A Convergence Rate for Extended-Source Internal DLA in the Plane

Internal DLA (IDLA) is an internal aggregation model in which particles perform random walks from the origin, in turn, and stop upon reaching an unoccupied site. Levine and Peres showed that, when particles start instead from fixed multiple-point distributions, the modified IDLA processes have deterministic scaling limits related to a certain obstacle problem. In this paper, we investigate the convergence rate of this"extended source"IDLA in the plane to its scaling limit. We show that, if $\delta$ is the lattice size, fluctuations of the IDLA occupied set are at most of order $\delta^{3/5}$ from its scaling limit, with probability at least $1-e^{-1/\delta^{2/5}}$.


Introduction
Internal Diffusion Limited Aggregation (IDLA) is a probabilistic growth process on the integer lattice Z d , first proposed by Meakin and Deutch [MD86] to model electro-chemical polishing.Namely, IDLA follows the growth of random sets A(n); we set A(1) = {0}, and A(n + 1) is obtained by adding to A(n) the point at which a centered simple random walk exits A(n).With the right scaling, this process resembles a stream of particles from the origin barraging (and thus smoothing) the inner surface of an origin-centered sphere.
In line with its applications in smoothing processes, the overall smoothness of IDLA has been an active area of investigation.Meakin and Deutch first studied this numerically, finding that variations of A(n) from the smooth ball were of magnitude log n in dimension 2 [MD86].Significant progress has also been made in proving these properties mathematically.In particular, Lawler, Bramson, and Griffeath [LBG92] proved that A(n) approaches the ball of radius d n/ω d -where ω d is the volume of the d-dimensional unit spherealmost certainly as n increases.Several groups [Law95, AG10] also found convergence rates for this process.Most recently, Asselah and Gaudillière proved that the fluctuations away from the disk are bounded by log 2 n in dimension 2 and √ log n in higher dimensions [AG13a,AG13b], and Jerison, Levine, and Sheffield independently proved a log n bound in dimension 2 and √ log n in higher dimensions [JLS12,JLS13b].Asselah and Gaudillière also proved lower bounds of √ log n on the maximum fluctuations of IDLA [AG11], showing that the recently proved results for d ≥ 3 are tight.
Of considerable interest is the extended-source case of IDLA, wherein particles start from a fixed point distribution rather than all from the origin.This generalizes the applicability of IDLA to a much wider range of surfaces, allowing us to see how different geometries interact with this smoothing process.This question was originally investigated by Diaconis and Fulton [DF91] in the context of a "smash sum" of two domains.Levine and Peres [LP10] reframed this notion as a generalized IDLA, proving deterministic scaling limits for a piecewise constant density σ : R d → Z ≥0 of starting points.It is worth noting, but beyond the scope of this paper, that another model with Poisson particle sources was proposed and studied by Gravner and Quastel [GQ00].
In this paper, we investigate the convergence rate of the extended-source IDLA of Levine and Peres to its scaling limit in dimension 2, adapting the techniques of Jerison et al. [JLS12].Under the additional assumptions that the initial mass distribution is "concentrated" (see Section 2) and the deterministic limit of its IDLA flow is smooth, we show that-if δ is the lattice size-the fluctuations of extended-source IDLA are of order δ 3/5 or below, with probability at least 1 − e −1/δ 2/5 .
There are several major difficulties in extending the argument of [JLS12] to a general source setting, and we introduce and apply some new technical tools to solve them.In particular, their proof relies heavily upon a specific formulation of the Poisson kernel, which only applies in the case of the disk.Although we are able to make use of the disk Poisson kernel in the first part of our proof, we replace it halfway through our paper with a more general formula-combining results on the discrete Green's function with a last-exit decomposition (both taken from [LL10]), we obtain an explicit formula for discrete Poisson kernels in general domains.This approach requires relatively fine control of the discrete Green's function in general domains; in fact, we obtain an L 1 convergence rate of the discrete Green's function to its continuum limit, of order δ 3/5 (see Lemma 5.2(c)); we have not seen this result in the literature before, and we imagine it may be useful in studying similar problems.
Finally, we believe that our overall δ 3/5 bound on the fluctuations of IDLA is non-optimal, and we discuss possibilities for improvement in Section 7.However, we will see in a sequel to this paper that our bound is strong enough to prove weak scaling limits of the IDLA fluctuations themselves; indeed, this subsequent result requires a bound of order δ 1/2+ε , for any ε > 0. The question of weak scaling limits of the IDLA fluctuations has been investigated in the single-source case by Jerison, Levine, and Sheffield [JLS14]-more recently, Eli Sadovnik [Sad16] has shown scaling limits for extended-source fluctuations integrated against harmonic polynomials.In the sequel, we will seek to generalize Sadovnik's result and apply it to fluctuations "through time", to better understand the covariances between fluctuations at different times.
In Sections 2 and 3, we will introduce our main result and provide a background on existing theory needed for our proof.The remaining sections are dedicated to the proof of Theorem 3.1.Sections 4 and 5 set up the necessary theory; the former shows that an early point implies a similarly late point, and the latter shows that a late point implies a different, very early point.Section 6 combines these results in an iterative argument, recovering the full theorem.
2 Background on extended-source IDLA We will focus on a specific sort of extended source-a concentrated mass distribution-slightly narrowing the definitions introduced in [LP10] in order to capture scaling limits for the partially-completed process.We will give details on various extensions in the final section.
