Internal DLA in Higher Dimensions David Jerison Lionel Levine∗ Scott Sheffield† 1 1 0 January 31, 2011 2 n a J 1 Abstract 3 Let A(t) denote the cluster produced by internal diffusion limited ] aggregation (internal DLA) with t particles in dimensi√on d ≥ 3. We R show that A(t) is approximately spherical, up to an O( logt) error. P . In the process known as internal diffusion limited aggregation (internal h t DLA)oneconstructsforeachintegertimet ≥ 0anoccupied setA(t) ⊂ Zd a m as follows: begin with A(0) = ∅ and A(1) = {0}. Then, for each integer t > 1, form A(t + 1) by adding to A(t) the first point at which a simple [ random walk from the origin hits Zd\A(t). Let Br ⊂ Rd denote the ball of 2 radius r centered at 0, and write B := B ∩Zd. Let ω be the volume of v r r d 3 the unit ball in Rd. Our main result is the following. 5 4 Theorem 1. Fix an integer d ≥ 3. For each γ there exists an a = a(γ,d) < 3 ∞ such that for all sufficiently large r, . 2 1 (cid:110) (cid:111)c 0 P Br−a√logr ⊂ A(ωdrd) ⊂ Br+a√logr ≤ r−γ. 1 : v We treated the case d = 2 in [JLS10] (see also the overview in [JLS09]), √ i X where we obtained a similar statement with logr in place of logr. To- r gether with a Borel-Cantelli argument, this in particular implies the follow- a ing [JLS10]: Corollary 2. The maximal distance from ∂B to a point in one (but not r √ both) of B and A(ω rd) is a.s. O(logr) when d = 2 and O( logr) when r d d > 2. ∗Supported by an NSF Postdoctoral Research Fellowship. †Partially supported by NSF grant DMS-0645585. 1 These results show that internal DLA in dimensions d ≥ 3 is extremely close to a perfect sphere: when the cluster A(t) has the same size as a ball √ of radius r, its fluctuations around that ball are confined to the logr scale (versus logr in dimension 2). In [JLS10] we explained that our method for d = 2 would also apply in √ dimensionsd ≥ 3withthelogr replacedby logr. Weoutlinedthechanges needed in higher dimensions (stating that the full proof would follow in this paper) and included a key step: Lemma A, which bounds the probability of “thin tentacles” in the internal DLA cluster in all dimensions. The purpose of this note is to carry out the adaptation of the d = 2 argument of [JLS10] to higher dimensions. We remark that in [JLS10] we used an estimate from [LBG92] to start this iteration, while here we have modified the argument slightly so that this a priori estimate is no longer required. One way for A(ω rd) to deviate from the radius r sphere is for it to d have a single “tentacle” extending beyond the sphere. The thin tentacle estimate [JLS10, Lemma A] essentially says that in dimensions d ≥ 3, the probability that there is a tentacle of length m and volume less than a small constant times md (near a given location) is at most e−cm2. By summing over all locations, one may use this to show that the length of the longest √ “thin tentacle” produced before time t is O( logt). To complete the proof of Theorem 1, we will have to show that other types of deviations from the radius r sphere are also unlikely. LemmaAof[JLS10]wasalsoprovedford = 2,albeitwithe−cm2 replaced bye−cm2/logm. However,whend = 2thereappeartobeothermore“global” fluctuations that swamp those produced by individual tentacles. (Indeed, we expect, but did not prove, that the logr fluctuation bound is tight when d = 2.) We bound these other fluctuations