Internal DLA on cylinder graphs: fluctuations and mixing

We use coupling ideas introduced in \cite{levine2018long} to show that an IDLA process on a cylinder graph $G\times \mathbb{Z}$ forgets a typical initial profile in $\mathcal{O}( \sqrt{\tau_N} N(\log \! N)^2 )$ steps for large $N$, where $N$ is the size of the base graph $G$, and $\tau_N$ is some measure of the mixing of $G$. The main new ingredient is a maximal fluctuations bound for IDLA on $G\times \mathbb{Z}$ which only relies on the mixing properties of the base graph $G$ and the Abelian property.


Introduction
Internal Diffusion Limited Aggregation (IDLA) is a mathematical model for corrosion phenomena, introduced by Meakin and Deutch [19] and independently by Diaconis and Fulton [9]. It models the growth of a cluster of particles as being governed by the harmonic measure on its boundary seen from an internal point. More precisely, let G = (V, E) denote an infinite, locally finite graph with a marked vertex Z 0 ∈ V, and assume that a simple random walk on G starting from Z 0 exits every finite set containing Z 0 in finite time almost surely. We define an increasing family (A(t)) t≥0 of subsets of V as follows. Set A(0) = {Z 0 }, and recursively define where Z t denotes the exit location from A(t − 1) of a simple random walk on G starting from Z 0 , independent of everything else. Then (A(t)) t≥0 is a Markov chain on the space of connected subsets of G, whose long-term behaviour is of interest. The case G = Z d , Z 0 = 0 is by now well understood: as t → ∞, A(t) has been shown to converge to a Euclidean ball [16] with logarithmic fluctuations away from it in dimension d ≥ 2 [3,5,11,14,4,12,13]. It was recently showed that the limiting shape is stable under the following perturbation: rather than starting random walks at 0, sample the starting point of the k th walk uniformly from A(k − 1) [6]. Bounding the fluctuations of IDLA with uniform starting point remains an open problem.
Let G = (V, E) be a finite, connected, non-oriented graph on N vertices. We take y) : v ∈ G, y ∈ Z} E = {((v, y), (v ′ , y ′ )) : v ∼ v ′ and y = y ′ , or v = v ′ and |y − y ′ | = 1}, where ∼ denotes the adjacency relation in G. We refer to the first and second coordinate as the horizontal and vertical coordinate respectively. For y ∈ Z we call the set G×{y} the y th level, while R y = G × {z ≤ y} will denote the rectangle of height y. A simple random walk on G is, for us, a discrete-time Markov chain which at each step moves either in the horizontal or in the vertical coordinate with probability 1/2. A vertical move consists in stepping from (v, y) to (v, y ± 1) with equal probability, while a horizontal move is a lazy simple random walk step on G, i.e. (v, y) → (v, y) with probability 1/2 and (v, y) → (w, y) Let π N denote the stationary distribution of a simple random walk on G, that is where deg v denotes the degree of v in G. We make the following assumption throughout.
for all N ≥ 1.
Graphs satisfying the above assumption are called quasi-regular. Example include the complete graph, expander graphs, the d-dimensional torus, and any graph with vertices of comparable degree. In particular, graphs in this class can have a wide range of mixing behaviour for large N (see the comments after the statement of Theorem 2).
We remark that Assumption 1 is not needed for our arguments to work: we choose to make it as it identifies the class of base graphs for which our approach gives best results.
Notation. All constants in this note are allowed to depend on δ, δ ′ , and we suppress this dependence from the notation.
We define IDLA on G as follows. Let A(0) be any connected subset of V containing the rectangle R 0 = {(v, y) : y ≤ 0} and finitely many sites above level 0. At each discrete time t ≥ 1, a simple random walk is released from level 0 at a random location sampled from π N , and its exit location Z t from A(t − 1) is added to the cluster by setting A(t) = A(t−1)∪{Z t }. Note that this is equivalent to adding a site to the cluster according to the harmonic measure on the cluster's external boundary seen from level −∞. When A(0) = R 0 we say that the process starts from flat. It is common to describe the IDLA dynamics as an interacting particle system. Particles are released one by one from level 0 according to π N , and perform simple random walks until they hit an unexplored location, where they settle. The cluster is then the set of sites containing settled particles.
