. Note that for η ∈ O N we have The main observation about ordering is the following.
This lemma is based on the following simple observation.
Proof of Lemma 3.4. To simplify notation assume η, σ ∈ O N . We have already observed that for i ≤ j, A j η ≺ A i η. Thus, we only need to prove that A j η ≺ A j σ. If d(η, j) = j, then d(η, j) ≥ d(σ, j), and the result is obvious. Assume henceforth that d(η, j) < j. If for all k = d(η, j), . . . , j − 1, we have that then the result is also obvious. In the opposite case, let k in [d(η, j), j[, be the first index for which we have then, Lemma 3.5 implies that k ≥ d(σ, j), and the lemma follows.

Comparing Polya's Urn with Random Allocation
By random allocation, we mean repeated draws of one out of N colors, labelled from 1 to N, uniformly at random. In other words, at each draw, the probability to pick up color i is 1/N . The law for Polya's urn and random allocation are denoted respectively q P and q U with ∀σ ∈ N N , q P i (σ) = Lemma 3.6 Polya's urn with N colors is more monopolistic than random allocation of N colors.
Proof. Note that for any σ, η ∈ O N such that η ≺ σ which clearly holds since σ ∈ O N . Thus, Corollary 3.3 implies the lemma.
Proof of Proposition 1.5. Note that if η σ and |η| = |σ|, we have This establishes Proposition 1.5 saying that Polya's evolution with N colors preserves the order.

Comparing deposition models with Polya's urn
Recall the definition of ballistic and diffusive deposition given in Section 1. Denote their law, respectively, by p B and p D . We show that both ballistic and diffusive deposition are more monopolistic than Polya's urn, which is one of the statements of Theorem 1.4. Assume for a moment the following lemma. (3.14) By Proposition 1.5, Lemma 3.7, and Proposition 3.1 we have that both ballistic and diffusive deposition are more monopolistic than Polya's urn.
To state a preliminary simple observation, we need more notation. For η ∈ O N , let p B,D i,k (η) be the probability that the explorer hits site i at height k, for k ∈ N.
Lemma 3.8 For any i, ∈ {1, . . . , N } and η(i) ≥ k > k ≥ 0 we have Proof of Lemma 3.7. We consider first the ballistic case. In view of Lemma 3.2, we need to show that . (3.17) In order to prove (3.17), we need to show that By Lemma 3.8, this inequality has the structure a 1 + · · · + a n n ≥ a m+1 + · · · + a n n − m for n > m ≥ 1 and a 1 ≥ a 2 ≥ · · · ≥ a n . The validity of such an inequality is immediate once we let µ = (a m+1 + · · · + a n )/(n − m), note a 1 ≥ · · · ≥ a m ≥ µ, and write a 1 + · · · + a n n = a 1 + · · · + a m + µ(n − m) Finally, by using (3.17) and Lemma 3.2 the first of equations (3.14) follows immediately. The diffusive case can be treated in the same way. This completes the proof of the lemma.
Proof of Lemma 3.8. First we prove the lemma for the ballistic case. The lemma follows, since p B i,k (η) = 1 N P (the explorer survives till k + 1) and P (the explorer survives till k) is an increasing function of k. Now, we consider diffusive deposition. First, assume k − k an even number. To each path s = {(x j , z j )} j=1,...,n hitting η in i at height k in a time n we associate uniquely a path s = {(x j , z j )} j=1,...,n , of the same length and, therefore, the same probability, hitting η in i at height k. Thus, the lemma follows since there exist other paths ending in i at height k. Given the path s , we construct s in the following way. Call n 1 the time of last passage of s through the intermediate height H = (k + k )/2. s is equal to s up to time n 1 while after n 1 it uses opposite height increments than the original s , i.e. z j+1 − z j = −(z j+1 − z j ) for all n 1 ≤ j < n keeping the same horizontal increments. We therefore obtain a path ending in i and height k. Note that such a path avoids η until it hits site i at height k, because η is a union of columns. If k − k is an odd number we do a similar construction but we have to associate a set of paths s of lenght n to a single path s of lenght n − 1. The set is obtained considering together all the paths s coinciding everywhere but the component x n 1 +1 , where n 1 is now the last hitting time of the level k+k +1 2 of the vertical process. Then the path s hitting η on site i at height k in a time n − 1 coicide with the paths s up to time n 1 , say x j = x j , z j = z j for any j ≤ n 1 and is specular to them after n 1 + 1, that is x j = x j+1 , z j − z j−1 = −(z j+1 − z j ) for any j = n 1 + 1, ..., n − 1 so that x n−1 = i and this is the first hitting to η. Clearly the probability of s is larger than or equal to the sum of the probabilities of the paths s , since s is one step shorter, and the sum on x n 1 +1 is done only on N − ζ n 1 +1 (η) sites.

