Internal Aggregation Models on Comb Lattices

The two-dimensional comb lattice $C_2$ is a natural spanning tree of the Euclidean lattice $\mathbb{Z}^2$. We study three related cluster growth models on $C_2$: internal diffusion limited aggregation (IDLA), in which random walkers move on the vertices of $C_2$ until reaching an unoccupied site where they stop; rotor-router aggregation in which particles perform deterministic walks, and stop when reaching a site previously unoccupied; and the divisible sandpile model where at each vertex there is a pile of sand, for which, at each step, the mass exceeding 1 is distributed equally among the neighbours. We describe the shape of the divisible sandpile cluster on $C_2$, which is then used to give inner bounds for IDLA and rotor-router aggregation.


Introduction
Let G be an infinite, locally finite and connected graph with a chosen origin o ∈ G. Internal diffusion limited aggregation (IDLA) is a random walk-based growth model, which was introduced by Diaconis and Fulton [DF91]. In IDLA n particles start at the origin of G, and each particle performs a simple random walk until it reaches a vertex which was not previously occupied. There the particle stops, and from now on occupies this vertex, and a new particle starts its journey at the origin. The resulting random set of n occupied sites in G is called the IDLA cluster, and will be denoted by A n .
IDLA has received increased attention in the last years. In 1992, Lawler, Bramson and Griffeath [LBG92] showed that for simple random walk on Z d , with d ≥ 2, the limiting shape of IDLA, when properly rescaled, is almost surely an Euclidean ball of radius 1. In 1995, Lawler [Law95] refined this result by giving estimates on the fluctuations. Recently several improvements have been obtained. Asselah and Gaudillière [AG10,AG11a] proved an upper bound of order log(n) for the inner fluctuation δ I and of order log 2 (n)

INTRODUCTION
for the outer fluctuation δ O in all dimensions d ≥ 2. In [AG11b] they improve the upper bound on the inner fluctuation to log(radius), for d ≥ 3. Independently, and by different methods, Jerison, Levine and Sheffield [JLS12,JLS11], proved also that both δ I and δ O are of order log(n) for IDLA on Z 2 and of order log(radius) for d ≥ 3 .
Rotor-router aggregation is a deterministic version of IDLA, where particles perform rotorrouter walks, which are deterministic analogues to random walks. They have been first introduced into the physics literature under the name Eulerian walks by Priezzhev, D.Dhar et al [PDDK96]. At each vertex of the graph G, we have an arrow (rotor) pointing to one of the neighbours of the vertex. A particle performing a rotor-router walk first changes the rotor at its current position to point to the next neighbour, in a fixed order chosen at the beginning, and then moves to the neighbour the rotor is now pointing at. In rotor-router aggregation each particle performs a rotor-router walk until it reaches an unoccupied site, where it stops. Then a new particle starts at the origin, without resetting the configuration of rotors. The resulting deterministic set R n of n occupied sites is called the rotor-router cluster.
Rotor-router aggregation on the Euclidean lattice Z d has been studied by Levine and Peres [LP09], who showed that the cluster R n is a ball in the Euclidean distance. On the homogeneous tree Landau and Levine [LL09] proved that, provided the start configuration of rotors is acyclic, the rotor-router cluster forms a perfect ball with respect to the graph metric, whenever it has the right amount of particles. Kager and Levine [KL10] studied the shape of the rotor-router cluster on a modified two dimensional lattice, which they call the layered square lattice. In each of the known examples the limiting shape of rotor-router aggregation is the same as the one for IDLA, but with much smaller fluctuations compared to IDLA.
In order to prove inner bounds for the above models, we use a third growth model, the so-called divisible sandpile, which has been introduced by Levine and Peres [LP09] as a tool for studying internal growth models on Z d . In the divisible sandpile model each vertex can have an arbitrary amount of mass. If a vertex has mass at least 1, it is called unstable and it can topple by distributing the mass exceeding 1 equally among its neighbours. At each timestep a vertex is chosen and toppled if it is unstable. Provided every vertex is chosen infinitely often, the masses converge to a limiting distribution ≤ 1. The set of vertices with limit mass equal to 1 is called the divisible sandpile cluster. If we start with a mass of n concentrated at the origin, the corresponding sandpile cluster will be denoted by S n .
All three growth models have very similar behaviour. This was first noticed by Levine and Peres [LP09,LP10] for the case when the state space is an Euclidean lattice. Computer simulations suggest that the connection between the three growth models holds in wide generality, but only partial results are available for other state spaces. All these three models have the so-called abelian property, which makes them amenable to rigorous analysis. In the case of IDLA and the rotor-router model this means that, if we let several particles run at the same time, instead of one after another, it is irrelevant for the end result in which order the particles make their moves. In the case of the divisible sandpile model, it means that the limiting distribution is independent of the order in which vertices topple.
The aim of this paper is to study the three aggregation models described above on the comb lattice C 2 , which is the spanning tree of the two-dimensional Euclidean lattice Z 2 , obtained by removing all horizontal edges of Z 2 except the ones on the x-axis.
The graph C 2 can also be constructed from a two-sided infinite path Z (the "backbone" of the comb), by attaching copies of Z (the "teeth") at every vertex of the backbone.
We use the standard embedding of the comb into Z 2 , and use Cartesian coordinates z = (x, y) ∈ Z 2 to denote vertices of C 2 . The vertex o = (0, 0) will be the root or the origin; see Figure 1. For functions g on the vertex set of C 2 we will often write g(x, y) instead of g(z), when z = (x, y). While C 2 is a very simple graph, it has some remarkable properties. For example, the so-called Einstein relation between the spectral-, walk-and fractal-dimension is violated on the comb, see Bertacchi [Ber06]. Peres and Krishnapur [KP04] showed that on C 2 and other similar recurrent graphs two independent simple random walks meet only finitely often. Random walks on C 2 have been studied by various authors, the first being Havlin and Weiss [WH86] and Gerl [Ger86].
The paper is organized as follows. In Section 2 we introduce some notations and basic facts which will be used through the rest of the work. Section 3 is dedicated to the study of the divisible sandpile on the comb C 2 . We show in Theorem 3.5 that the sandpile cluster S n on C 2 has up to constant fluctuations the shape . Section 4 deals with IDLA on C 2 . Using the results obtained for the sandpile model, we prove an inner bound for IDLA, which is of the type (1). Finally, in Section 5, we give an inner estimate for the rotor-router model on C 2 which is weaker than the result obtained for IDLA. For a fixed initial configuration of rotors the exact shape of the rotor-router cluster on the comb has been obtained by the authors in [HS11] using a purely combinatorial approach.