Let for all t ≥ 0, and refer to (A * (t)) t≥0 as the shifted IDLA process associated to (A(t)) t≥0 .
Shifted IDLA is itself a Markov chain on the countable state space Ω. It is easy to see that this chain is irreducible (all states communicate with the flat configuration R 0 ). As in [18], it can be shown that shifted IDLA is positive recurrent, so it has a stationary distribution µ N (cf. Remark 1 below). If a random cluster A is distributed according to µ N , we say that A is a stationary cluster.
The aim of this note is to address the following questions: • What do stationary clusters look like?
• How long does shifted IDLA take to forget a stationary initial state?
The first question concerns fluctuation bounds for IDLA clusters, while the second one looks as the mixing properties of shifted IDLA as a Markov chain. Both questions were answered in [18] for the cycle base G = Z N , which is by now well understood. Here we generalise the analysis to quasi-regular base graphs, for which far less is known. In particular, while [18] makes use of optimal fluctuation bounds which are specific to the case G = Z N , here we observe that near-optimal fluctuation bounds are not needed to obtain a near-optimal mixing result (see comments after the statement of Theorem 2).
In order to state our results, let us introduce the following measure of mixing for a simple random walk on the base graph G. Recall that a coupling of two simple random walks on G is a process (ω(k), ω ′ (k)) k≥0 such that both marginals are simple random walks on G, possibly with different initial distributions. A coupling of two simple random walks on G is said to be Markovian if (ω(k), ω ′ (k)) k≥0 is itself a Markov chain. Clearly, the trivial coupling (independent random walks) is Markovian. If ω, ω ′ are coupled according to a Markovian coupling we can, and do, always assume that, if Definition 2. Fix an arbitrary Markovian coupling of two simple random walks ω, ω ′ on G, with ω(0) = v ∈ V and ω ′ (0) ∼ π N . We denote the joint distribution of ω, ω ′ under the coupling by P v,π N . For ε > 0, define Write τ N for τ N (1/4). Note that τ N differs from the usual notion of mixing time since, in general, the optimal coupling (i.e. the coupling that achieves the mixing time) of two Markov chains is not Markovian [10]. Here we force Markovianity in order to be able to take advantage of the Abelian property of IDLA [9]. The meeting time of independent Markov chains on finite graphs has been studied under several assumptions (see [2,1,20,7,15,21] and references therein). In particular, for simple random walks on quasi-regular graphs it is known that τ N = O(N 3 ) [8,7]. Markovian couplings achieving the order of the total variation mixing time (up to multiplicative constants) are known for many graphs (see [17]). Examples in the quasi-regular case include, but are not limited to, the complete graph on N vertices (τ N = O(1)), the hypercube {0, 1} n (N = 2 n , , and the binary tree on N vertices (τ N = O(N )). To the best of our knowledge, understanding how large can τ N be compared to the Total Variation mixing time of the same Markov chain is an interesting open problem.
Our first result bounds the height of stationary clusters in terms of τ N , thus providing non-trivial information on the stationary distribution µ N . (1). Then for any γ > 0, there exists a constant c γ , depending only on γ, δ, δ ′ , such that for N large enough.
A similar result holds for any lukewarm start ν N for shifted IDLA (cf. [18], Section 1), we omit the details. Note that the faster the mixing of the base graph, the smaller the height bound provided by Theorem 1. We remark that the proof of Theorem 1 only relies on the mixing properties of the base graph G and the Abelian property of IDLA, but does not require any Green's function estimate for the underlying random walks.
We use the above control on stationary clusters to bound the time it takes for shifted IDLA to forget any such initial state.
Theorem 2. Assume (1). Then for any γ > 0 there exist a constant d γ and a set Ω γ ⊆ Ω, depending only on γ, δ, δ ′ , such that for N large enough, and such that the following holds. For any where (A(t)) t≥0 and (A ′ (t)) t≥0 are IDLA processes starting from A 0 and A ′ 0 respectively, and P (t) and P ′ (t) denote the laws of A(t) and A ′ (t).