Finally, set T = k T k , and fix increasing functions We can also consider infinite concentrated mass distributions, allowing s ∈ [0, ∞); we define these by requiring that the restriction s ∈ [0, T ] is a finite concentrated mass distribution, and we suppose we have fixed some such T ≥ 0.
Geometrically, a concentrated mass distribution is a set of domains Q s i i ⊂⊂ int(D 0 ) growing at a rate s i (s).There are several differences and restrictions of this definition as compared to that in [LP10], which require comment.Most notably, we define the mass distribution to grow in time, as s → σ s , such that we can study the partially complete process (i.e., that at s < T ).After discretization, this will correspond to a prescribed order of IDLA sources; this allows us to relate the IDLA process to a smoothly growing deterministic set, to recover uniform bounds (in time) on the IDLA fluctuation, and, in the sequel, to study correlations between fluctuations at different points in space and in time.
That the arclength of ∂Q s i exists and is bounded uniformly allows us to discretize the process s → σ s in a natural way.That is, viewing σ s as a multiset and taking any m ≥ 1 and any time s 0 ∈ [0, T ), we can find the "first" point in 1 m Z 2 added to σ s after the time s 0 ; this procedure will give the overall ordering of IDLA sources.As such, this hypothesis could be weakened, so long as the discretization remained possible.
The requirement that Q s i ⊂⊂ int(D 0 ) is necessary for the proof of Lemma 4.1(c), which in turn is necessary for the estimate 4.2(c).In short, we make heavy use of discrete Poisson kernels on sets cut out by IDLA, and in particular, their pointwise convergence (away from the pole) to continuous Poisson kernels; this hypothesis guarantees a minimum distance between this pole and the source points, which in turn guarantees a strong rate of convergence of these Poisson kernels at source points.
As we will see, IDLA processes begun from finer and finer discretizations of a concentrated mass distribution approach a smooth, deterministic flow s → D s , where the set D s is defined in terms of the Diaconis-Fulton "smash" sum, as defined in [LP10]: , we define the discrete smash sum A ⊕ B as follows.Let C 0 = A ∪ B, and for each x i ∈ {x 1 , ..., x n } = A ∩ B, start a simple random walk at x i and stop it upon exiting C i−1 .Let y i be its final position, and define As proven in [LP10], if we instead take domains A, B ⊂ R 2 , the smash sums A m ⊕ B m of approach a deterministic limit, which we label A ⊕ B. An example is pictured in Figure 1.
Figure 1: The smash sum A ⊕ B is the deterministic limit of an IDLA-type growth process starting from the sets A and B, representing the dispersal of particles in A ∩ B (in dark blue above) to the edges of A ∪ B (in yellow above).In our setting, we see that IDLA also converges to an iterated smash sum.
Given a mass distribution σ s and using the notation of Definition 2.1, we define the sets D s , s ∈ [0, T ] to be the smash sums Importantly, these are deterministic sets, depending only on the mass distribution.
Visually, D s is a smooth outward flow from D 0 , with Vol(D s ) = s + Vol(D 0 ); we can think of D s as the result of allowing the mass at {Q s i i } to diffuse (in the sense of Brownian motion) to-and accumulate at-the edge of D 0 .These sets satisfy the following key property: Lemma 2.3.For (D s , σ s ) arising from a mass distribution, we have This property, which identifies D s as a quadrature domain, is well-known; for instance, see [Sak84].
Example.Suppose that we take D 0 = B 1 to be the unit disk, and we set Q s i = B √ s/π for s ∈ [0, π/4] and i = 1, ..., N , where B r = B r (0) is generally the origin-centered disk of radius r.We further define s i (s) = s/N , so that all sets Q s i are growing at the same rate.Visually, particles are emanating evenly (with density N ) from outwardly moving rings of radius 0 ≤ r ≤ 1/2, as shown in Figure 2.
Here, T = N π/4.From symmetry considerations, it is clear that D s = B √ 1+s/π are outwardly expanding disks, as in the case of a point-source; we omit the proof here, as it is not critical to our results.The property of Lemma 2.3 is simply the mean value property in this setting.To define discrete processes on a mass distribution, we first need to discretize the distribution itself; fortunately, there is a natural way to discretize any mass distribution.Fix an integer m, and note that and such that f is constant near s m,k for s m,k = 0, T .Define the sequence S m = {z m,1 , ..., z m,nm } inductively as follows: 1. Let s m,n be the smallest s m,i ≥ s m,n−1 such that ) exceeds the number of times z m,n occurs in {z m,i } i<n .
Intuitively, we allow the sets Q s i i to expand a slight (i.e., O(m −2 )) amount, and then we add all new points to the sequence of z m,i .It is possible that multiple points may satisfy this condition for a given time-in the limit m → ∞, the order in which these "nearby" points appear will not matter.
Given these sequences S m , the (resolution m) internal DLA (IDLA) associated to the mass distribution is the following process: Definition 2.5 (Internal DLA).Suppose we have a concentrated mass distribution with initial set D 0 giving rise to the sequences S m .The IDLA A m (t) associated to the mass distribution is as follows.Define the initial set A m (0) = 1 m Z 2 ∩ D 0 .Then, for each i ≥ 1, start a random walk at z m,i , and let z i be the first point in the walk outside the set Importantly, the law of A m (i) does not depend on the order of {z m,1 , ..., z m,i }, as proven by Diaconis and Fulton [DF91].
We know from Levine and Peres [LP10] that the sets A m (m 2 s) approach their deterministic limits D s almost surely.That is, for any ε > 0, we know that d H (∂A m (m 2 s), ∂D s ) ≤ ε almost surely for sufficiently large m, where the boundary ∂A m (m 2 s) denotes the points in A m (m 2 s) adjacent to A m (m 2 s) c .Here and below, we use d H to denote the Hausdorff distance between sets: In the following sections, we will use this result along with an iterative argument to recover a stronger convergence rate on A m (m 2 s).

