AA SHAPE THEOREM FOR EXPLODING SANDPILES
AHMED BOU-RABEE
Abstract.
We study scaling limits of exploding Abelian sandpiles using ideas from per-colation and front propagation in random media. We establish sufficient conditions underwhich a limit shape exists and show via a family of counterexamples that convergence maynot occur in general. A corollary of our proof is a simple criteria for determining if a sandpileis explosive; this strengthens a result of Fey, Levine, and Peres (2010). Introduction
Overview.
We consider the Abelian sandpile growth model on the integer lattice. Startwith a background of indistinguishable chips, η : Z d → Z , add n chips at the origin, andattempt to stabilize via parallel toppling :(1) v t +1 = v t + 1 { s t ≥ d } s t +1 = s t + ∆( v t +1 − v t ) , where ∆ v ( x ) = (cid:80) y ∼ x ( v ( y ) − v ( x )) is the Laplacian on Z d , v = 0 is the initial odometer ,and s = η + n · δ is the starting sandpile. We say s is stabilizable if there is T < ∞ sothat v t = v T for all t ≥ T . A background is robust if η + n · δ is stabilizable for all n ≥ explosive . When s = η + n · δ is not stabilizable, it is explosive and theinfinite sequence { s t } t ≥ is an exploding sandpile. See Figures 1 and 2.Fey, Levine, and Peres coined these notions in [FLP10] (see also [FR05]) and providedsufficient conditions for determining if a background is explosive or robust: backgrounds η ≤ (2 d −
2) are always robust, but otherwise can be robust or explosive, depending on thearrangement of sites with (2 d −
1) chips. In fact, they showed that if η ≤ (2 d − n chips are added to the origin of such η , the diameter ofthe set of sites which topple grows like n /d .Pegden and Smart used this bound together with the theory of viscosity solutions to showthat the terminal odometer for n · δ , after a rescaling, converges to the solution of a fully Figure 1. s t · { v t > } for η ∼ Bernoulli(2 , , /
2) and t = 50 , , , , , , , ( ≥ a r X i v : . [ m a t h . P R ] F e b igure 2. Snapshot of the support of an exploding sandpile 1 { v t > } for η ∼ Bernoulli(2 d − , d − , p ) the first time it exits a box of side length 500.The top row is d = 2 and the bottom d = 3. From left to right, p =1 / , / , / d −
2) [BR19].These results explain the phenomena of scale-invariance in sandpiles which have a compact , n /d , growth rate - large, compact-growth sandpiles look like high-resolution versions ofsmaller sandpiles. Simple models of growth are of interest to the mathematics and physicscommunities - see, for example, [DS13, DF91, GG98, PW85] and the references therein. TheAbelian sandpile in particular has a rich history, [LKG90, Dha99, LBR02, Ost03, Red05,FdBR08, HLM +
08, FMR +
09, LP09, LP10, Pao13, LP17, J´ar18, Kli18, HJL19, AS19, LS19,HS19, CKF20, Mel20, AM20].In this paper, we study limit shapes of sandpiles in the explosive regime. The techniquesused differ fundamentally from the existing compact-growth theory. Indeed, as we willdemonstrate, some explosive sandpiles (both random and deterministic) do not converge.On the other hand, compact-growth sandpiles essentially always have limits - the argumentthere is ‘soft’ and applies in wide generality. Our proof below is quantitative and involvesestablishing specific, finite-scale estimates. We identify sufficient conditions under whichexploding sandpiles converge to the level set of an asymmetric norm - much like in first-passage percolation [CD81] and threshold growth [Wil78, GG93].1.2.
Main results.
For expositional clarity, we consider one family of random, explosivebackgrounds with limit shapes. The reader interested in generalizations may consult Section5. Suppose η : Z d → Z is drawn from a product measure P with(2) (cid:40) P ( η (0) = (2 d − − p P ( η (0) = (2 d − p. ey, Levine, and Peres showed the following. Theorem 1.1 (Proposition 1.4 in [FLP10]) . In all dimensions d ≥ , if p > , η is explosivewith probability 1. Fix p > explosion threshold by(3) M η := min { n ≥ η + n · δ is not stabilizable } . We prove that the support of the infinite sequence of parallel toppling odometers, { v t } t ≥ ,for the explosive sandpile s = η + M η · δ converges in the Hausdorff topology. Theorem 1.2.
There exists a convex domain B ( p ) ⊂ R d so that on an event of probability1, |{ ¯ v t > }(cid:52)B ( p ) | → , where ¯ v t ( x ) := v t ([ tx ]) . In fact, we prove something stronger. Not only does the support of the explosion converge,but the rate at which the explosion spreads also converges.
Theorem 1.3.
Let T η ( x ) := min { t ≥ v t ( x ) > } . On an event of probability 1, therescaled arrival times ¯ T η ( x ) := n − T η ([ nx ]) converge locally uniformly to N p , a continuous,convex, one-homogeneous function on R d . During our proof of Theorem 1.3, we introduce a quantitative criteria for determining if asandpile is explosive. The criteria asserts, roughly, that if a sandpile explodes quickly on afinite box, then it must do so on the entire lattice. We use this and a coupling with bootstrappercolation to extend the aforementioned Theorem 1.1.
Theorem 1.4.