Thus IDLA forgets any stationary initial state in O(N √ τ N (log N ) 2 ) steps, with high probability. The reader should compare this result with Theorem 1.3 of [18] for G = Z N , stating that (3) holds with t γ = d γ N 2 log N . Although our stationary height bound (2) is far from optimal in this case, we remark that the conclusion of Theorem 2 only differs from the near-optimal result in [18] by a logarithmic factor. We conjecture that this is the case for all quasi-regular graphs.
Conjecture 1. Theorem 2 is optimal up to logarithmic factors.
Structure of the paper. The remainder of the paper is organised as follows. We start with a preliminary result on the mixing properties of a simple random walk on G in Section 2. In Section 3 we recall the water level coupling introduced in [18], and adapt it to bound the maximal fluctuations for IDLA on G × Z for polynomial times (cf. Theorem 3). This is the core of the article. Finally, in Section 4 we briefly explain how to deduce Theorems 1 and 2 along the lines of [18], leaving the details to the reader.
Acknowledgement. I am thankful to Lionel Levine for several insightful discussions, and for a careful reading of this note.

Preliminaries
We collect here two preliminary results from [18]. Although they are only proved in [18] for the case G = Z N , it is straightforward to adapt the proofs to this setting by replacing the reflection coupling with the Markovian coupling used to define τ N .
To start with, note that, for any γ > 0, This follows from a standard argument (see [17], Chapter 4), which carries out to our definition of τ N . Let ω = (ω(t)) t≥0 be a simple random walk on G, as described in the introduction, and let τ y n := inf{t ≥ 0 : ω(t) ∈ G × {n}} denote the first time the walk reaches level n.
For the proof see [18], Lemma 3.1. This tells us that, by the time a simple random walk on G has travelled √ τ N log N in the vertical coordinate, it has mixed well in the horizontal one. Proposition 1 can be used to show that, as long as the walkers have time to mix in the horizontal coordinate before exiting the cluster, the IDLA process does not depend too much on their initial positions.
. Then there exists a coupling of A, A ′ such that the following holds. For any γ > 0 there exists a finite constant b γ,m , depending only on γ and m, such that if This tells us that, as long as an IDLA cluster is filled up to level b γ,m √ τ N log N , releasing the next T walkers from fixed initial locations below level 0 or from π N -distributed locations at level 0 results in the same final cluster with high probability. We refer the reader to [18], Proposition 3.1, for the proof, which also shows that the constant b γ,m is linear in γ and m.

IDLA maximal fluctuations
In this section we adapt the water level coupling introduced in [18] to prove the following result on the maximal fluctuations of IDLA on G.
The above theorem, of independent interest, tells us that the maximal fluctuations for IDLA on G × Z from the rectangular shape are O( √ τ N (log N ) 2 ) for polynomially many steps, when starting the process from flat. We split the proof into several lemmas, keeping Assumption 1 in force throughout.
Let us start with the following coupon-collector inequality stating that, starting from flat, after O(N log N ) releases level 1 is filled with high probability. Lemma 1. Let (A(t)) t≥0 be an IDLA process on G with A(0) = R 0 . For arbitrary γ > 0, let a γ = (γ + 1)/δ. Then we have that Proof. By Assumption 1, each time a new random walk is released from level 0 each vertex at level 1 has probability at least We use this simple observation to implement the water level coupling. This proof generalises the one of Theorem 1.1 in [18], bypassing the lack of the logarithmic fluctuations result via Lemma 1.

Proof of Theorem 3.
It suffices to show that the statement holds for any fixed T ≤ N m . The result will then follow by replacing γ with γ + m and using the union bound over all T ≤ N m .
The proof consists of two steps. In Step 1, we use Lemma 1 together with the Abelian property of IDLA to reduce the case T = O(N √ τ N (log N ) 2 ). In Step 2 we then bound the fluctuations of A(T ) for such small T by a standard argument due to Lawler et al [16].