Main result
We will write (A) ε and (A) ε for the outer-and inner-ε-neighborhoods of a set A ⊂ R 2 , respectively.That is, Our primary result is the following convergence rate on the IDLA occupied sets A m (t) to their deterministic scaling limits D s : Theorem 3.1.Suppose D τ is a smooth flow arising from a concentrated mass distribution.For large enough m, the fluctuation of the associated IDLA A m (t) is bounded as for a constant C 5 depending on the flow.Equivalently, where d H is the Hausdorff distance.
As mentioned in the introduction, we have reason to believe that the m −3/5 convergence rate so described is non-optimal.Indeed, we will see in Lemma 5.2 that this results from a relatively rudimentary L 1 bound on the convergence rate of discrete Green's functions, rather than from the geometry of IDLA itself.We will discuss suggestions for further research in Section 7.

Overview of notation
Henceforth, we will assume we have fixed a smooth, concentrated mass distribution, and we will use the language of Section 2 to refer to it.That is, T will always refer to the total volume of our source sets, z m,i to the i th source point in the resolution-m discretization of our mass distribution, and D s to the scaling limit of IDLA started on the density σ s .
To discuss fluctuations of IDLA away from its scaling limit, we define the following notions of "earliness" and "lateness": As introduced in the preceding subsection, we will write (A) ε and (A) ε for the outer-and inner-εneighborhoods of a set A ⊂ R 2 : Finally, for convenience and visual clarity, we will use m 2 s [resp., m 2 T , etc.] in place of m 2 s in places where the meaning is clear.In particular, A m (sm 2 ) := A m ( sm 2 ).

Required lemmas
A number of existing results are necessary in the proof of Theorem 3.1; we collect many of them here.Firstly, we use the following two estimates on IDLA.The first bounds the probability of so-called "thin tentacles"-shown in Figure 3-and is simply a transcription of Lemma 2 of [JLS12] in our setting.The second is a part of the estimate of Levine and Peres [LP10] earlier described, demonstrating that extremely late points are unlikely.