in higher dimensions via the same scheme introduced in [JLS09, JLS10], which involves constructing and estimating certain martingales related to the growth of A(t). It turns out the quadratic variations of these martingales are, with high probability, of order logt when d = 2 and of constant order when d ≥ 3, closely paralleling what one obtains for the discrete Gaussian free field (as outlined in more detail in [JLS10]). The connection to the Gaussian free field is made more explicit in [JLS11]. Section 1 proves Theorem 1 by iteratively applying higher dimensional analogues of the two main lemmas of [JLS10]. The lemmas themselves are proved in Section 3, which is the heart of the argument. Section 2 contains preliminary estimates about random walks that are used in Section 3. 2 A brief history of internal DLA fluctuation bounds ThehistoryoffluctuationboundssuchastheoneinCorollary2isasfollows. In1991,Lawler,Bramson,andGriffeathprovedthatthelimitshapeofinter- nal DLA from a point is the ball in all dimensions [LBG92]. In 1995 Lawler gave a more quantitative proof, showing that the fluctuations of A(ω rd) d from the ball of radius r are at most of order O(r1/3log4r) [Law95]. In December 2009, the present authors announced the bound O(logr) on fluctuations in dimension d = 2 [JLS09] and gave an overview of the argument, making clear that the details remained to be written. In April 2010, As- selah and Gaudillière [AG10a] gave a proof, using different methods from [JLS09], of the bound O(r1/(d+1)) in all dimensions, improving the Lawler bound for all d ≥ 3. In September 2010, Asselah and Gaudillière improved thistoO((logr)2)inalldimensionsd ≥ 2withanO(logr)boundon“inner” errors [AG10b]. In October 2010 the present authors proved the O(logr) bounds(announcedinDecember2009)fordimensiond = 2andoutlinedthe √ proof of the O( logr) bound for dimensions d ≥ 3 [JLS10]. In November √ 2010, Asselah and Gaudillière gave a second proof of the O( logr) bound [AG10c]. Their proof uses methods from [AG10b] along with Lemma A of [JLS10] to bound “outer” errors and a new large deviation bound (in some sense symmetric to Lemma A) to bound “inner” errors. More references and a more general discussion of internal DLA history appear in [JLS10]. 1 Proof of Theorem 1 Let m and (cid:96) be positive real numbers. We say that x ∈ Zd is m-early if x ∈ A(ω (|x|−m)d), d where ω is the volume of the unit ball in Rd. Likewise, we say that x is d (cid:96)-late if x ∈/ A(ω (|x|+(cid:96))d). d Let E [T] be the event that some point of A(T) is m-early. Let L [T] be m (cid:96) the event that some point of B is (cid:96)-late. These events correspond (T/ω )1/d−(cid:96) d to “outer” and “inner” deviations of A(T) from circularity. Lemma 3. (Early points imply late points) Fix a dimension d ≥ 3. For each γ ≥ 1, there is a constant C = C (γ,d), such that for all sufficiently √ 0 0 large T, if m ≥ C logT and (cid:96) ≤ m/C , then 0 0 P(E [T]∩L [T]c) < T−10γ. m (cid:96) 3 Lemma 4. (Late points imply early points) Fix a dimension d ≥ 3. For each γ ≥ 1, there is a constant C = C (γ,d) such that for all sufficiently √ 1 1 large T, if m ≥ (cid:96) ≥ C logT and (cid:96) ≥ C ((logT)m)1/3, then 1 1 P(E [T]c∩L [T]) ≤ T−10γ. m (cid:96) T =m 0 m 1 m m m 2 ‘ ‘ ‘ ‘ =C(TlogT)1/3 1 0 Figure 1: Let mT be the smallest m(cid:48) for which A(T) contains an m(cid:48) early point. LetlT bethelargest(cid:96)(cid:48) forwhichsomepointofB is(cid:96)(cid:48)-late. (T/ω )1/d−(cid:96)(cid:48) d By Lemma 3, ((cid:96)T,mT) is unlikely to belong to the semi-infinite rectangle in the left figure if (cid:96) < m/C . By Lemma 4, ((cid:96)T,mT) is unlikely to belong 0 to the semi-infinite rectangle in the second figure if (cid:96) ≥ C ((logT)m)1/3. 