Estimating Unit Growth
In this section we discuss how heights grow. We consider the random walk S n = (X n , Z n ), and for an integer k, we call H k the first time the walk reaches height k. In other words, Since the X-component is uniform on the base, giving the configuration σ, the ordered onē σ, or the height occupation ζ (defined in (1.6)) is equivalent, and we use P g (σ) or P g (ζ) indifferently to denote the probability an explorer hits the ground. Lemma 2.1 is obtained as a simple application of Jensen's inequality whereas Lemma 2.2 requires Kesten-Kozlov-Spitzer representation of the local times [17].
Proof of Lemma 2.1. For an integer k ≤σ 1 let l(k) be the number of visits of height k by the random walk before H 0 . We have the represenation Our hypothesis |σ| < N/2 implies that for k ≥ 1, ζ k ≤ ζ 1 ≤ N/2, and since log(1 − x) ≥ −x − x 2 for 0 ≤ x ≤ 1/2, we have using Jensen's inequality Now note that E[l(k)] = 2k. Indeed, the height of the random walk being a simple random walk on N, we have for k ≤σ 1 , by conditionning on the first step (4.4) The equality E[l(k)] = 2k for k ≤σ 1 follows at once. Note now that Finally, (2.1) follows as we note that On Kesten-Kozlov-Spitzer representation. Let u(h) be the number of up-crossings of height h before touching the base. In other words, we define u(0) = 0 and for h > 0 Similary, down-crossings of height h correspond to jumps from h to h − 1 before time H 0 . One way to realize the random walk n → Z n is to assign the sequence of up and downcrossings on each height. Thus, we consider {{ξ k i , i ∈ N}, k ∈ N} a collection of i.i.d. geometric variables, with law P (ξ = n) = 1/2 n+1 for n ∈ N. Now, the sequence of up and down-crossings at height k is as follows: ξ k 0 up-crossings, then one down-crossing, then ξ k 1 up-crossings, the one down-crossing, then ξ k 2 up-crossings... and so on and so forth. The key observation is that each ξ k i , for i ≥ 1, is preceded by an up-crossing of the height k − 1. In other words, We set G(h) = σ(ξ k i , k ≤ h, i ∈ N) the σ-field representing the choices of moves on the first h heights. Kesten-Kozlov-Spizter representation expresses the local times of Z in terms of the u. Thus, if l(k) represents the number of visits of height k before H 0 , for a walk with starting level aboveσ 1 , then ∀k ≥ 1, l(k) = u(k) + u(k − 1) + 1. (4.7) Then, with notation We set a(k) = 0 for k ≥σ 1 , whereas for any k <σ 1 so that and by induction, we obtain Note that (4.10) reads for 1 ≤ k ≤σ 1 and a(k − 1) ≥ a(k) ≥ 0 follows by induction from (4.10). Inequality (2.1) implies that (4.14) Proof of Lemma 2.
2. An explorer settling on the pile at site i, hits the i-th pile at a height between 1 and σ i . Knowing that it settles at height k, it has chance 1/ζ k to settle on (i, k) since we are on the complete graph. We underestimate the probability of settling on σ i , if we only consider trajectories hitting only one of the {ζ 1 , ζ 2 , . . . } before H 0 . Thus, .