Lemma 3.2 (Thin Tentacles).
There are positive absolute constants b, C 0 , and c 0 such that for all z Proof.The proof can be taken verbatim from Jerison et al. [JLS12], with our scaling in mind.
Figure 3: It is conceivable that the IDLA set A m extends out from its limit D s in thin tentacles, as pictured; we quantify this "thinness" near a point ζ by the number of filled spaces in disks centered at ζ.In Lemma 3.2, we show that a fixed positive fraction of each such disk (for small enough radii) is very likely filled in.
Lemma 3.3.There are absolute constants C 0 , c 0 > 0 such that for all real ε > 0, T ≥ τ ≥ 0, and large enough m, Proof.By Levine and Peres [LP10, p. 49], the probability for large enough m, where c depends only on S m , D τ , and ε.
), so we can bound the total probability of L ε [τ m 2 ] as for some C 0 .Choosing some c 0 < c, we have m 2 e −cm 2 / log m ≤ e −c 0 m 2 / log m for large enough m, and the lemma follows.
The next two lemmas control the flow s → D s .In short, the first shows that the arclength of ∂D s is uniformly bounded on both sides, and the second shows that D s grows at a linear rate at all points.The first follows directly from the smoothness of D s .
Lemma 3.4.For s ∈ [0, T ], the arclength of ∂D s is bounded as where u, U > 0 are constants depending only on the flow.Lemma 3.5.For a smooth flow D s and any times where v, V > 0 are constants depending only on S m .
Proof.The upper bound follows from the smoothness of D s and the compactness of the interval [0, T ].
For the lower bound, we will exploit the fact that D s is also the scaling limit of divisible sandpile processes on {z m,1 , ..., z m,nm } with starting set D 0 .We will not give details on the divisible sandpile process here; see [LP10] for more details on scaling limits of divisible sandpiles.
Choose an s ∈ [0, T ], and let D 1/m s (t) be the fully occupied set of the divisible sandpile on the lattice 1 m Z 2 with starting density In the interval [s, s + ε], a total of εm 2 particles are released-in fact, one particle is started at z m,n at each time n/m 2 .From Lemmas 5.1(d) and 5.2(a,b), we can bound the exit probability as which tells us that, in the divisible sandpile model, we need m/c particles to ensure that the new set . Now, we can apply the same estimate to the expanded set and thus Continuing in this manner, we find that , and in particular that Now, D s+ε is the scaling limit of these sets D √ 2 for all ε > 0, which implies the claim.
Finally, the following two lemmas control the exit times of Brownian motion from an interval [a, b].These are restatements of Lemmas 5 and 6 in [JLS12], so we omit the proofs here.Below, let B(s) be centered, one-dimensional Brownian motion, and denote Lemma 3.7.For any k, s > 0,

The recurrent potential kernel
Key to much of our analysis will be the so-called recurrent potential kernel g : Z 2 → R, which acts as a free Green's function for the discrete Poisson equation.We define it in probabilistic terms as where P n (z) is the probability that an n-step simple random walk from the origin in Z 2 ends at z. Importantly, ∆ h g(x) := 1 4 (g(x + 1) + g(x − 1) + g(x We will also use the first few terms of the asymptotic expansion of g: A complete expansion was discovered by Kozma and Schreiber [KS04], but we will not use it here.We also consider discrete derivatives of g.Without loss of generality, choose a unit vector n = α 1 x + α 2 (x + ŷ) in the "east-northeast" half-quadrant-i.e., with 1 ≥ α 1 , α 2 ≥ 0. Then define which is discrete harmonic away from {0, 1, 1 + i}.Now, extend both g and ∂ ng by linear interpolation to the grid ] is compact, we can assume without loss of generality that c holds this property for all n.Numerical calculations show that we can take c 1/5.
For an integer m ≥ 1, let B − R 0 = B mR 0 (mR 0 n) be the radius mR 0 disk tangent to the origin in the direction n.By Lemma 8(a) of [JLS12], we know that By the above discussion, this means that for any