Suppose β : Z d → Z is drawn from a product measure P . If β ≥ d and P ( β (0) = (2 d − > , then β is explosive with probability 1. Proof outline.
An exploding sandpile may be thought of as a heterogeneous, discretereaction-diffusion equation. This perspective leads us to the literature for stochastic homog-enization of reaction-diffusion equations [ZZ20, Fel19, LZ19, AC18, CM14, GG06, IPS99,BS98, GG96, GG93, Wil78]. These works suggest two methods of proof. The first, whichwe do not pursue, is half-space propagation - a limit shape can be completely describedby those starting with a half-space initial condition - an early example of this techniqueappears in [Wil78]. Another method is to identify a subadditive quantity similar to the first-passage time, [CD81], which (directly or indirectly) describes the limit shape, then applythe subadditive ergodic theorem.Our proof of Theorem 1.3 follows the second outline, however, there are several hurdles toovercome. A fundamental one is the nonlinear diffusion of the sandpile. This nonlinearity cancause explosions to propagate irregularly. In fact, an arbitrary exploding sandpile may spreadquickly in certain cells, but slowly in others, causing convergence to fail. We demonstrate anexplicit family of counterexamples to that effect in Section 6. A major part of our argumentis showing that this irregularity cannot happen if η ≥ (2 d −
2) and there are ‘enough’ siteswith (2 d −
1) chips.From now until Section 5, take η as in the statement of Theorem 1.3 and fix p >
0. Webegin in Section 2 by showing that explosions on η spread quickly. This is done by estab-lishing a high-probability bound on the ‘crossing-speed’ of η in a finite, but large cube and assing to a coarsened lattice. On the coarsened lattice, there is an infinite cluster of ‘goodcubes’ upon which explosions are guaranteed to spread quickly. We use this together withlarge-deviations for the chemical distance of supercritical Bernoulli percolation to get uni-form, linear bounds on the arrival times. (This portion of the proof shares some similaritieswith Schonmann’s argument for bootstrap percolation [Sch92].)At this stage, if the arrival times, T η , were subadditive, we could apply the subadditiveergodic theorem and conclude. However, T η is not, in general, subadditive: roughly, whenan explosion started at the origin reaches some site x for the first time, it mixes up thebackground and so cannot be compared directly to the explosion originating from x .We overcome this lack of subadditivity by shifting our focus to a related, but simplerprocess, the last-wave - an exploding sandpile where the origin is constrained to topple afixed number of times. In Section 3, we use the established regularity of explosions to showthat the last-wave can be approximated by a quantity which is exactly subadditive, andhence converges. This can also be viewed as a shape theorem for a bootstrap percolationtype process.The proof of Theorem 1.3 is completed in Section 4 where we show that the last-waveis an approximation to the expanding front of an exploding sandpile. The argument forthis is a deterministic comparison which requires η ≥ (2 d − Acknowledgments.
Thank you to Charles K. Smart for motivating, helpful discussionsduring this project. Thank you to Lionel Levine for several inspiring conversations and forsuggesting this question. Thank you to Dylan Airey for asking if exploding sandpiles have alimit shape.
Code.
Julia code which can compute the figures in this article is included in the arXivupload and may be freely used and modified.
Notation and conventions. • Functions on Z d are extended via nearest-neighbor interpolation to R d and vice-versa. • d will always refer to the dimension of the underlying space. • e , . . . , e d are the d unit directions in Z d . • x i = x · e i is the i th coordinate of vector x and x d − i = ( x i +1 , . . . , x d ). • | x | refers to the Manhattan norm and | x | ∞ the (cid:96) ∞ norm. • y ∼ x if | y − x | = 1. • Scalar operations on vectors/functions/sets are interpreted pointwise. • | · | is either the counting measure or Lebesgue measure depending on the input. • For A ⊂ Z d , A c := { x ∈ Z d : x (cid:54)∈ A } , ∂A := { x ∈ A c : ∃ y ∼ x ∈ A } , and ¯ A := A ∪ ∂A . • C, c , are positive constants which may change from line to line. Dependence is indi-cated by, for example, C d . • The act of firing or toppling a site, x , removes 2 d chips from x and adds one chip toeach y ∼ x . η ∼ Bernoulli( a, b, p ) is shorthand for: η : Z d → Z is drawn from a product measure P with P ( η (0) = a ) = p and P ( η (0) = b ) = (1 − p ).2. Regularity of explosions
We use η ≥ (2 d −
2) together with the i.i.d. assumption to establish almost sure regularityof explosions. The main result of this section is a quantification of [FLP10]’s Theorem 1.1recalled above. The method is a static renormalization (see Chapter 7 in [Gri13]) inspiredby Schonmann’s proof for boostrap percolation [Sch92].2.1.
Parallel toppling preliminaries.