Preliminaries
Let (G, E(G)) be an infinite, undirected and connected graph, with vertex set G, equipped with a symmetric adjacency relation ∼, which defines the set of edges E(G) (as a subset of G × G). We write (x, y) for the edge between the pair of neighbours x, y. In order to simplify the notation, instead of writing (G, E(G)) for a graph, we shall write only G, and it will be clear from the context whether we are considering edges or vertices. Let o ∈ G be some fixed reference vertex called the origin. For x, y ∈ G, let d(x, y) be the length of the shortest path from x to y. Also, write d(x) for the degree of x, i.e, the number of neighbours of x. For a subset A ⊂ G we denote by ∂A = x ∈ G \ A : ∃y ∈ A with x ∼ y and ∂ I A = x ∈ A : ∃y ∈ A with x ∼ y the (outer) boundary respectively the inner boundary of A.
Let P = p(x, y) x,y∈G be the one-step transition probabilities of the simple random walk on G, i.e., p(x, y) = 1/d(x) if y ∼ x and 0 otherwise. We write X t for the position of the random walker at the discrete timestep t. Probabilities will be written as P, in particular P x denotes the probability of a random walk which starts at x ∈ G. Similarly E and E x will denote expectations using the same convention. For y, z ∈ G the Green function is defined as and represents the expected number of visits to z of the random walk X t started at y. For a subset A ⊂ G, write G A for the Green function of the random walk stopped upon leaving the set A. That is, if τ = min{t ≥ 0 : X t / ∈ A}, then For a function f : G → R, its Laplace operator ∆f is defined as .
A function f : G → R is called superharmonic on a set A ⊂ G if ∆f ≤ 0, and harmonic if ∆f = 0, for all x ∈ A. For a function g : G → R, define its least superharmonic majorant as Remark that the function s is itself superharmonic on G. The following is widely known.
Lemma 2.1 (Minimum principle.). If f is a superharmonic function on G and there exists x ∈ G such that f (x) = min G f , then f is constant.