1 Theorem 1 will follow because mT > m = T is impossible and the other 0 rectangles on the right are all (by Lemmas 3 and 4) unlikely. WenowproceedtoderiveTheorem1fromLemmas3and4. Thelemmas themselveswillbeprovedinSection3. LetC = max(C ,C ). Westartwith 0 1 m = T. 0 Note that A(T) ⊂ B , so P(E [T]) = 0. Next, for j ≥ 0 we let T T (cid:112) (cid:96) = max(C((logT)m )1/3,C logT) j j and m = C(cid:96) . j+1 j By induction on j, we find P(E [T]) < 2jT−10γ mj P(L [T]) < (2j +1)T−10γ. (cid:96)j 4 To estimate the size of (cid:96) , let K = C4logT and note that (cid:96) ≤ (cid:96)(cid:48), where j j j (cid:96)(cid:48) = (KT)1/3; (cid:96)(cid:48) = max((K(cid:96)(cid:48))1/3,K1/2). 0 j+1 j Then (cid:96)(cid:48) ≤ max(K1/3+1/9+···+1/3jT1/3j,K1/2) j so choosing J = logT we have T1/3J < 2 and (cid:112) (cid:96) ≤ 2K1/2 ≤ C logT. J The probability that A(T) has (cid:96) -late points or m -early points is at J J most (4J +1)T−10γ < T−9γ < r−γ. Setting T = ω rd, (cid:96) = (cid:96) and m = m , we conclude that if a is suffi- d J J ciently large, then (cid:110) (cid:111) P B √ ⊂ A(ω rd) ⊂ B √ ≤ P(E [T]∪L [T]) < r−γ r−a logr d r+a logr m (cid:96) which completes the proof of Theorem 1. 2 Green function estimates on the grid This section assembles several Green function estimates that we need to prove Lemmas 3 and 4. The reader who prefers to proceed to the heart of the argument may skip this section on a first read and refer to the lemma statements as necessary. Fix d ≥ 3 and consider the d-dimensional grid G = {(x ,...,x ) ∈ Rd : at most one x ∈/ Z}. 1 d i In many of the estimates below, we will assume that a positive integer k and a y ∈ Zd have been fixed. We write s = |y| and Ω = Ω(y,k) := G ∩B \{y}. s+k For x ∈ ∂Ω, let P(x) = P (x) y,k be the probability that a Brownian motion on the grid G (defined in the obvious way; see [JLS10]) starting at x reaches y before exiting B . Note s+k 5 that P is grid harmonic in Ω (i.e., P is linear on each segment of Ω\Zd, and for each x ∈ Ω∩Zd, the sum of the slopes of P on the 2d directed edge segments starting at x is zero). Boundary conditions are given by P(y) = 1 and P(x) = 0 for x ∈ (∂Ω)\{y}. The point y plays the role that ζ played in [JLS10], and P plays the role of the discrete harmonic function H . One ζ difference from [JLS10] is that we will take y inside the ball (i.e., k ≥ 1) instead of on the boundary. To estimate P we use the discrete Green function g(x), defined as the expectednumberofvisitstoxbyasimplerandomwalkstartedattheorigin in Zd. The well-known asymptotic estimate for g is [Uch98] (cid:12) (cid:12) (cid:12)g(x)−a |x|2−d(cid:12) ≤ C|x|−d (1) (cid:12) d (cid:12) for dimensional constants a and C (i.e., constants depending only on the d dimension d). We extend g to a function, also denoted g, defined on the grid G by making g linear on each segment between lattice points. Note that g is grid harmonic on G \{0}. Throughout we use C to denote a large positive dimensional constant, and c to denote a small positive dimensional constant, whose