Before proceeding, we mention some basic prop-erties of parallel toppling which we use below. Recall that { v t } t ≥ and { s t } t ≥ are theinfinite sequence of parallel toppling odometers and sandpiles for initial conditions v = 0and s = η + M η · δ . An induction argument ((4.4) in [BG07]) shows(4) v t +1 ( x ) = min {(cid:98) s ( x ) + (cid:80) y ∼ x v t ( y )2 d (cid:99) , v t ( x ) + 1 } s t +1 ( x ) = s ( x ) + ∆ v t +1 ( x ) . Another induction shows that when s ≤ · (2 d ) −
1, the minimum in (4) is unnecessary.In the sequel we also consider a version of parallel toppling where the odometer on someset, S , (the complement of a cube, the origin) is ‘frozen’ at the initial value w but the initialsandpile s (cid:48) = s is the same,(5) w t +1 ( x ) = (cid:40) w t ( x ) + 1 { s (cid:48) t ( x ) ≥ d } if x (cid:54)∈ S w ( x ) if x ∈ S s (cid:48) t +1 = s (cid:48) t + ∆( w t +1 − w t ) . We call this S -frozen parallel toppling. (Recently, and after this paper was written, [GMP21]was posted - therein so-called ‘freezing sandpiles’ are studied in the context of computationalcomplexity.) If s + ∆ w ≤ · (2 d ) − S c , then as above:(6) w t +1 ( x ) = (cid:98) s ( x ) + (cid:80) y ∼ x w t ( y )2 d (cid:99) for x (cid:54)∈ S s (cid:48) t +1 ( x ) = s ( x ) + ∆ w t +1 ( x ) . Also, if s ≤ (2 d − w t ≤ max x ∈S w ( x ). The definitions allow us to compare the twoversions of parallel toppling,(7) v t + t ≥ w t where t := min { t ≥ v t ≥ w } w t ≥ v t if w ≥ v and w ( S ) ≥ sup t v t ( S ) . Crossing speeds.
To provide a global upper bound on the arrival times, T η ( x ), weshow a local upper bound. In particular we study the following ‘cell problem’, a term fromhomogenization denoting a simple problem which describes the local behavior of a morecomplicated one. e consider sandpile dynamics on a box of side length k , Q k := { x ∈ Z d : 1 ≤ x ≤ k } .For a point z ∈ ¯ Q k and direction 1 ≤ i ≤ d , denote the line passing from one side of the boxto the other(8) L ( i,z ) k := (cid:91) j =1 ,...,k ( z , . . . , z i − , j, z i +1 , . . . , z d ) . Let w t be the parallel toppling odometer for { Q ck ∪ L ( i,z ) k } -frozen parallel toppling (definedin (5)) with initial conditions w ( x ) = 1 { x ∈ L ( i,z ) k } , s (cid:48) = η . Denote the crossing time ,(9) C ( i,z ) k := min { t ≥ w t ( Q k ) = 1 } , where the right-hand side is ∞ if w ∞ ( Q k ) (cid:54) = 1. We show that if k is sufficiently large, thecrossing time is bounded with high probability. Proposition 2.1.
For every δ > , there is a k so that (10) max i,z C ( i,z ) k ≤ k d with probability at least (1 − δ ) .Proof. For each k ≥
1, we construct an event for which (10) occurs with probability ap-proaching 1 in k . By Harris’ inequality and symmetry, it suffices to show (10) for lines inone direction, say i = 1. Let z ∈ ¯ Q k be given.Write C k = C (1 ,z ) k . We show that if all lines, L (1 ,y ) k , y ∈ Q k , contain at least one site with(2 d −
1) chips, then every site in the cube eventually topples. Denote the event upon whichthis happens by,(11) Ω (cid:48) := (cid:92) y ∈ Q k Ω y := (cid:92) y ∈ Q k { η : Z d → Z : η ( x ) = (2 d −
1) for some x ∈ L (1 ,y ) k } . Recall p > d −
1) chips. Fix 0 < (cid:15) < p , and note, byHoeffding’s inequality, for each y ∈ Q k , P ( (cid:88) x ∈L (1 ,y ) k η ( x ) = 2 d − ≤ ( p − (cid:15) ) k ) ≤ exp( − (cid:15) k ) . Therefore, by the union bound, (deleting duplicates), P (Ω (cid:48) c ) ≤ k d − P (Ω (cid:48) c ) ≤ k d − exp( − (cid:15) k ) , and so for every δ >
0, there is k sufficiently large so that P (Ω (cid:48) ) ≥ − δ .It remains to check that for η ∈ Ω (cid:48) , C k ≤ k d . We do so by constructing a topplingprocedure which is dominated by w t . After firing L (1 ,z ) k , all sites in neighboring lines, L (1 ,y ) k , y d − ∼ z d − have at least (2 d −
1) chips. In fact, since η ∈ Ω (cid:48) , at least one site in eachneighbor L (1 ,y ) k ⊂ Q k has 2 d chips, causing all sites in the line to topple. Iterating shows thisprocedure will terminate with every site in Q k toppling in at most k d steps. (cid:3) .3. A static renormalization scheme.
We exhibit a coarsening of the lattice upon whichexplosions are guaranteed to spread quickly. A cube, Q k , is good if max i,z C ( i,z ) k ≤ k d . Foreach i ∈ Z d , let(12) Q k ( i ) := Q k + i · k. The cubes { Q k ( i ) } i ∈ Z d define a macroscopic lattice with edge set { ( Q k ( i ) , Q k ( j )) , | j − i | = 1 } .For k sufficiently large, Proposition 2.1 implies that the set of good cubes is dominated bya supercritical site percolation process on the macroscopic lattice. This together with largedeviations bounds for supercritical percolation [AP96, GM07] imply the following. (See, forexample, Section 5 in [Mat08] for an explicit proof.) Proposition 2.2.