Divisible Sandpile
Let C 2 be the comb as in Figure 1, and let µ 0 be an initial mass distribution on C 2 , i.e., a function µ 0 : C 2 → R + with finite support. The divisible sandpile is a sequence (µ k ) k≥0 of mass distributions, which are created according to the following rule. At each time step k, choose a vertex x ∈ C 2 . If µ k (x) ≥ 1, the pile of sand at x is unstable and topples, which means that x keeps mass 1 for itself and the remaining mass µ k (x) − 1 is distributed equally among the neighbours y of x, that is, according to the transition probabilities p(x, y) of the simple random walk on C 2 . Given a mass distribution µ k at time k and a vertex x ∈ C 2 , the toppling operator can be defined as where δ x (y) = δx(y) d(y) and α k (y) = max{µ k (y) − 1, 0}. Let (x k ) k≥0 be a sequence of vertices in C 2 called the toppling sequence, which contains each vertex of C 2 infinitely often. Then the mass distribution of the sandpile after k steps is defined as Hence, µ k+1 (y) is the amount of mass present at y after toppling the sites x 0 , . . . , x k in succession. One of the tools that will be used throughout this work in various incarnations is the so-called odometer function, which was introduced by Levine and Peres [LP09].
and represents the total mass emitted from y during the first k topplings.
For simple random walks on C 2 it is easier to work with the normalized odometer function d(x) . Lemma 3.1 of Levine and Peres [LP09] can be easily adapted to our case, in order to show that, as k goes to infinity, µ k and u k converge to limit functions µ and u respectively. Define The set S is called the sandpile cluster with initial mass distribution µ 0 . The limit functions µ and u satisfy and From (2) and (3) it follows that and The following result provides a method for solving this free boundary problem. For a proof, see once again Levine and Peres [LP09, Lemma 3.2].
Lemma 3.2. Consider a function γ : Then the normalized odometer function u of the sandpile satisfies u = γ + s, where s is the least superharmonic majorant of −γ.
Lemma 3.2 gives a representation of the odometer function which is independent of the toppling sequence.
Remark 3.3 (Abelian property). The limit u of the normalized odometer function and the sandpile cluster S are independent of the toppling sequence (x k ) k≥0 .

Divisible Sandpile on the Comb
With the help of Lemma 3.2, we shall next describe the limit shape of the sandpile cluster on the two-dimensional comb C 2 . Consider an initial mass distribution µ 0 concentrated at the origin o, that is µ 0 = n · δ o , and denote by the sandpile cluster, and by u n the limit of the normalized odometer function for this choice of initial distribution. We use another simple fact about u n ; for a proof see [LP09, Lemma 3.4].
By (4), the normalized odometer function satisfies The odometer function u n can be reduced to odometer functions of suitable divisible sandpiles on Z, which are easy to compute. Letũ n be the normalized odometer function of the divisible sandpile on Z, with initial mass distributionμ 0 concentrated at 0, that is, µ 0 = n · δ 0 . By Remark 3.3 it is clear that the sandpile clusterS n on Z in this case is a symmetric interval around the origin 0. In order to computeũ n , by Lemma 3.2, we need to construct a functionγ n : Z → R with Laplacian given in (5). It is easy to check that γ n defined byγ satisfies the required property. Sinceγ n is nonnegative, the constant function 0 is a superharmonic majorant of −γ n . Hence, by Lemma 3.2, we haveũ n ≤γ n . Now, consider γ n : where n x ∈ R for all x ∈ Z. The quantities n x can be interpreted as the total amount of mass that ends up in the copy of Z that is attached to the vertex (x, 0). Then γ n satisfies (5) if and only if holds for all x ∈ Z. From (7) and (9), and using the fact that n x = n −x by symmetry, we get the following recursion for the numbers n x Equation (11) has strictly positive solutions as quadratic polynomials of the form Figure 2: Two plots of γ n for n = 1000. The graphic on the left is superimposed with contour lines representing the sets B n for various values of n. In the density plot on the right, dark areas represent small values. By construction, the finite area which is surrounded by the local minima of γ n coincides with the region S n covered by the sandpile.
By the initial condition (10), the parameter t satisfies the equation which has one real root given by with T (n) = 8 √ 3 243 √ 2187n 2 + 125 + 8 3 n 1 3 . By a series expansion around n = ∞, one obtains t = 2 2 3 Therefore, the function γ n (x, y) =γ nx (y) with n x defined by (12) and (13) satisfies the conditions of Lemma 3.2, with µ 0 = n · δ o . See Figure 2 for a graphical representation of γ n . We are now ready to prove the limit shape for the divisible sandpile on C 2 .
Theorem 3.5. Let S n be the divisible sandpile cluster on C 2 , with µ 0 = n · δ o . Then there exists a constant c ≥ 0 such that, for n ≥ n 0 : and Proof. The upper bound S n ⊂ B n+c : The mass distributions n x are nonnegative for all x, therefore γ n is nonnegative, and this implies that the constant function 0 is a superharmonic majorant of −γ n . Thus, by Lemma 3.2, γ n is an upper bound of the odometer function u n . Moreover, Lemma 3.4 implies that u n decreases by a fixed amount on the sandpile cluster S n when we move away from the origin. Therefore, in order to get an upper bound for S n , it suffices to calculate the minima of γ n along each infinite ray starting at o = (0, 0). By the symmetry of C 2 , it is sufficient to consider only the first quadrant.
Consider the rays which lies entirely on the positive x-axis. We have γ n (x, 0) = 1 8 n 2 x . The minimum of this function is attained at x min = 3 4 t, with t given in (13). Using the series expansion (14) of t we get which is also an upper bound of S n on the x-axis by Lemma 3.4, since γ n ( x min , 0) is bounded by a constant which is independent of n, and smaller than 1/10.
To calculate the extent of the sandpile cluster on the "teeth", we need to compute the minima of γ n in the y-direction. On each "tooth" of the comb, γ n is a quadratic polynomial which attains its minimum at y min (x) = nx 2 . Moreover, γ x, y min (x) ≤ 1/2. Using (12) and a series expansion around infinity we get x min . Thus, using the expansion (16) for x min we obtain (x, y) ∈ S n if |x| ≤ kn 1/3 + O(1) and |y| ≤ l n 1/3 − x k 2 + O(1), for n ≥ n 0 . This proves the upper bound S n ⊂ B n+c .
The lower bound B n−c ⊂ S n : On each infinite ray the minimum of γ n (z) is smaller than a constant a > 0, independent of n. Also, from the upper bound, we have u n (z) = 0 for all z ∈ ∂B n+c . Hence u n (z) − γ n (z) ≥ −a for all z ∈ ∂B n+c . Since the function u n − γ n is superharmonic, by the Minimum Principle, it attains its minimum on the boundary and the inequality u n (z) − γ n (z) ≥ −a holds for all z ∈ B n+c . Thus γ n − a is also a lower bound of the odometer function on B n+c , which gives the inner estimate B n−c ⊂ S n , for some constant c.
The next corollary follows directly from the proof of the theorem.
Corollary 3.6. Let u n be the normalized odometer function of the divisible sandpile on C 2 , with initial mass distribution µ 0 = n · δ o , and B n ⊂ C 2 defined as in (15). Then there exists a constant 0 < a < 2, such that, for all n > n 0 and all z ∈ C 2 γ n (z) − a 1 Bn ≤ u n (z) ≤ γ n (z).