Early points imply late points
The following sections make up the proof of Theorem 3.1, split into three parts.First, we will show that the existence of an early point at time t implies that of a similarly late point by the same time.For this, we use a harmonic function H ζ (z) that has a pole at the proposed early point, ζ ∈ 1 m Z 2 , and we define a martingale M ζ (t) (roughly) by summing the values of H is large, the martingale takes a much larger value than expected at time t; we finish up by using Lemma 3.7 to show that this is unlikely.
In the following two sections, we set up the theory necessary for this first proof.
4.1 The discrete harmonic function , and let τ > 0 be such that ζ ∈ ∂D τ .This is possible because the sets ∂D s for s > 0 form a foliation of D T \ D 0 .
Without loss of generality, suppose the outward normal vector n to ∂D τ at ζ is pointing into the "eastnortheast" half-quadrant, or equivalently that n • x ≥ n • ŷ ≥ 0. This subsumes other cases by reflecting the plane appropriately.Now, write n = α 1 + α 2 (1 + i).Because of the direction of n, both α 1 and α 2 are positive and bounded below 1. Define We can view this as a directional derivative of the potential kernel in the direction opposite n.We will extend this by linear interpolation to the grid This function is designed to be a discrete-harmonic approximation of the continuum function pictured in Figure 4, where we view n as a complex number.Now, choose R 0 = R 0 (τ ) such that the two disks B + and B − of radius R 0 tangent to ∂D τ at any point lie entirely inside and outside D τ , respectively.Note that R 0 is bounded away from zero, as [0, T ] is compact and R 0 > 0 for all time.Let R 0 = cR 0 /4C 2 , as in Equation 2, and define the following subsets of 1 m Z 2 : ζ is the discretized version of the Hele-Shaw level set D τ , and Ω 2 ζ is an approximation of the "inner" radius R 0 circle tangent to ∂D τ at ζ.We will combine these as (c) There is an absolute constant In particular, if Proof.(a) There is an absolute constant C 2 < ∞ such that Proof.
(a) As shown in [JLS12], the level sets of H ζ differ from the level curves of F ζ by at most a fixed distance , where B + is the disk of radius R 0 contained within D τ and tangent to ∂D τ at ζ.
By construction, (B + ) C 2 /m ⊂ (D τ ) C 2 /m .Thus, by adding Ω 2 ζ , we never modify points in Ω 1 ζ outside the narrow strip (D τ ) C 2 /m \ D τ .(b) The proof of this fact is the same as that of Lemma 8(b) of [JLS12], but now using the fact that sup where Now, choose R > 0 with the following properties: ζ Note that, for any r > R, this implies B r (0 Importantly, by Lemma 3.5, and similarly, By Lemma 2.3, however, and we thus find Finally, we must show that the contribution of B (if it is nonempty) to the sum is negligible.From Equation 3, we know that Similarly, there are at most 4V U C 1 /v source points between times s 0 and s, so we bound the final term as Putting these contributions together implies the lemma.

The martingale M ζ (t)
The harmonic function H ζ gives rise to a natural martingale associated to our IDLA process, using the concept of a grid Brownian motion: Definition 4.3.A grid Brownian motion starting at the point x ∈ 1 m Z 2 is a random process t → W t ∈ G m defined as follows.
Let B t be an origin-centered Brownian motion, and for each integer n ≥ 1, let τ * n > 0 be the n th time that B t visits a point in 1 m Z.For each n, choose a uniform random direction ûn ∈ {1, i}.For t ∈ [τ * n , τ * n+1 ], define In short, W t is simply the process B t , but turning in a random direction at each lattice point.
For k ∈ {0, 1, 2, ...}, let βk (s) be independent Brownian motions on the grid G m , starting at the source points z m,k .We will define a modified IDLA process Then set β k (s) = βk min s 1−s , s * k , and set We now proceed with the technique mentioned at the beginning of this section.That is, we will use the martingales M ζ to detect the presence of a late or early point at ζ; if either is the case, then M ζ will be either much larger or much smaller than its mean.In turn, Lemma 3.7 will imply that this scenario is unlikely for small times S ζ .With the following two lemmas, we will be able to show that S ζ is small on the event E a/m (t) c , allowing the above argument to go through.Lemma 4.5.Suppose D s is a smooth flow arising from an initial mass distribution.For where K is a constant depending only on the flow.
Lemma 4.6.Suppose D s is smooth, and fix a ≥ 2C 2 + 2, ≤ a, and s ∈ [0, T ].For where K is as in Lemma 4.5 and K > 0 is another absolute constant. Proof.
. By Lemma 3.5, we can take t 0 to satisfy and thus As discussed in the proof of Lemma 4.5, for m log E e τn(−1/mR 2 ,5/2)