(4.15)
Fix h > 0, and write ζ h for the height occupation such that ∀k = h, ζ h k = ζ k , and ζ h h = 0.
Define also κ(A) to be κ D exp(−2A).  Proof of Lemma 4.3. If σ i = 0 then (4.26) is immediate with κ = 1. If σ i ≥ 1 the chances an explorer attaches to column i, in configuration σ, is smaller than if all columns distinct from i were set to zero. This is seen by coupling. First, for configuration σ, let σ i denote the configuration where we annihilate all columns distinct from i. In other words Therefore, in σ i , the highest column is i with height σ i ≥ 1. Now, the event hit column i in σ i is the complement of the event hit the base first. Since P g (σ i ) is the probability the explorer hits first the base in configuration σ i , by Lemma 2.1, we have that for some κ > 0 P An explorer attaches to i σ ≤P An explorer attaches to i σ i which completes the proof. Proof of Lemma 4.5. To get attached to site i, the particle has to survive up to the time it reaches height σ i , and then at each step-down has a chance 1/N to fall on column i provided it has avoided the other columns. Thus P An explorer attaches to site i σ which completes the proof.

Stochastic domination
Both in ballistic and diffusive deposition, we have a simple upper bound on the probability of attaching to a given column (see Section 4.1), which depends only on its height. This, in turn, is used to bound the number of explorers necessary to increase the height by one unit in terms of a geometric random variable, for which everything can be computed explicitly.
In other words, call τ 1 the number of explorers needed so that column 1 reaches height 1. Let τ 2 be the additional number of explorers needed to reach a height 2, and so on. Note that {τ 1 > k} means that out of k explorers none of them has reached site 1. These times are used to control the height of column 1 after k explorers have been sent, ∀k ≥ 1, ∀H ≥ 1, P 0 (σ 1 (k) > H) = P (τ 1 + · · · + τ H < k).
We need to estimate the sum of the {τ i } with the following general lemma.

Very Early Regime
One important step in the cluster growth is to reach height log(N ). We cover this intermediary step in the following proposition, even if it is included in Theorem 1.1.
Proposition 5.1 Consider diffusive deposition. There exist positive constants b and γ such that almost surely, for N large This proposition is concerned with what we call the very early regime, and it is based on comparison with urn models.
Two scales play an important role in this section: the time scale N/ log(N ), and the space scale log(N )/ log log(N ). We therefore introduce notation , and H N = log(N ) log log(N ) .
The set of configurations σ with maximal height lower than γ log N and |σ| ≤ βT N is called the very early regime and is denoted by X ve (γ, β). In other words, and note that If τ A is the hitting time of set A, we show in this Section that there are constants b < β and δ > 0 such that Strategy of the proof. We divide time in two periods. In the first, of length T N , a large number of columns, of order N a with 0 < a < 1, reach a height δH N . This is the content of Lemma 5.3, whose main ingredient is a coupling between diffusive deposition and random allocation. In the second period, we use the estimate of Corollary 2.3 to control the growth of these columns together with sending Poisson waves of explorers to ensure the growth of each column independently.
Step 1: Reaching height H N . The random allocation evolution is denoted by n → η(n). Our first lemma deals exclusively with random allocation.
In Lemma 3.7, we establish that diffusive deposition, denoted t → σ(t) is more monopolistic than random allocation. Thus, there is a coupling such that with probability 1, when σ(0) = η(0), we have for any t ≥ 0 Assume now that σ(T N ) ∈ X ve (γ, β), and that Then, by our coupling Thus, for any α > 1 − 2δ, we have that L > N α . We therefore state the result as follows.