values may change from line to line. Lemma 5. There is a dimensional constant C such that (a) P(x) ≤ C/(1+|x−y|d−2). (b) P(x) ≤ Ck(s+k+1−|x|)/|x−y|d, for |x−y| ≥ k/2. (c) maxP(x) ≤ Ck/(s−r−k)d−1 for r < s−2k. x∈Br Proof. Themaximumprinciple(forgridharmonicfunctions)impliesCg(x− y) ≥ P(x) on Ω, which gives part (a). The maximum principle also implies that for x ∈ Ω, P(x) ≤ C(g(x−y)−g(x−y∗)) (2) where y∗ is the one of the lattice points nearest to (s+2k+C )y/s. Indeed, 1 both sides are grid harmonic on Ω, and the right side is positive on ∂B s+k by (1), so it suffices to take C = (g(0)−g(y−y∗))−1. Combining (1) and (2) yields the bound Ck P(x) ≤ , for |x−y| ≥ 2k. |x−y|d−1 6 Next, let z ∈ ∂B be such that |z −y| = 2L, with L ≥ 2k. The bound s+k above implies Ck P(x) ≤ , for x ∈ B (z) Ld−1 L Let z∗ be one of the lattice points nearest to (s+k+L+C )z/|z|. Then 1 F(x) = a L2−d−g(x−z∗) d is comparable to L2−d on ∂B (z∗) and positive outside the ball B (z∗) (for 2L L a large enough dimensional constant C — in fact, we can also do this with 1 C = 1 with L large enough). It follows that 1 P(x) ≤ C(k/Ld−1)(Ld−2)F(x) on ∂(B (z∗) ∩ Ω) and hence by the maximum principle on B (z∗) ∩ Ω. 2L 2L Moreover, F(x) ≤ C(s+k+1−|x|)/Ld−1 for x a multiple of z and s+k−L ≤ |x| ≤ s+k. Thus for these values of x, P(x) ≤ C(k/L)F(x) ≤ Ck(s+k+1−|x|)/Ld We have just confirmed the bound of part (b) for points x collinear with 0 and z, but z was essentially arbitrary. To cover the cases |x−y| ≤ 2k one has to use exterior tangent balls of radius, say k/2, but actually the upper bound in part (a) will suffice for us in the range |x−y| ≤ Ck. Part (c) of the lemma follows from part (b). The mean value property (as typically stated for continuum harmonic functions) holds only approximately for discrete harmonic functions. There are two choices for where to put the approximation: one can show that the average of a discrete harmonic function u over the discrete ball B is r approximately u(0), or one can find an approximation w to the discrete r ball B such that averaging u with respect to w yields exactly u(0). The r r divisiblesandpilemodelof[LP09]accomplishesthelatter. Inparticular, the following discrete mean value property follows from Theorem 1.3 of [LP09]. Lemma 6. (Exact mean value property on an approximate ball) For each real number r > 0, there is a function w : Zd → [0,1] such that r • w (x) = 1 for all x ∈ B , for a constant c depending only on d. r r−c • w (x) = 0 for all x ∈/ B . r r 7 • For any function u that is discrete harmonic on B , r (cid:88) w (x)(u(x)−u(0)) = 0. r x∈Zd The next lemma bounds sums of P over discrete spherical shells and discrete balls. Recall that s = |y|. Lemma 7. There is a dimensional constant C such that (cid:88) (a) P(x) ≤ Ck for all r ≤ s+k. x∈Br+1\Br (cid:12) (cid:12) (cid:12)(cid:88) (cid:12) (b) (cid:12) (P(x)−P(0))(cid:12) ≤ Ck for all r ≤ s. (cid:12) (cid:12) (cid:12) (cid:12) x∈Br (cid:12) (cid:12) (cid:12) (cid:12) (c) (cid:12)(cid:12) (cid:88) (P(x)−P(0))(cid:12)(cid:12) ≤ Ck2. (cid:12) (cid:12) (cid:12)x∈B (cid:12) s+k Proof. Part (a) follows from Lemma 5: Take the worst shell, when r = s. Then the lattice points with |x−y| ≤ k, s ≤ |x| ≤ s+1 are bounded by Lemma 5(a) (cid:90) k s2−dsd−2ds = k 0 (volumeelementondiskwiththickness1andradiuskinZd−1issd−2ds.) For theremainingportionoftheshell, Lemma5(b)hasnumeratork(s+k−s) = k2, so