First estimate
Choose constants Lemma 4.7.For large enough m, s ∈ [0, T ], 3a + C 2 ≥ a ≥ C 3 m 2/5 , and ≤ αa, we have Step 1.For each integer 1 ≤ t ≤ m 2 s and each lattice point z, let be the event wherein z first joins the cluster at time t and is the first a/m-early point.Now, , and let ζ = ζ(z, t) be the nearest point to z in the annulus /m , we have by Lemma 4.5 that Let M = 6m 2/5 , so that Markov's inequality gives Step 2. On the event Q z,t , we know that as no points are (m + 1)-early.However, we also know that which implies by Lemma 3.5 that In turn, Equation 5 implies that This means that A m (t) ⊂ D τ , and thus that A m (t) does not meet ∂Ω ζ by Lemma 4.2(a).This means that we can replace A m (t) by A ζ (t), which we partition as where t 0 is chosen such that D t 0 /m 2 ⊂ (D t/m 2 ) /m .By Lemma 3.5, we can satisfy d On the event where z m,i(z ) is the source point that initially generated the point z ∈ A m (t).Next, we try to estimate the equivalent sum over A 1 .By the discussion above, we know that only 4 U V v −1 m points can be outside the bounds of A 1 , meaning that {z m,i(z Adding up the contributions from A 1 and A 3 gives from the definitions of C 3 and above.Now, since z / ∈ (D t/m 2 ) a/m ⊃ (D 0 ) a/m , Lemma 3.2 tells us that for large enough m, using the facts that a ≥ C 3 m 2/5 and C 3 ≥ 3/c 0 .On the event Q z,t , the point z is a/m early but not (a + 1)/m-early, so a/m ≤ d(z, D t/m 2 ) ≤ (a + 1)/m.We know that ζ is the nearest point to z in the annulus of Equation 4, which means that (for a > 2C 2 ) we have md(z, ζ) ≤ 5V a/v.Then F ζ (z) = v/5V a + O(a −2 ), and so by Lemma 4.1, for all z ∈ B(z, a), as long as m (and hence a) is large enough.On the event {#A 2 > ba 2 }, this means and hence (from Equation 6) on event and so Step 3. Since C 3 ≥ 72V vb , we know that with M as in part 1.Using Lemma 3.7, we find Finally, we bound ) that occurs in the expression for M ζ is even negative, let alone a large negative number.The problem that occurs in the general source (i.e., non-disk) setting is that we cannot obtain a positive lower bound on H ζ (z m,i ), as the source point z m,i may be "behind" the pole ζ, as shown in Figure 7.To remedy this issue, we introduce a second harmonic function Hζ , defined to be the discrete Poisson kernel on a slightly modified domain Ωζ ≈ Ω ζ .We will see that the difference Hζ (z) − Hζ (z m,i ) is negative and bounded away from zero, so our program will go through roughly as mentioned above.

Late points imply early points
On the other hand, we will not be able to get a strong replacement for Lemma 4.1(c), which tells us that H ζ closely approximates a continuum harmonic function.This leads to an overall m 2/5 error-rather than the logarithmic errors we saw in Lemma 4.2(c)-when summing Hζ over the set D s , and it eventually creates the m −3/5 error of Theorem 3.1.

The Poisson kernel on Ω ζ
We introduce a new, positive harmonic function on the new set Namely, if W z (t) is a (grid) Brownian motion in Ωζ starting at z, and τ * is the first exit time of W ζ (t) from Ωζ , we define Hζ (z We can recognize this as the Poisson kernel associated to the set Ωζ .In particular, it satisfies the following key properties: Lemma 5.1.For any m, Hζ satisfies the following: (d) This follows from the last-exit decomposition for simple random walks [LL10, Prop.4.6.4].
Lemma 5.2.Suppose D s is smooth.Then, where G Dτ is the continuous Green's function of D τ .
, 1] depends only on ζ and J Dτ is the Poisson kernel on D τ .
(c) The following mean-value property holds: Proof.(a) For this, we use the estimate as the log m and λ terms cancel out.Fixing z, we see that E log |W z (τ * ) − z| is a discrete harmonic function of z , with boundary values log |z − z| for z ∈ ∂ Ωζ .With the possible exception of the points ζ ± im −2 , all boundary points of Ωζ also lie on the boundary of ∂D τ ; then we can compare f 0 (z ) = E log |W z (τ * )−z| with the continuous harmonic function f 1 (z ) = G Dτ (z , z) + log |z − z|.Indeed, the latter has fourth derivative bounded above by C/d(z, z ) 3 , so we know where ∆ h is the five-point stencil Laplacian.Furthermore, E log |W z (τ * ) − z| and G Dτ (z , z) + log |z − z| differ by at most O(m −2 ) on the boundary (at ζ ± im −2 ), so the maximum principle gives

This allows us to bound
with C 2 large enough, using the fact that Len(∂A α ) ≤ Len(D τ ), and using |∇G Dτ (z)| = O(|z − ζ| −2 ) to relate the initial integral to a sum.More precisely, we could integrate C 2 /d(z, ζ ) over an encompassing shape as in Figure 6 to retrieve the bound (Dτ )α 0 . This immediately gives For the remaining sets, we introduce slice coordinates (x, y) for ∂D τ near ζ, such that ζ = (0, 0).These points are bounded outside the disk B − ζ , so the probability of their associated random walks exiting Ωζ at ζ is bounded by Hζ Then we find Finally, for z ∈ D α 0 , we can bound which gives Hζ The error is dominated by the second order part: The proof finishes as does Lemma 4.2(c), but including the extra factor c ζ from Lemma 5.1(d).