Internal Diffusion Limited Aggregation
Let X i t i∈N be a sequence of independent and identically distributed simple random walks on the comb C 2 , with common starting point X i 0 = o. Then X i t represents the position of the i-th particle at time t.
Definition 4.1. Internal diffusion limited aggregation (IDLA) is a stochastic process of increasing subsets A i i∈N of C 2 , which are defined recursively as A 1 = {o} and for i ≥ 2 The IDLA cluster is build up one site at a time.
That is, suppose that we already have the cluster A i−1 after i − 1 particles stopped, and we want to get A i . For this, the i-th particle X i t starts at o, and evolves as long it stays inside the IDLAcluster A i−1 . When X i t leaves A i−1 for the first time, it stops, and the point outside of the cluster that is visited by X i t is added to the new cluster A i . The set A i is called the IDLA-cluster of i particles. Figure 3 shows IDLA clusters on C 2 with 100, 500 and 1000 particles.
We will prove the following shape theorem for IDLA on C 2 . . The set B n is the same as the limit shape of the divisible sandpile from Theorem 3.5. The proof of Theorem 4.2 uses ideas of Lawler, Bramson and Griffeath [LBG92] and of Levine and Peres [LP10]. Following [LBG92], we introduce the stopping times τ i n = min{t ≥ 0 : X i t / ∈ B n } and τ i z = min{t ≥ 0 : X i t = z}, for z ∈ B n . Consider the probability that a fixed vertex z ∈ B n does not belong to the IDLA cluster A n , which can be written in terms of the stopping times defined above as Hence, by the Borel-Cantelli Lemma, convergence of the series is a sufficient condition for Theorem 4.2. Fix now n and z ∈ B n and consider the random variables Then N represents the number of particles that visit z before leaving the cluster. The variable M counts the number of particles that visit z before leaving B n , and L is the number of particles that visit z after leaving the cluster A i but before leaving B n . Remark that if L < M , then z ∈ C 2 belongs to A n . Moreover N ≥ M − L. Therefore, in order to estimate P[z / ∈ A n ], we just need an upper bound for P[M = L]. For any fixed number a We shall show that for a suitable choice of a, the probabilities P[M ≤ a] and P[L ≥ a] are small enough, such that the series (18) converges. The derivation of a suitable value of a will be done differently, not like in the case of Euclidean lattices, studied by Lawler, Bramson and Griffeath in [LBG92], who used classical asymptotics for the Green function stopped on a ball. Since in our case, the Green function G Bn stopped on B n is not directly available, we use the odometer function of the divisible sandpile as a replacement, as suggested by Levine and Peres in [LP10]. For simplicity of notation, we shall write G n (y, z) instead of G Bn (y, z). The random variable M is a sum of i.i.d. indicator variables, with Even though L is a sum of dependent indicator variables, following [LBG92], L can be bounded by a sum of independent indicators as follows. Only those particles with X i σ i ∈ B n contribute to L and for each y ∈ B n there is at most one index i with X i σ i = y. The corresponding post-τ y random walks are independent. In order to avoid dependencies in L, enlarge the index set to all of B n and definẽ where the indicators 1 y correspond to independent random walks starting at y. Then L ≤L, and the expectation ofL is given by Now (19) can be rewritten as We shall relate the random variablesL and M with the odometer function of the divisible sandpile. For this, consider the function f n : ∂B n ∪ B n → R, Set h x (y) = Gn(x,y) d(y) , then for x, y ∈ B n one has By linearity of the Laplace operator, f n solves the following Dirichlet problem Recall that the odometer function u n of the divisible sandpile with initial distribution µ 0 = n · δ o solves the same Dirichlet problem (6) on the sandpile cluster S n (whose shape is given by B n by Theorem 3.5).
The uniqueness of the solution of a Dirichlet problem gives that f n = u n on the set B n , and u n is approximated (up to an additive constant) by the function γ n defined in (8).
Since u n > 0, it follows that f n (z) > 0, for all z ∈ B n , which is equivalent to E[M ] > E[L].
We will use the following large deviations estimate for sums of independent indicators. For a proof, see Alon and Spencer [AS92, Cor. A.1.14].
Lemma 4.3. If N is a sum of finitely many independent indicator random variables, then for all λ > 0, where c λ is a constant depending only on λ.
In order to find an upper bound for the right hand-side of (22) we use the previous Lemma and choose λ > 0 and a such that Hence λ has to satisfy the relation and g n defined as To obtain λ, we have to bound f n (z)/g n (z) away from 0. For this, we first have to calculate the function g n which is, like f n , defined on ∂B n ∪ B n .