For fixed k large enough, there are constants c, C so that the followinghold on an event of probability 1. (1) There is an infinite cluster C ∞ of good cubes on the macroscopic lattice { Q k ( i ) } i ∈ Z d . (2) There is n so that for n ≥ n , any connected component of C c ∞ that intersects [ − n, n ] d has volume smaller than (log n ) / . (3) There is n so that for n ≥ n , for any x, y ∈ C ∞ with | x | ≤ n and | x − y | ≥ (log n ) , c | x − y | ≤ d ( x, y ) ≤ C | x − y | , where d is the chemical (graph) distance on C ∞ . The definition of C ∞ ensures that once Q k ( i ) ∈ C ∞ is overlapped by the support of theodometer - Q k ( i ) ∩ { v t > } contains a straight line - an explosion will occur. This togetherwith Proposition 2.2 controls the speed at which the explosion propagates. We show nextthat the explosion spreading in C ∞ also quickly fills holes in the cluster.2.4. A path-filling property.
For a set of points A ⊂ Z d , let m i := min z ∈ A z i and M i :=max z ∈ A z i for i = 1 , . . . , d . Denote the bounding rectangle of A as(13) br ( A ) := { z ∈ Z d : m ≤ z ≤ M } . We show, using η ≥ (2 d − i.e. , an untoppled site with two neighbors which havetoppled, a contradiction. Our proof uses this idea together with a slightly technical induction(which, it seems, we cannot avoid as the claim is needed in all dimensions). See Figure 3 foran illustration of this result. Lemma 2.1.
Let A := { z ( i ) } be a finite path z ( i ) ∼ z ( i +1) . Let w t denote A -frozen paralleltoppling with initial conditions w ( A ) = 1 , s (cid:48) = η . Then, w t ( br ( A )) = 1 , for all t ≥ | br ( A ) | .Proof. By monotonicity of parallel toppling, we may take η = (2 d − br ( A ) eventually topples, as if no site topples at time t , then nosite topples at time ( t + 1).We say A contains a ( ± i, ± j ) turn if z ( m ) = z ( m − ± e i and z ( m +1) = z ( m ) ± e j for some i (cid:54) = j where z ( m − ∼ z ( m ) ∼ z ( m +1) are in A . If A does not contain a turn, then br ( A ) = A .Hence, we may suppose it contains at least one turn. igure 3. Terminal A -frozen parallel toppling odometer, w ∞ , on Z for initialconditions w = 1 { x ∈ A } and s = 2. Red pixels are sites in A and blackpixels are sites which eventually topple. Case 1 - one-turn path.
By shifting coordinates, we may suppose A contains only a (1 , A = { , e , . . . , k e , k e + e , . . . , k e + k e } for k ≥ k . We induct on k . If k = 1, then after firing every site in A , all sites in A + e get one chip, while the corner site, (( k − e + e ) gets 2 chips. Since η = (2 d − k − e + e ) ∼ (( k − e + e ) ∼ · · · ∼ e to fire. Continuing the induction shows that every site in br ( A ) = { x ∈ Z d : 0 ≤ x ≤ ( k e + k e ) } eventually fires. Case 2 - cubic path.
We call A a cubic path if, after an isometry,(15) A = { , e , . . . , k e , k e + e , . . . , d (cid:88) i =1 k i e i } for k ≥ · · · ≥ k d ≥
0. Let d := max { i ≤ d : k i > } . We induct on d , the base case d = 2established in Case 1. For notational convenience, suppose the claim holds for d = ( d − d = d .Consider the ( d − P := { , e , . . . , d − (cid:88) i =1 k i e i }P := { k e , k e + e , . . . , d (cid:88) i =1 k i e i } . By the inductive hypothesis, after P i fire, both ( d − F := { x ∈ Z d : 0 ≤ x ≤ d − (cid:88) i =1 k i e i }F := { x ∈ Z d : k e ≤ x ≤ d (cid:88) i =1 k i e i } re. We then ‘fill in’ the cube by identifying newly fired ( d − P (cid:48) j := { je , je + e , . . . , je + k e ,je + k e + e , . . . , je + k e + k e , · · · je + (cid:88) i (cid:54) =2 k i e i } which are in F ∪ F for j = 0 , . . . , k . By the inductive hypothesis, the firing of each P (cid:48) j makes every ( d − L j := { x ∈ Z d : je ≤ x ≤ je + (cid:88) i (cid:54) =2 k i e i } , fire and br ( A ) = ∪ k j =0 L j . Case 3 - general path.