Second estimate
Lemma 5.5.There is an absolute constant C 4 > 0 such that, for large enough m, if s ∈ [0, T ], ≥ C 4 m 2/5 , and a ≤ 2 /C 4 m 2/5 , then Proof.Without loss of generality, let a = 2 /C 4 m 2/5 ≥ .We can further suppose that m ≥ max(3a + C 2 , 5a/ inf ζ R 1 ).Indeed, otherwise we have a = εm for a constant ε = inf(1/4, inf R 1 /5); by Lemma 3.3, we know that we can choose m large enough that Then we know that d(ζ, ∂D T 1 ) = /m-by Lemma 3.5, this implies that On the event L[ζ], we know that any particles in A ζ (T 1 ) that hit the boundary must do so away from ζ; that is, Hζ ≡ 0 for these particles.As in [JLS12], this implies that Mζ (T 1 ) is maximized if the interior of Ωζ ∩ 1 m Z 2 is fully occupied by A ζ (T 1 ), so we can bound Mζ (T 1 ) as follows: where z m,i(z) is the source point from which the particle landing at z ∈ Ω ζ started, and weighting each term of the first sum by its number of occurrences in the multiset A ζ (T 1 ).First, we reorganize the source terms of the two sums: for a constant c > 0 depending only on the flow, using Lemmas 5.2(a,b) to deduce that inf i Hζ (z m,i ) = Θ(m −1 ).Next, notice that the two right-hand sums are the same that appear in Lemma 5.2(c), implying that Choose m large enough that e a ≥ e C 4 m 2/5 ≥ m.By Lemma 5.4, Let M = (K + K + 1)a, so Markov's inequality implies Since Mζ (m 2 T 1 ) ≤ −c on the event L[ζ], this means that for C 4 ≥ 4(K + K + 1)/c 2 , using Lemma 3.7 and the fact that Mζ (t) = Bζ ( Sζ (t)) for a centered Brownian motion Bζ .We conclude that where α, C 4 > 0 are as in Sections 4.3 and 5.2, respectively.Now, if k ≥ α −1 C 4 m 2/5 , we know that and thus that k−1 = αa k−1 ≥ α −1 C 4 m 2/5 .Thus, if k ≥ α −1 C 4 m 2/5 , we know that n ≥ C 4 m 2/5 and that a n ≥ C 3 m 2/5 for all n ≤ k, assuming without loss of generality that α −2 C 4 ≥ C 3 .We also know (from the choice of ε) that a 0 ≤ m/4; in general, if a k ≤ m/4, then for large enough m.Then the pair ( n , a n ) satisfies the hypothesis of Lemma 4.7, and similarly for ( n , a n−1 ) and Lemma 5.

Concluding Remarks
There are a number of possible improvements to the results proven here.Most importantly, it would be interesting to improve the m −3/5 bounds on the fluctuations; we expect that fluctuations are truly of order m −1 log m, as in the point-source case.Hypothetically, this result could be proven using our techniquethe primary obstacle is that we need a stronger version of Lemma 4.2(c), which quantifies how closely Hζ approximates a continuum harmonic function.In general, if we can replace the m 2/5 in 4.2(c) with m ε [resp., log m], we could derive bounds on the fluctuations of order m −1+ε [resp., m −1 log m].On the flip side, this also means that we could significantly weaken both 4.1(c) and 4.2(c) and still prove a non-trivial convergence rate of IDLA.Furthermore, it would be interesting to lift some of the hypotheses we set on the flow.However, we imagine that it is less likely our technique would apply without the requirements of a concentrated mass distribution or a smooth flow.Indeed, both hypotheses are necessary to guarantee that R 0 is bounded away from 0, and thus that H ζ is small enough on the boundary.However, if an independent bound on Hζ could be obtained, showing that it satisfies Hζ (z) ≤ 1 md(z,ζ)−C 2 without comparing it to H ζ , it could be used in place of H ζ for both parts of the proof.
There are also closely related settings that have not been studied extensively.An interesting example would be to replace the "solid" initial sets Q s i with submanifolds of D 0 .Since these would be zero volume, they could eject particles evenly from all points rather than having a moving source i ∂Q s i i .Another example would be a collection of point sources; in fact, the theorem corresponding to Lemma 3.3 in this setting has already been proved by Levine and Peres [LP10, Theorem 1.4], so it would likely not be too difficult to adapt our argument to this case.
Finally, a question we will investigate in the sequel is that of the scaling limits of the fluctuations themselves.Jerison, Sheffield, and Levine [JLS14] studied this question for same-time fluctuations in the pointsource case, and they found that, when the fluctuations are scaled up by a factor of m d/2 (in dimension d), they have a weak limit in law of a certain Gaussian random distribution.They found a similar result in the case of a discrete cylinder Z × Z/mZ with source points along a fixed-height circumference [JLS13a]; here, they further studied the correlations between fluctuations at different times in the flow.The same question has been studied by Eli Sadovnik [Sad16] in the extended-source case, focusing on same-time fluctuations and using harmonic polynomials as test functions; we are interested in strengthening his result to allow smooth test functions and to investigate correlations between fluctuations at different times.