The function g n
By (20) and (21), the function g n is the solution of the Dirichlet Problem and can therefore be obtained by solving some linear recursions. For simplicity, we first shift the set B n by kn 1/3 in the direction of the positive x-axis. This shifted set will be denoted by B t n , which is the set of all (x, y) ∈ C 2 with 0 ≤ x ≤ 2kn 1/3 and On the shifted set we define the function g t n : B t n → R, by g t n (x, y) = g n (x + kn 1/3 , y), which solves the same Dirichlet problem (28) on B t n with the origin moved to (kn 1/3 , 0). By symmetry of g n , it is enough to compute g t n for vertices (x, y) with 0 ≤ x ≤ kn 1/3 and y ≥ 0. For z = (x, y) ∈ B t n , with y = 0, the Laplace ∆g t n (z) is equal to −1/2, hence on each "tooth" of the comb, g t n satisfies the linear recursion 2g t n (x, y) = g t n (x, y + 1) + g t n (x, y − 1) + 1, which has the general solution g t n (x, y) = 1 2 (y − y 2 ) + c 1 (x) + yc 2 (x), where c 1 (x) and c 2 (x) are functions of x, to be determined. For (x, 0), (x, 1) ∈ C 2 , we have g t n (x, 0) = c 1 (x) and g t n (x, 1) = c 1 (x) + c 2 (x).