It suffices to show that if there is a path of firings between any twodistinct points x, y , then br ( { x, y } ) eventually fires. Before showing this, we suppose it weretrue and demonstrate sufficiency. Take q ∈ br ( A ) and observe by definition there are points( z (1) , Z (1) ) , . . . , ( z ( d ) , Z ( d ) ) in A with z ( i ) i ≤ q i ≤ Z ( i ) i . Then, q (1) := ( q , q (cid:48) d − ) ∈ br ( { z (1) , Z (1) } )for some ( d − q (cid:48) d − . Continue and let q (2) := (cid:40) ( q , q , q (cid:48)(cid:48) d − ) ∈ br ( { q (1) , z (2) } ) if q (1)2 ≥ q ( q , q , q (cid:48)(cid:48)(cid:48) d − ) ∈ br ( { q (1) , Z (2) } ) otherwisefor some ( d − q (cid:48)(cid:48) d − , q (cid:48)(cid:48)(cid:48) d − . After iterating, we find q ( d ) = q , which shows thateventually q will fire.Now fix two points x, y ∈ A and decompose a path between them into a sequence of cubicpaths P (1) := m (cid:91) i =1 P (1) i , where P (1) i := { p i − , . . . , p i } is cubic and p := x and p m := y . (This can be done by, forexample, starting at x and exploring the path but cutting whenever the cubic condition isviolated.) Case 2 shows that eventually every site in A (1) := (cid:83) mi =1 br ( P (1) i ) will fire. If m = 1,we are done, otherwise we construct a new cubic path from p to p passing through A (1) .Once we have shown this, we iterate to conclude.Suppose p = 0, p = (cid:80) dj =1 k j e j , and p = d (cid:88) j =1 ( k j − k (cid:48) j ) e j + d (cid:88) j =( d +1) ( k j − k (cid:48) j ) e j + d (cid:88) j =( d +1) ( k j + k (cid:48) j ) e j , for some 1 ≤ d ≤ d ≤ d and k j , k (cid:48) j ≥ k j − k (cid:48) j ) < j ≤ d and ( k j − k (cid:48) j ) ≥ d + 1) ≤ j ≤ d . After this coordinate change, it suffices to exhibit a path from p to p with differences constrained to be − e j for j = 1 , . . . , d and + e j for j = ( d + 1) , . . . , d . here is a cubic path (only positive moves) from p to w := d (cid:88) j =( d +1) ( k j − k (cid:48) j ) e j + d (cid:88) j =( d +1) ( k j ) e j contained within br ( { p , p } ) as p = 0 ≤ w ≤ p . Then, since w ∈ br ( { p , p } ) there is acubic path (only positive moves) from w to w := w + d (cid:88) j =( d +1) k (cid:48) j e j contained in br ( { p , p } ) and similarly there is a cubic path (only negative moves) from w to p = w + d (cid:88) j =1 ( k j − k (cid:48) j ) e j . Our new cubic path is the concatenation of these three paths: p → w → w → p . (cid:3) The last-wave
In this section we study a simplified parallel toppling procedure closely related to boostrappercolation (see Section 7 for an explicit connection, we do not utilize the coupling here). Thissimplified process has an inherent subadditive structure which allows us to prove convergenceusing the subadditive ergodic theorem. In the next section we show that this process is agood approximation to an exploding sandpile.3.1.
The n -wave process. Fix n ≥ z ∈ Z d , and consider the following modified paralleltoppling process(19) u ( z )0 := n · δ z u ( z ) t +1 ( x ) := (cid:98) (cid:80) y ∼ x u ( z ) t ( y ) + η ( x )2 d (cid:99) for x (cid:54) = z. This is a { z } -frozen parallel toppling process (see (5)) which we call the n -wave for η startingat z . Overloading terminology, the n -wave is stabilizable if there is T < ∞ so that u ( z ) t = u ( z ) T for all t ≥ T . Let(20) ˆ M η ( z ) := min { n ≥ n -wave for η starting at z is not stabilizable } . We write u ( z ) t for the ˆ M η ( z )-wave starting at z and call this the last-wave . We also considerthe penultimate-wave starting at z as the terminal odometer for the ( ˆ M η ( z ) − u ( z ) .The set of sites touched by the penultimate wave is its penultimate-cluster ,(21) P ( z ) := { z } ∪ { x ∈ Z d : there is y ∼ x with ˜ u ( z ) ( y ) > } (we included the point, { z } , as ˆ M η ( z ) may be 1). When z is the origin, we omit thesuperscripts.The arrival time for the last-wave starting at site s to site x is(22) ˆ T ( s, x ) := min { t ≥ u ( s ) t ( x ) > } nd the penultimate-cluster arrival time is(23) ˜ T ( s, x ) := min { t ≥ u ( s ) t ( P ( x )) > } . We write ˆ T ( x ) := ˆ T (0 , x ) and ˜ T ( x ) := ˜ T (0 , x ) (choice of the same letter T for all arrivaltimes was intentional - we will see they are asymptotically close).3.2. Basic properties of the last-wave.
We derive some basic properties of the last-wave. Throughout this section and the next, let η be drawn from the event of probability 1in Proposition 2.2 and let k be the (deterministic but large) side length of a good cube. Thefollowing is a consequence of Theorem 4.1 in [FdBR08]. Lemma 3.1.
There is a constant C := C d so that for all R ≥ , the support of the ( C · R d ) -wave contains [ − R, R ] d .Proof. From Theorem 4.1 in [FdBR08], we know that if n chips are placed at the origin ona background of (2 d − v , contains a cube ofradius r n ≥ ( n /d − /
2. Moreover, ˜ v (0) ≤ Cn d . This implies, by (7) that a ( C · n d )-wavecontains [ − r n , r n ] d . (cid:3) Lemma 3.2.
The last-wave is well-defined, ˆ M η (0) < ∞ .Proof. By Lemma 3.1 and Proposition 2.2, if the origin is fired a sufficient number of times,the support of the odometer contains a good cube, Q k ⊂ C ∞ . (cid:3) Lemma 3.3.