Step 2). Fix γ > 0 as in the statement. Let a γ+2m be defined as in Lemma 1, and b γ+m+1,m be as in Proposition 2. We write a, b in place of a γ+2m , b γ+m+1,m for brevity. Define Recall that the IDLA process starts from the flat configuration A(0) = R 0 . To start with, release aN log N walkers from level 0 according to π N . We stop the walkers upon reaching level 1. A walker that reaches level 1 at an empty site settles there and never moves again. Walkers that reach level 1 at occupied sites, on the other hand, are stopped and declared frozen. As the name suggests, the motion of frozen walkers will be resumed at a later stage. Then, by Lemma 1, after aN log N releases level 1 is filled with probability at least 1 − N −(γ+2m) and all walkers, whether settled or frozen, are at level 1. On this high probability event, we repeat this procedure by releasing aN log N additional walkers which stop, with the same rule as before, at level 2, and so on. Let W 1 denote the cluster of occupied sites (i.e. sites containing a settled walker) after releasing t 0 walkers from level 0 in groups of aN log N , stopping walkers of the k th group at level k as described above.
for N large enough. On this high probability event, all the frozen walkers at levels 1 to ℓ − b √ τ N log N are at distance at least b √ τ N log N from the boundary of W 1 , so by Proposition 2 releasing them has, with high probability, the same effect as releasing the same number of walkers from level 0 according to π N . To make this precise, note that on the high probability event {W 1 = R ℓ } there are exactly frozen walkers up to level ℓ − b √ τ N log N . Let W 1 (t 1 ) denote the cluster obtained by releasing these t 1 frozen walkers. Then, if (A ′ (t)) t≥0 is an auxiliary IDLA process with A ′ (0) = W 1 , by Proposition 2 we can couple A ′ (t 1 ) and W 1 (t 1 ) so that for N large enough. This reduces the problem of bounding the fluctuations of an IDLA cluster with N m particles to the one of bounding the fluctuations of an IDLA cluster with It therefore suffices to iterate this procedure at most N m−1 times, as long as ℓ ≥ b √ τ N log N , to reduce to the case with probability at least 1 − N −γ .
Step 2. Assume that T ≫ N log N , otherwise the conclusion is trivial, and let (A(t)) t≥0 be an IDLA process starting from A(0) = R 0 . We show that, if T ≤ αN √ τ N (log N ) 2 for some α (which may depend on γ, m), then there exists β ∈ (0, 1), independent of N , such that for N large enough. To see this, let Z k (t) := |A(t) ∩ {y = k}| denote the number of sites in A(t) at level k, and set µ k (t) := E(Z k (t)). Then for all k > 0 and t ≥ 0. Indeed, clearly µ 1 (t) ≤ N and µ k (0) = 0 for all k > 0. For other values of k, j we have where in the above inequality we have used that the probability that the (t + 1) th walker reaches level k − 1 inside A(t) is maximised when A(t) is completely filled up to level k − 2. Thus for k > 0 and (7) follows by a simple iteration. Now take t = T , k = 3T N + 1 and recall that k! ≥ k k e −k to get for any β ∈ e 3 3 , 1) and N large enough.
This proves (6), hence concluding the proof of Theorem 3.

Proof of Theorems 1 and 2
We sketch here the proof of Theorems 1 and 2. We give precise statements of the several lemmas leading to the proof, but choose to leave the proof details to the reader, since these are straightforward generalisations of the arguments in [18]. This section also serves as a survey of the ideas introduced for the case G = Z N in [18]. We split the argument into several steps, for ease of reading.
Step 1: Decay of excess height. For an IDLA cluster A ∈ Ω, we define the excess height of A by where h(A) denotes the height of A. Thus E(R k ) = 0 for all infinite rectangles R k , while E(A) is large if A is tall and sparse. To start with, we show that if the excess height of a cluster is too large, it tends to decrease under IDLA dynamics.
for N large enough.
To see this, note that if the excess height is large, then there are many empty sites below level h(t). Call a level below h(t) bad if it contains at least one empty site. If the excess height is high enough, then there are many bad levels at distance at least 20 √ τ N log N one another. Thus a random walker has time to mix in the horizontal coordinate between bad levels, and each time it reaches a new bad level it has probability at least δ/(2N ) to settle at an empty site. It follows that, if there are enough bad levels, the random walk will settle below height h(t) with high probability, which will cause the excess height to decrease. Lemma 2 shows that an IDLA process spends a small proportion of time in clusters with high excess height. From this it is standard to deduce that stationary clusters have low excess height, which gives the following result.