IDLA inner bound
We are now able to conclude the proof of Theorem 4.2.
Lemma 4.4. For all ε > 0 there exists n ε , such that for all n ≥ n ε and all z ∈ B n(1−ε) Proof. By (26), one needs to study the function λ n (x, y) = fn(x,y) gn(x,y) . We have where c 1 (x), c 2 (x) and n x are defined in (34), (32) and (12), respectively. It suffices to consider the first quadrant. For every fixed x, the function λ n (x, y) is decreasing in y for 0 ≤ y ≤ nx 2 . From the proof of Theorem 3.5 we already know that n x 2 = l n 1/3 − x k 2 + O(1).
Obviously B n,ε ⊂ B n for all ε, hence fn gn is well defined on this set. Furthermore, by (35), fn gn is also decreasing on B n,ε as a function of y, for all ε > 0 and n big enough. This means that it is enough to study fn gn at the inner boundary of B n,ε . For this, let z = (x, y) ∈ B n,ε with |y| = (1 − ε)l n 1/3 − |x| k 2 be such a boundary point. Then The statement follows from the fact that for each ε > 0 one can find an ε > 0 such that Proof of Theorem 4.2. Recall that we need to show the convergence of the series (18). Fix z ∈ B n(1−ε) . We set λ = ε 4 > 0 in Lemma 4.4, and choose in equation (25). Apply now Lemma 4.3 to M andL. Recall also that where c λ is a constant depending only on λ. Hence, for all n ≥ n ε , we have n≥nε z∈B n(1−ε) In order to estimate the stopped Green function G n (z, z) upon exiting B n , with z = (x, y), note that We have the trivial upper bound G n (z, z) ≤ 2G A (y, y) where G A is the Green function of the simple random walk on the integer line, stopped at the interval A = − b n (x), b n (x) . Using Proposition 1.6.3 and Theorem 1.6.4 from Lawler [Law91], this can be bounded by For every ε > 0, the function g n (z) − f n (z) is again decreasing on every non-crossing path which starts at o and stays inside B n(1−ε) . Hence, it attains its minimum on the inner boundary ∂ I B n(1−ε) of B n(1−ε) . Taking limits, we get for every sequence z n = (x, y n ) with x fixed and z n ∈ ∂ I B n(1−ε)

The recurrent potential kernel
and for the sequence z n = (x n , 0) with x n = kn 1/3 (1 − ε) 1/3 Hence for all ε > 0 and n big enough for a constant C ε which depends only on ε. Since, by (37) the stopped Green function

Hence, (36) can be bounded by
which concludes the proof.

The recurrent potential kernel
In the recurrent lattice case Z 2 , the limiting shape of IDLA is derived using estimates for the recurrent potential kernel which is defined as follows In all previously known cases, the limiting shape of the IDLA-cluster is determined by the level sets of the potential kernel in the recurrent case and by level sets of the Green function in the transient case. Nevertheless, this is not the case for comb lattices C 2 , even if the simple random walk is recurrent. In order to show this, let us consider the generating function of the potential kernel where G(x, y|z) = ∞ t=0 P x [X t = y]z t is the generating function of the Green function of simple random walk on C 2 . Using standard techniques for generating functions one gets for x = (x 1 , See Bertacchi and Zucca [BZ03] for details. Therefore and lim z→1 − A(x, o|z) = 2|x 1 |. By Abel's theorem for power series, the potential kernel on the comb C 2 , if it exists, has to be equal to A(x, o) = 2|x 1 |. Hence we have an example of an IDLA cluster whose behaviour is not given by the level sets of the potential kernel.

Rotor-Router Aggregation
A rotor-router walk on a graph G is defined as follows. For each vertex x fix a cyclic ordering c(x) of its neighbours, i.e., c(x) = (x 0 , x 1 , . . . , x d(x)−1 ), where x ∼ x i for all i = 0, 1, . . . , d(x) − 1. The ordering c(x) is called the rotor sequence of x. A rotor configuration is a function ρ : G → G, with ρ(x) ∼ x, for all x ∈ G. Hence ρ assigns to every vertex one of its neighbours. A particle configuration is a function σ : G → N 0 , with finite support. If σ(x) = m > 0, we say that there are m particles at vertex x. A particle located at a vertex x with current rotor ρ(x) = x i , performs a rotor-router walk like this: it first sets ρ(x) = x i+1 , where addition is modulo d(x), and then it moves to x i+1 .
Rotor-router aggregation is a deterministic process of increasing subsets (R i ) i∈N of G defined recursively as R 1 = {o}, and where z i is the first vertex outside of R i−1 that is visited by a particle performing a rotorrouter walk, started at o. The particle stops at z i , and a new particle starts its tour at the origin, but without resetting the rotor configuration. The set R n of the occupied sites in G is called the rotor-router cluster of n particles. The odometer function u R (x) of rotor-router aggregation is defined as the total number of particles which are sent out by the vertex x during the creation of the rotor-router cluster R n .
In this section we study rotor-router aggregation on C 2 , and we give an inner bound for the cluster R n which holds for arbitrary initial configuration of rotors and is independent of the rotor sequence. The approach below relies on an idea of Holroyd and Propp [HP10], who use rotor weights in order to prove a variety of inequalities concerning rotor-walks and random walks.