The last wave is bounded by one outside the interior of the penultimate-cluster,for all t ≥ and x, z ∈ Z d , u ( z ) t ( x ) ≤ (1 + ˜ u ( z ) ( x )) . Proof.
To reduce clutter, we take z = 0. We prove this by induction on t . The base case t = 0 follows by definition. For all t ≥
1, the definition also ensures it holds at the origin.So, we may take x (cid:54) = 0 and check: u t +1 ( x ) = (cid:98) (cid:80) y ∼ x u t ( y ) + η ( x )2 d (cid:99)≤ (cid:98) (cid:80) y ∼ x (1 + ˜ u ( y )) + η ( x )2 d (cid:99) = 1 + ˜ u ( x ) + (cid:98) (cid:80) y ∼ x (˜ u ( y ) − ˜ u ( x )) + η ( x )2 d (cid:99) = 1 + ˜ u ( x )as ∆˜ u ( x ) + η ( x ) ≤ (2 d −
1) for x (cid:54) = 0. (cid:3) Lemma 3.4.
The penultimate-cluster arrival times are subadditive: for all a, b, c ∈ Z d , ˜ T ( a, c ) ≤ ˜ T ( a, b ) + ˜ T ( b, c ) . Proof.
Suppose P ( c ) (cid:54)⊆ P ( b ), otherwise the claim is immediate. It suffices to check(24) w t ( x ) := u ( a )˜ T ( a,b )+ t ( x ) ≥ u ( b ) t ( x ) for all t ≥ x ∈ P ( b ) c , hich we do by induction. By Lemma 3.3, u ( b ) t ( x ) ≤ u ( b ) ( x ) = 0. In particular, for all t ≥ x ∈ ∂ ◦ P ( b ), u ( b ) t ( x ) ≤ ≤ w t ( x ) (where the interior boundary of a set S ⊂ Z d is ∂ ◦ S := { x ∈ S : there is y ∼ x in S c } ). Using this and the inductive hypothesis, if x ∈ P ( b ) c ∩ { a } c , w t +1 ( x ) = (cid:98) (cid:80) y ∼ x w t ( y ) + η ( x )2 d (cid:99)≥ (cid:98) (cid:80) y ∼ x u ( b ) t ( y ) + η ( x )2 d (cid:99) = u ( b ) t +1 ( x ) . If x ∈ P ( b ) c ∩ { a } , then w t ( x ) ≥ ≥ u ( b ) t ( x ) as ˆ M η ( a ) ≥ (cid:3) Lemma 3.5.
There is a constant γ > so that for all n sufficiently large and | x | ≤ n , P ( x ) ⊂ [ − r n , r n ] d , where r n ≤ (log n ) γ .Proof. By Lemma 2.1, 1 { ˜ u ( x ) > } is a rectangle, therefore it suffices to bound the maximalside length. By Proposition 2.2, if any side length of the rectangle exceeds (2 k log n ) / thenit must overlap a good cube, contradicting stability. (cid:3) Lemma 3.6.
There are constants γ and C so that on an event of probability 1, for all n sufficiently large and | x | ≤ n , ˆ T ( x ) ≤ C | x | + (log n ) γ . Proof.
Let x ∈ Z d be given. Since the sandpile is exploding, at some constant time C thesupport of the odometer will overlap the infinite cluster at a good cube near the origin, Q k ( z )for some z ∈ Z d . Once this occurs the arrival time to any site is at most a constant times thechemical distance in the infinite cluster. Let Q k ( y ) for y ∈ Z d be one of the nearest cubes in C ∞ to x . There are now two cases to consider. Case 1: | z − y | < (log n ) Choose nearby points M i ∈ C ∞ so that (log n ) c ≤ d ( z, M i ) ≤ (log n ) C and x, y ∈ br ( { M i } ). Case 2: | z − y | ≥ (log n ) By the chemical distance bound and the definition of C ∞ , within at most C | z − y | ≤ C ( | z | + | y | )steps, Q k ( y ) will topple. Once this happens, P ( x ) is surrounded in at most (log n ) C moresteps and P ( x ) will fire in at most |P ( x ) | additional steps. (cid:3) Convergence of the last-wave.
We show that the arrival time for the last-waveconverges.
Lemma 3.7.
There exists a constant γ > so that on an event of probability 1, for all n sufficiently large and | x | ≤ n , ˆ T ( x ) ≤ ˜ T ( x ) ≤ ˆ T ( x ) + (log n ) γ . roof. The first inequality is immediate. For the second, suppose v t ( x ) > t, | x | ) >C . We must show that there is a nearby good cube Q k ( y ) ⊂ C ∞ which has already fired.Once we have shown this, the same argument as in Lemma 3.6 allows us to conclude. This is,however, a consequence of Lemma 2.1 and Proposition 2.2. Any path of topplings of lengthat least (2 k log n ) / must overlap a good cube. (cid:3) Proposition 3.1.
On an event of probability 1, the rescaled last-wave arrival times n − ˆ T η ([ nx ]) converge locally uniformly to N p , a continuous, convex, one-homogeneous function on R d .Proof. In light of Lemma 3.7 it suffices to prove the result for ˜ T . Convergence in integer di-rections follows from the subadditive ergodic theorem. Everywhere convergence then followsfrom continuity and approximation. The properties of N p are immediate from the scalingand microscopic subadditivity. (cid:3) Remark 1.