Lemma 3. For any γ > 0 there exists a constant C E,γ , depending only on γ, δ, such that, Step 2: From low excess height to low height. Let (A(t)) t≥0 be an IDLA process starting from a stationary configuration A(0) ∼ µ N , and denote the associated shifted IDLA process by (A * (t)) t≥0 . While Lemma 3 shows that A(0) has low excess height with high probability, this does not imply low height, since |A(0)| is random. It gives, on the other hand, an upper bound on the number of empty sites, that is the number of sites in R h(A(0)) \ A(0). This allows us to show that the height of the associated shifted IDLA process becomes O(N 4 √ τ N ) within polynomial time, as stated below.
Lemma 4. Let (A * (t)) t≥0 be a shifted IDLA process with A(0) ∼ µ N , and let Then for any γ > 0 there exists a constant C * E,γ , depending only on γ, δ, such that The above result is proved as follows. On the high probability event E(A(0)) ≤ E * γ+2 there are at most N E * γ+2 empty sites below level h(A(0)). Each time we add a new particle to the cluster, the lowest empty site has probability at least δ/N to be filled, independently of everything else. Hence it will take at most a geometric number of releases of parameter δ/N to fill it. We fill empty sites sequentially while the excess height does not exceed 2E * γ+2 . When it does, we wait for the dynamics to bring the excess height below E * γ+2 , and try again to fill all the empty sites. It is easy to see that, with high probability, this procedure succeeds within N 6 √ τ N particle releases.
Remark 1. The same argument shows that shifted IDLA defines a positive recurrent Markov chain, since starting from the flat configuration R 0 the expected return time to R 0 is finite.
Step 3: Forgetting polynomially high initial configurations. Lemma 4 tells us that stationary clusters have O(N 6 √ τ N ) height with high probability. We use the polynomial height bound to show that an IDLA process starting from a stationary configuration forgets that it did not start from flat within polynomially many steps.
for some absolute constant α < ∞. Let (A(t)) t≥0 and (A ′ (t)) t≥0 denote two IDLA processes starting from A 0 and A ′ 0 respectively, and denote the laws of A(t), A ′ (t) by P (t), P ′ (t). Then for any γ > 0 there exists a constant β γ , depending only on γ, δ, δ ′ , such that, writing s γ = αN 7 √ τ N + β γ N √ τ N (log N ) 2 for brevity, we have This follows from the water level coupling introduced in Section 3. Indeed, let W 0 be a new IDLA cluster, independent of everything else, built by adding s γ particles to the flat configuration R 0 . By Theorem 3 we can choose β γ large enough so that W 0 is completely filled up to height αN 6 √ τ N with high probability. We take W 0 to be the initial configuration of two auxiliary processes (W (t)) t≤|A 0 | and (W ′ (t)) t≤|A 0 | , that we think of as water flooding the clusters. Water falling in A 0 (respectively A 0 ) freezes, and it is only released at a later time. Frozen water particles are released in pairs, and their trajectories are coupled so to make the particles meet with high probability before exiting the respective clusters. By Proposition 2 we can take β γ large enough to ensure that all pairs of frozen particles meet with high probability before exiting their respective clusters, in which case we have W (t) = W ′ (t) for all t ≤ |A 0 |. The above proposition then follows by invoking the Abelian property of IDLA for the equalities in law = W ′ (|A 0 |).
Step 4: Stationary height: proof of Theorem 1. We use Proposition 3 to argue that a stationary shifted IDLA process has, after polynomially many steps, both stationary law and O( √ τ N (log N ) 2 ) fluctuations, since it forgot hat it did not start from flat with high probability. This tells us that stationary clusters have height O( √ τ N log N ), and it thus proves Theorem 1. More precisely, let (A(t)) t≥0 denote a shifted IDLA process started at stationarity A(0) ∼ µ N . Then A(t) ∼ µ N for all t ≥ 0. In particular, if for arbitrary γ > 0 we define s γ as in Proposition 3, we have that On the other hand, since h(A 0 ) = O(N 6 √ τ N ) with high probability by Lemma 3, Proposition 3 tells us that A(s γ ) is close in distribution to a shifted IDLA process started from height at most 1, and hence, using that s γ ≤ N 9 for N large enough, with high probability by Theorem 3.
Then µ N (Ω c γ ) ≤ N −γ , and Theorem 2 follows by exactly the same argument used in the proof of Proposition 3.