that (cid:90) ∞ k2s−dsd−2ds = k k Next, for part (b), let w be as in Lemma 6. Since P is discrete harmonic r in B , we have for r ≤ s s (cid:88) w (x)(P(x)−P(0)) = 0. r x∈Zd Since w equals the indicator 1 except on the annulus B \ B , and r Br r r−c 8 |w | ≤ 1, we obtain r (cid:12) (cid:12) (cid:12)(cid:88) (cid:12) (cid:88) (cid:12) (P(x)−P(0))(cid:12) ≤ |w (x)||P(x)−P(0)| (cid:12) (cid:12) r (cid:12) (cid:12) x∈Br x∈Br\Br−c (cid:88) ≤ (P(x)+P(0)) x∈Br\Br−c ≤ Ck. In the last step we have used part (a) to bound the first term; the second term is bounded by Lemma 5(b), which says that P(0) ≤ Ck/sd−1. Part(c)followsbysplittingthesumoverB intok sumsoverspherical s+k shells B \B for j = 1,...,k, each bounded by part (a), plus a sum s+j s+j−1 over the ball B , bounded by part (b). s Fix α > 0, and consider the level set U = {x ∈ G | g(x) > α}. For x ∈ ∂U, let p(x) be the probability that a Brownian motion started at the origin in G first exits U at x. Lemma 8. Choose α so that ∂U does not intersect Zd. For each x ∈ ∂U, the quantity p(x) equals the directional derivative of g/2d along the directed edge in U starting at x. Proof. We use a discrete form of the divergence theorem (cid:90) (cid:88) divV = ν ·V. (3) U U ∂U where V is a vector-valued function on the grid, and the integral on the left isaone-dimensionalintegraloverthegrid. Thedotproductν ·V isdefined U as e ·V(x−0e ), where e is the unit vector pointing toward x along the j j j unique incident edge in U. To define the divergence, for z = x+te , where j 0 ≤ t < 1 and x ∈ Zd, let d ∂ (cid:88) divV(z) := e ·V(z)+δ (z) (e ·V(x+0e )−e ·V(x−0e )). j x j j j j ∂x j j=1 If f is a continuous function on U that is C1 on each connected compo- nent of U −Zd, then the gradient of f is the vector-valued function V = ∇f = (∂f/∂x ,∂f/∂x ,...,∂f/∂x ) 1 2 d 9 withtheconventionthattheentry∂f/∂x is0ifthesegmentisnotpointing j in the direction x . Note that ∇f may be discontinuous at points of Zd. j Let G = −g/2d, so that div∇G = δ . If u is grid harmonic on U, then 0 div∇u = 0 and div(u∇G−G∇u) = u(0)δ . 0 Indeed, on each segment this is the same as (uG(cid:48) −u(cid:48)G)(cid:48) = u(cid:48)G(cid:48) −u(cid:48)G(cid:48) + uG(cid:48)(cid:48)−u(cid:48)(cid:48)G = 0 because u and G are linear on segments. At lattice points u andGarecontinuous,sothedivergenceoperationcommuteswiththefactors u and G and gives exactly one nonzero delta term, the one indicated. Let u(y) be the probability that Brownian motion on U started at y first exits U at x. Since u is grid-harmonic on U, we have div∇u = 0 on U, hence by the divergence theorem (cid:90) (cid:88) u(0) = div(u∇G−G∇u) = uν ·∇G. U U ∂U Next we establish some lower bounds for P. Lemma 9. There is a dimensional constant c > 0 such that (a) P(0) ≥ ck/sd−1. (b) Let k = 1, and z = (1− 2m)y. Then s min P(x) ≥ c/md−1. x∈B(z,m) Proof. By the maximum principle, there is a dimensional constant c > 0 such that P(x) ≥ c(g(x−y)−a (k/2)2−d) d for x ∈ B (y). In particular, k/2 P(x) ≥ ck2−d for all |x−y| ≤ k/4 Now consider the region U = {x ∈ G : g(x) > a (s(cid:48))2−d} d where s(cid:48) is chosen so that |s(cid:48) −(s−k/8)| < 1/2 and all of the boundary points of U are non-lattice points. (A generic value of s(cid:48) in the given range will suffice.) 10

Internal DLA in Higher Dimensions

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