Convergence of the last wave may be viewed as a sort of bootstrap percolationshape theorem. Sites are initially randomly assigned two thresholds, 1 or 2. A site withthreshold l becomes infected when at least l of its neighbors are infected. Infected sites remaininfected. The above shows that if you start off with a large enough cluster of infected sitesat the origin, every site will eventually become infected and the speed at which the infectionspreads converges.For more on the relationship between sandpiles and bootstrap percolation, see Section 7below. Similar shape theorems include [GM12, KS08, AMP02, CD81] and especially [Wil78,GG93, FL11] . Proof of Theorem 1.3
Let η be drawn from the event of full probability in Proposition 3.1. It suffices to showthat the last-wave is a good approximation of the original process. Proposition 4.1.
On an event of probability 1, there are constants C , C so that for all x ∈ Z d , (25) T ( x ) ≤ ˆ T ( x ) + C and (26) ˆ T ( x ) ≤ T ( x ) + C , where the last-wave arrival time ˆ T is defined in (22) and T ( x ) := min { t ≥ v t ( x ) > } .Proof. Recall that v t is the parallel toppling odometer for η + M η · δ and u t from Section 3.1.We first observe that (25) follows immediately from (7): since ˆ M η (0) < ∞ , and η + M η · δ is not stabilizable, for t ≥ C η , v t (0) ≥ ˆ M η (0) = u (0).We now verify (26). We first consider the special case where only one firing at the originis needed to have an infinite last-wave. tep 1: Special case, ˆ M η (0) = 1 . Denote the reachable sets up to time t for the last waveand exploding sandpile as(27) R t := { x ∈ Z d : v t ( x ) > } ˆ R t := { x ∈ Z d : u t ( x ) > } . Note, by minimality, if ˆ M η (0) = 1, then η (0) + M η (0) = 2 d . This together with η ≥ (2 d − t ≥ R t ⊆ ˆ R t and(29) | v t ( x ± e i ± e j ) − v t ( x )) | ≤ x ∈ Z d and e i (cid:54) = e j | v t ( x ± e i ) − v t ( x )) | ≤ x ∈ Z d and e i and(30) v t (0) ≥ max x ∈ Z d v t ( x ) . The base case is immediate, so suppose (28), (29), and (30) hold at t and we check ( t + 1). Inductive step for (30) . (4) and (30) at time t imply if x (cid:54) = 0, v t +1 ( x ) ≤ (cid:98) d · v t (0) + η ( x )2 d (cid:99) = v t (0) + (cid:98) η ( x )2 d (cid:99) = v t (0) ≤ v t +1 (0) , as η ≤ (2 d − Inductive step for (29) . We first check the origin. By (30), if v t ( y ) = v t (0) − y ∼
0, then ∆ v t (0) + 2 d ≤ (2 d − v t ( e i + e j ) = v t (0) − e i (cid:54) = e j and the origin is unstable. Then, v t ( e i ) = v t (0) and v t ( e j ) = v t (0). This implies,by (29) applied to e i and e j , that all other neighbors y ∼ ( e i + e j ) have a lower bound, v t ( y ) ≥ v t (0) −
1. Hence, ∆ v t ( x + e i + e j ) ≥
2, which implies that ( x + e i + e j ) is unstableusing η ≥ (2 d − x (cid:54) = 0 and suppose for sake of contradiction(31) ∆ v t ( x ) + η ( x ) ≥ d but for some adjacent neighbor ( x + e j ),(32) v t ( x + e j ) = v t ( x ) − v t ( x + e j ) + η ( x + e j ) ≤ d − . At least two other adjacent neighbors, y (cid:48) ∼ x must satisfy v t ( y (cid:48) ) = v t ( x ) + 1. Indeed,otherwise by (32) and (29), ∆ v t ( x ) ≤
0, violating our assumption (31). However, one ofthose neighbors must be y (cid:48) = x ± e i for some e i (cid:54) = e j . This contradicts (29) at time t since v t ( x + e i ) = v t ( x + e i + ( − e i + e j )) + 2 . ext, take a diagonal neighbor, ( x + e i + e j ) for i (cid:54) = j , and suppose for sake of contradiction(31) but(33) v t ( x + e i + e j ) = v t ( x ) − v t ( x + e i + e j ) + η ( x + e i + e j ) ≤ (2 d − . By (31) there must be at least one adjacent neighbor y ∼ x with v t ( y ) = v t ( x ) + 1. Thisneighbor cannot be ( x + e i ) or ( x + e j ) as it would contradict (29) for ( x + e i + e j ). Possibly y = ( x ± e i (cid:48) ) for i (cid:48) (cid:54)∈ { i, j } , y = x − e i , or y = x − e j . In these cases,(34) v t ( x + e i ) = v t ( x + e j ) = v t ( x ) = v t ( x + e i + e j ) + 1 . Indeed, if not, then, say, v t ( x + e i ) = v t ( x ) −
1, and so there must be an additional neighbor, y (cid:48) ∼ x , y (cid:48) (cid:54) = y , with v t ( y (cid:48) ) = v t ( x ) + 1. But, either y (cid:48) or y is diagonal to ( x + e i ), whichcontradicts (29).Assuming (34), the same argument implies v t ( y ) ≥ v t ( x + e i + e j ) for all y ∼ ( x + e i + e j ).This together with (34) shows ∆ v t ( x + e i + e j ) ≥
2. which contradicts (33).