Figure 2 :
Figure 2: An illustration of the occupied set D s (dark blue) and the remaining source pointsQ T i i \ Q s i i (multicolored) inExample 2, at two different times.Here, our starting set D 0 = B 1 is the unit disk, and our source points Q s i = B √ s/π are identical radially-expanding disks within it (shown in 3D for visual clarity).From symmetry, we see that the occupied sets D s are also growing disks.

Figure 4 :
Figure 4: A plot of F ζ , with an example domain D τ marked out by dark blue curves.Importantly, F ζ has a large positive pole within D τ at ζ, but its negative pole lies entirely outside D τ .The discrete harmonic function H ζ closely approximates this function away from the pole ζ.

Figure 5 :
Figure 5: We form our domain Ω ζ by combining the subsets Ω 1 ζ and Ω 2 ζ ; the latter guarantees that H ζ is not too large on the boundary, but it may also affect the regularity of the boundary.

( a )
By definition, H ζ is grid-harmonic everywhere except for ζ, ζ + 1/m, and ζ + (1 + i)/m.Firstly, ζ itself lies on the boundary of Ω ζ by definition.As the normal vector n to ∂D τ at ζ points into the eastnortheast half-quadrant, for large enough m, neither of the remaining points can lie in D τ (and thus in Ω 1 ζ ).Furthermore, H ζ is negative at both points, as in [JLS12], so they cannot lie in Ω 2 ζ .Thus, they cannot lie in Ω ζ , so H ζ is grid-harmonic in that set.The lower bound follows from Equation 2. (b) As in part (a), the lower bound H ζ (z) ≥ −1/2mR 0 is clear from Equation 2. The upper bound follows from the inclusion of Ω 2 ζ , as the boundary of Ω ζ must lie at or outside the boundary of Ω 2 ζ .(c, d) The last points are exactly Lemma 7(c, d) in [JLS12], as our notions of H ζ and F ζ are simply rotations of theirs.Lemma 4.2.

Figure 6 :
Figure 6: An illustration of U R (yellow).Note that the disks of radius R do not get "too close" to ζ, so the sum z∈U R 1 m 2 |z−ζ| 2 is of order log m.
By the Dubins-Schwarz theorem [RY91, Theorem V.1.6],we can write M ζ (t) = B ζ (S ζ (t)), where S ζ (t) = M ζ , M ζ t is the quadratic variation of M ζ and B ζ is a standard Brownian motion.For each k, S ζ (k) is a stopping time w.r.t. the filtration {F T ζ (s) } s≥0 , where T ζ (s) = inf{t | S ζ (t) > s}.Further, B ζ (s) is adapted to this filtration.By the strong Markov property, the processes Bk ζ (u) := B ζ (S ζ (k) + u) − B ζ (S ζ (k)) are independent Brownian motions started at zero.Finally, for −a < 0 < b, write τ k (−a, b) = inf{u > 0 | Bk ζ (u) / ∈ [−a, b]}.We will use these exit times in accordance with the following lemma, which is just a restatement of Lemma 9 of [JLS12] in our setting: Lemma 4.4.Fix ζ ∈ 1 m Z 2 \ D 0 , and let Very roughly, we would like the proof of the second part of Theorem 3.1 to go as follows.If ζ is the first ( /m)-late point in A ζ (t), then at the time T ∼ m 2 τ + m , the set A ζ (t) has several particles at every boundary point z = ζ in ∂Ω ζ .Since H ζ (ζ) is much larger than H ζ (z = ζ), this would tell us in turn that M ζ would have a much lower value than expected.Combined with Lemmas 4.5 and 4.6, we would be able to recover a strong upper bound of the probability of L /m [T ] ∩ E a/m [T ] c .Unfortunately, we are unable to say that the difference H ζ (z) − H ζ (z m,i

ζ H ζ < 0 H ζ > 0 Figure 7 :
Figure 7: The original harmonic H ζ is negative on a half-plane cut out by the pole ζ.While this does not come into play in the case of a point-source, it is critical in extended-source IDLA, forcing us to define a new harmonic function Hζ to continue with the proof.
5, from Lemma 5.1(d) and part (b) above.Next, we control the sum over B α 0 .Since ∂D τ is smooth, the probability of a point z near the boundary to exit Ωζ at ζ is bounded by C