Rotor Weights
Let G be a locally finite and connected graph, σ 0 : G → N an initial particle configuration with finite support, and ρ 0 : G → G an initial rotor configuration with ρ 0 (x) = x 0 for all x ∈ G, that is, all initial rotors point to the first element in the rotor sequence c(x).
Routing particles in the system, such that at each time step t exactly one particle makes one step of a rotor-router walk, gives rise to a sequence (ρ t , σ t ) t∈N 0 of rotor-and particleconfigurations. To each of the possible states (ρ t , σ t ) of the system, we will assign a weight.
Fix a function h : G → R. Define the particle weights at time t to be and the rotor weights of single vertices x ∈ G as where x i is the i-th neighbour of x in the rotor sequence c(x). Notice that, for k ≥ d(x), The total rotor weights at time t are given by where u t (z) is the odometer function of this process, that is, the number of particles sent out by x in the first t steps. Note that ρ 0 is chosen in such a way that, if i ≡ u t (x) mod d(x), then x i = ρ t (x) for all t ≥ 0 and x ∈ G.
It is easy to check that the sum of particle-and rotor-weights are invariant under routing of particles, i.e., for all times t, t ≥ 0 For rotor-router aggregation on G we start with n particles at the origin, that is, σ 0 = n·δ o , and we route a particle only if there is at least one other particle at the same position. The process terminates when no two particles are at the same position. Denote by t = t (n) the number of steps it takes to finish the process, and by (σ t , ρ t ) the final configuration. By the abelian property, the configuration (σ t , ρ t ) does not depend on the order the particles made their steps, and by definition σ t (x) = 1 {x∈Rn} .
We use the following weight function: for some y ∈ G, let h y : G → R given by where G n is the Green function of the simple random walk on G, stopped upon exiting the sandpile cluster S n with initial mass distribution µ 0 = n · δ o on G. Take now some y ∈ S n . Here S n is the sandpile cluster of the divisible sandpile on some general graph G. Recall that the Laplace of h y (x) on G is given by (24). The particle weights at the beginning are while the rotor weights are W R (0) = 0. At the end of the process, i.e., at time t when the rotor-router cluster R n is formed, we have since h y is equal to 0 outside of S n . For the rotor weights we get from (40) where u R is the rotor odometer function and k x = u R (x) mod d(x). By (24) and (39) Hence by the invariance of the total weights (41), we obtain Denote by v(y) the lefthand side of (47). Then v(y) solves the Dirichlet problem ∆v(y) = 1 d(y) 1 − nδ 0 (y) , for y ∈ S n , v(y) = 0, for y ∈ S n . By (4), the normalized odometer function u n of the divisible sandpile on G with initial mass distribution µ 0 = n · δ o satisfies exactly the same Dirichlet problem, hence v(y) = u n (y) and we get the following result, which compares the odometer function u R of rotor-router aggregation with the odometer function u n of the divisible sandpile. This result holds for any locally finite and connected graph G.
Proposition 5.1. Let u n be the normalized odometer function of the divisible sandpile with initial mass distribution µ 0 = n · δ o , and u R the odometer function of rotor-router aggregation with n particles starting at the origin o ∈ G. Then, for all y ∈ G, with Levine and Peres [LP09] derived an inequality similar to (48) in the case of Z d using a different method. For trees, W R (y) can be expressed in terms of the expected distance from the starting point of a random walk to the point where it first exits S n .
Proposition 5.2. If G is a tree and d(·, ·) is the graph distance on G, then where T = inf t ≥ 0 : X t ∈ S n , and (X t ) is the simple random walk on G.
Proof. For y ∼ z let N yz be the number of transitions from y to z before the random walk exits S n . Then See also [LP, Proposition 2.2] for more details. Since G is a tree, the net number of crossings of each edge is smaller or equal to one, i.e., We consider G as a tree rooted at x, and denote by π x,z the shortest path from x to z. For y = x, write y − for the parent of y, i.e., the unique neighbour of y that lies on the shortest path π x,y . With this notation we get y,z∈Sn y∼z where the last equality is due to the antisymmetry of N yz − N zy . Let C y = z ∈ S n : y ∈ π x,z be the cone of y. The random variable N y − y − N yy − is either zero or one, the latter if the random walk exits S n in the cone C y , hence For all z ∈ ∂S n we have # y ∈ S n \ {x} : z ∈ C y = d(x, z) − 1, therefore which completes the proof.