Lemma 5.3
For α ∈ [ 1 2 , 1), and δ < (1 − α)/2, we have almost surely, for N large enough, and for the diffusive deposition t → σ(t), By applying Markov's property at time T N (5.12) Step 2: Poisson waves. We realize diffusive deposition for times in [T N , bT N ] by a sequence of Poisson waves, the k-th wave made of X (k) explorers, and {X (k) , k ≥ 1} an i.i.d sequence of Poisson random variables with parameter x N going to infinity with N . Our starting configuration denoted σ (0) satisfies Let σ (k) be the configuration of diffusive deposition starting from σ (0) after the k-th wave is sent, i.e., Define now, using κ D of Corollary 2.3, We have for the diffusive deposition process for any k = 1, 2, . . . , if σ (k) ∈ X ve (γ, β) then This immediately follows from Corollary 2.3. Consider an auxiliary growth processσ(t) which evolves on the sites of Λ N ∪ {0}, defined iteratively as follows. Set σ (0) 0 = 0, and for i ∈ Λ N , setσ Each explorer in the k-th wave is attached to site i ∈ Λ N with probability whereas site 0 grows by one with probability 1 − i∈Λ N p A i (k). The following result is crucial. Lemma 5.4 There exists a coupling between t → σ(t) and t →σ(t) such that if |σ (k) | ≤ τ Xve(γ,β) c and t is within the k-th wave, i.e., t ∈ [|σ (k−1) |, Proof. The coupling part is simple and we omit it here. We denote by {Y 1 i , i ∈ Λ N } independent Poisson variables of parameter x N p A i (1). We denote by G 1 the sigma-field generated by X (1) and by {Y 1 i , i ∈ Λ N }. We now build G k by induction. Assume that G k−1 has been built. Then conditioned on G k−1 , we fix the height of all sites after the k − 1-th wave. Draw a Poisson variable X (k) independent of G k−1 , and denote by {Y k i , i ∈ Λ N } the independent Poisson variables of parameter p A i (k)x N which is itself G k−1 measurable. Note that Y k i depends only on X (k) and on the past through Y 1 i + · · · + Y k−1 i , the height of site i after the k − 1-th wave. In other words, for any real function f i , there is a function φ i such that ). (5.16) Note also, that if we integrate only over X k , and for any real functions f i , for i ∈ Λ N we have ). (5.17) This means that what happens on different sites of Λ N is independent. Now, each Poisson wave we send has about x N explorers, and we expect to send about (b − 1)T N /x N waves. Recall that for N large, we have a.s. that σ (0) ∈ X ve (γ, β) and |Λ N | > N α . Therefore, if t N denotes the integer part of bT N /(2ex N ), and for simplicity (5.18) Step 3: Dealing with one site. We show that for a function (γ) going to 0 with γ, for We define the successive wave numbers at which the column at 1 grows. Let τ be the number of waves needed so as to increase by at least one the height of site 1. Then, let τ 1 = τ and τ n = τ • θ(τ 1 + · · · + τ n−1 ). Note that for any integer n where we used (5.14). Note that at the number of waves t = τ 1 + · · · + τ k−1 , the configuratioñ σ (t) i is larger or equal than H + k − 1. We have, using Lemma 5.4 Then, we are in the setting of Lemma 4.6, and have a comparison with independent geometric random variables {τ k , k ≥ 1} with Then, (with the abuse of taking γ log(N ) to be integer) .
The proof of the Proposition 5.1 is completed if α > (γ).

Growing Columns
In this section we present a simple way to bound the height of the maximal pile, based on stochastic domination, for both ballistic and diffusive deposition. Indeed, for both models we construct a sequence of inter-arrival times of explorers on a given column, say column number 1, stochastically dominated by independent geometric variables. We now state three propositions that are crucial in the proofs of Theorems 1.2 and 1.1. The propositions bound the probabilities of building a high pile, and are proven at the end of this section.

2)
with κ = 1 + O(N −1/2 ). Proposition 6.2 implies that for α small, αN/ log(N ) particles are not enough to reach a maximal height of order log(N ). Finally, we consider diffusive deposition, with a configuration in X e (A), and with one distinguished site, say i, above height γ log(N ). We show that as long as we do not leave the early regime, see equation (4.24) we have a fast growth. Let τ Xe c the time at which you exit the early regime. Proposition 6.3 Let χ, γ, C be any positive constants with χ < 1/2. Assume that there is a distinguished site, say i * , with σ i * ≥ γ log(N ). Then, we have

Growing a Column in ballistic deposition
Proof of Proposition 6.1. By lemma 4.5 and in general This implies that (6.1) is a large deviation event since More precisely by Lemma 4.6 we have with {τ i , i = 1, . . . , H} independent geometric variables of mean N i . By the exponential Chebyshev's inequality we get, for every λ > 0 Note that for a geometric variable X of mean 1/p .
When λ is positive, exp(λ) − 1 ≥ λ, and we have We choose λ = i 0 /N so that by the asymptotic of the harmonic series So, we obtain Now, if we choose i 0 to be the integer part of αH, for some constant α, then If α is sufficiently small, say α = 1 2 , then log((1+α)/α)−1 > 0, and Lemma 6.1 is established.

Growing a Column in diffusive deposition
Proof of Proposition 6.2. We follow the arguments of the previous proof. By using Lemma 4.3 for any i ≥ 1, and integer n By Lemma 4.6, we have for any H, X P τ 1 + ... + τ H < X) ≤ P τ 1 + ... +τ H < X), Then, for every λ > 0, by Chebyshev's inequality We set λ = κH 2 /N and we note that . (6.7) We conclude obtaining (6.2).
Proof of Proposition 6.3. Again, let {τ 1 , τ 2 , . . . , } be the random number of explorers linked with growing a column at i * from σ with an initial state with σ i * = γ log(N ). By Corollary 4.2 we have for any integer m < X Therefore, by Lemma 4.6 By Chebyshev inequality, assuming a := 1 − e −λ < p k . Thus, Hence, We choose a = κ(A)(γ 2 log 2 N )/N (and a ≥ λ/2) and X = CN/ log N , and we have

Growing a Tower in diffusive deposition
In this section, we bound the probability of forming a high tower of explorers in diffusive deposition. We define n C (σ) = x∈σ∧C σ x , so that, for N large enough we have that P Explorer attaches to σ ∧ C σ ≤ κ A N [L + Hn C (σ)]. (6.11) This allows us to define, as before, geometric random variables stochastically smaller than the number of explorers needed to settle one of them in C. Let τ 1 be the number of explorers needed in order that one settles in C, when we start with the empty configuration. By induction, when k − 1 explorers are settled in C, define τ k to be the number of explorers needed to settle the k-th explorer in C, and we do this up to time LH. Then for any configuration σ with n C (σ) = k − 1, for any positive integer m P (τ k > m | σ) ≥ 1 − kH + L N m = P (τ k > m). (6.12) We invoke again Lemma 4.6 to obtain P τ 1 + · · · + τ HL ≤ X ≤P τ 1 + · · · +τ HL ≤ X ≤ e λX  This concludes the proof since the exponent of N is less than −1 when 9cκ < (3π/4 − 2). (recall that κ = 1 + O(N −1/2 )).
Proof of (1.10). Recall that from Proposition 5.1 there is b > 0 (and (5.2) for the quantitative estimate), so that very likely τ Xve(γ,β) c < bT N , where T N = N/ log(N ). We therefore condition on the evolution up to τ Xve(γ,β) c .
Proof of Theorem 1.2. The statement follows immediately by Proposition 6.1 with H = γ log(N ) with γ > 2/κ. Indeed, by Proposition 6.1, there exists a constant κ such that P (∃i : σ i (N ) > γ log N ) ≤ N e −κγ log N = N 1−κγ , and the proof concludes.
Acknowledgements. A.A. Thanks Robin Pemantle for discussions on urns. A.A. and E.S. thank the CIRM for a friendly atmosphere during their stay as part of a research in pairs program. This work has been carried out thanks to the support of A * MIDEX grant (ANR-11-IDEX-0001-02) funded by the French Government "Investissements d'Avenir" program. E.S. thanks Université Paris-Est, Créteil. Finally, we thank two anonymous referees for their careful reviewing.