Rotor-router Aggregation on the Comb
Since C 2 is a tree, by Proposition 5.1 and 5.2, one needs an upper bound for the expected distance from the starting point of a random walk (X t ) on C 2 to the point where it first exits S n , in order to derive an inner estimate of the rotor-router cluster. Recall that on C 2 , the sandpile cluster S n has the shape B n given in Theorem 3.5. Using the trivial upper estimate E z d(z, X T ) ≤ max d(z, w) : w ∈ ∂S n = |x| + |y| + ln 2/3 , with z = (x, y) and l = 1 2 3 2 1/3 as in Theorem 3.5, we can show the following inner bound.
Theorem 5.3. Let R n be the rotor-router cluster of n particles on C 2 . Then, for n ≥ n 0 and for any initial rotor configuration and choice of rotor sequence, we havẽ B n ⊂ R n , whereB n = (x, y) ∈ C 2 : |x| ≤ kn 1/3 − c 1 n 1/6 , |y| ≤ l n 1/3 − Proof. By the definition of rotor-router aggregation, z ∈ C 2 : u R (z) > 0 ⊂ R n , and by Proposition 5.1 together with Proposition 5.2, we have for vertices z = (x, y) where the last inequality is due to (50). By Corollary 3.6, we have a lower bound of the sandpile odometer u n for z ∈ S n γ n (z) − a ≤ u n (z), where a is a positive constant smaller than 2, and γ n is the function defined in (8). Thus, to derive an inner bound, it suffices to check for which z = (x, y) ∈ S n the inequality γ n (x, y) − 2 |x| + |y| + ln 2/3 > 0 (51) holds. By symmetry it is enough to consider x, y ≥ 0. We first check inequality (51) on a "tooth" of the comb, that is, for a fixed x. The function γ n is given as where n x is the amount of mass that ends up in the x-"tooth" of the sandpile. Since x is fixed, we can treat n x as a constant. Hence the right hand side of (51) is a quadratic polynomial in y with smallest root y x = 2 + n x 2 − 4 + k l n 2/3 + 2n x + 4x.
Since (x, y) ∈ S n , we have the bound x ≤ kn 1/3 , hence y x = l n 1/3 − x k 2 + 2 + √ 6 3 x − cn 1/3 , for n ≥ n 0 , and a positive constant c. To get a bound on the x-axis, we calculate for which x > 0 the inequality y x > 0 is satisfied. Since y x is a polynomial of degree 2 in x this is easy to do, and again by series expansion around n = ∞ we obtain x ≤ k · n 1/3 − c 3 n 1/6 , for n ≥ n 0 . The inner bound for R n now follows from (53) together with (52). Figure 4 shows the inner estimate of the rotor-router cluster from Theorem 5.3 in comparison to sandpile cluster S n , for n = 1000. The white area is the area where the inequality (51) is valid, and corresponds to the setB n of Theorem 5.3. The colouring is based on the value of the right-hand side of (51).
The inner bound could be improved if one has a substantially better upper bound for E z [d(z, X T )]. For regular graphs, one can also give an universal inner estimate for rotor-router aggregation, which relates the rotor-router cluster to a divisible sandpile cluster with a smaller mass. Using the methods of Levine and Peres in [LP09] one can deduce the following.
Proposition 5.4. Let G be a regular graph with degree d and root o, and let R n be the rotor-router cluster of n particles starting at o. Further, let S n be the divisible sandpile cluster with mass distribution µ 0 (x) = n · δ o (x). Then S n/(2d−1) ⊂ R n .

Figure 4: Inner bound
Remarks. In the IDLA model, for an upper bound of the type A n ⊂ B n(1+ε) , we still do not have sufficient tools. In all previous studied cases, the IDLA cluster grows uniformly, and this makes easy the study of random walks. This is of course violated in our case, since the set B n in (1) grows with rate n 1/3 in the x-direction, and with rate n 2/3 in the y-direction. The harmonic measure for random walks stopped when exiting the set B n , was studied in [HS11], by making use of a special rotor-router type process. This is the first example where IDLA aggregate is not a set of uniform harmonic measure. In [HS11], we also give subsets of C 2 with uniform harmonic measure.
Recently Duminil-Copin et al. [DLYY11] developed a method for proving an outer bound for IDLA without needing a harmonic measure estimate, provided an inner bound is known. Nevertheless, their method cannot be applied in our setting, since one of the required regularity conditions (weaker lower bound) is not satisfied on the comb.