Inductive step for (28) . It suffices to check this for x (cid:54) = 0 as u t (0) = 1. Suppose for sake ofcontradiction there is some site x with(35) ∆ v t ( x ) + η ( x ) ≥ d but(36) ∆ u t ( x ) + η ( x ) ≤ (2 d − u t ( x ) = 0. By (28), u t ( x ) = v t ( x ) = 0 and hence(37) v t ( y ) ≤ y ∼ x, by (29). However, (28) and (37) imply that ∆ v t ( x ) + η ( x ) = ∆ u t ( x ) + η ( x ) ≤ (2 d − Step 2: General case.
In the general case, we introduce a pair of approximations to whichwe can apply the arguments of the special case. Let ˜ v be the terminal (unfrozen) odometerfor η + ( M η (0) − · δ and let(38) ˜ P := { } ∪ { x ∈ Z d : there is y ∼ x with ˜ v ( y ) > } . Let ˜ w t be the ˜ P -frozen parallel toppling odometer with initial conditions(39) ˜ w ( x ) = 1 { x ∈ ˜ P} s (cid:48) = η. Let w t be the parallel toppling odometer for s : Z d → Z where(40) s ( x ) := (cid:40) η ( x ) if x (cid:54)∈ ˜ P (2 d −
1) + δ otherwise . Denote the reachable sets for these processes by(41) R t := { x ∈ Z d : w t ( x ) > } ˜ R t := { x ∈ Z d : ˜ w t ( x ) > } . The same argument as in Step 1 shows that(42) R t ⊆ ˜ R t . e claim that we can conclude after proving the following inequalities,(43) ˜ w t ≤ u t + c (44) v t ≤ ( w t + ˜ v ) . Indeed, if v t ( x ) >
0, then (44) implies ˜ v ( x ) > w t ( x ) >
0. In both cases, using either(42) or (39), ˜ w t ( x ) > u t + c ( x ) > Proof of (43) . We know that | ˜ P| = C < ∞ . Hence, at some finite time u c ( ˜ P ) ≥ w t ≤ u t + c . Proof of (44) . This is true at t = 0, we use (4) and induct, v t +1 ( x ) ≤ (cid:98) (cid:80) y ∼ x v t ( y ) + η ( x ) + M η · δ d (cid:99)≤ (cid:98) (cid:80) y ∼ x ( w t ( y ) + ˜ v ( y )) + η ( x ) + M η · δ d (cid:99) = ˜ v ( x ) + (cid:98) (cid:80) y ∼ x w t ( y ) + (cid:80) y ∼ x (˜ v ( y ) − ˜ v ( x )) + η ( x ) + M η · δ d (cid:99)≤ ˜ v ( x ) + (cid:98) (cid:80) y ∼ x w t ( y ) + s ( x )2 d (cid:99) = ˜ v ( x ) + w t +1 ( x ) . The third inequality used ∆˜ v ( x ) + ( M η − · δ + η ≤ (2 d − (cid:3) Remark 2.
Some qualitative features of the limit shape are immediate. For example, theorigin is an interior point and the limit is invariant with respect to symmetries of the lattice(symmetry may fail in the periodic case introduced in Section 5). Also, a coupling withoriented percolation as in [DL81, Mar02] can be used to establish a ‘flat-edge’ for p sufficientlyclose to one in all dimensions. We omit the details since it is routine - see, for example, theproof of Theorem 1.2 in [AMP02] or Theorem 6.3 in [GM04] . A generalization
Sufficient hypotheses.
We present sufficient hypotheses on η under which the argu-ments above go through seamlessly. Fix η min ∈ Z and let Ω denote the set of all boundedfunctions η : Z d → Z , η min ≤ η ≤ (2 d − σ -algebra F generated by { η → η ( x ) : x ∈ Z d } . Denote the action of integer translation by T : Z d × Ω → Ω, T ( y, η )( z ) = ( T y η )( z ) := η ( y + z ) , and extend this to F by defining T y E := { T y η : η ∈ E } . Let L ⊆ Z d be a sublattice , a finiteindex subgroup of Z d . Let P be a stationary and ergodic probability measure on (Ω , F ) withrespect to L ,(45) Stationary: for all E ∈ F , y ∈ L : P ( T y E ) = P ( E ) , (46) Ergodic: E = (cid:84) y ∈L T y E implies that P ( E ) ∈ { , } . We refer to the probability measure P as explosive if P ( η is explosive) = 1. tationarity and ergodicity are the weakest hypotheses under which a convergence resultis proved - straightforward counterexamples can be constructed. However, we do not expectexploding sandpiles to have a limit shape without an additional independence hypothesis.At the very least, our proof will not work, as domination by a coarsened product measurewas essential. Our first hypothesis is hence a quantification of ergodicity. Hypothesis 1 (Finite range of dependence) . There exists a constant
K < ∞ so that for all x, y ∈ Z d , η ( x ) and η ( y ) are independent if | x − y | > K . Next, fix a finite (rectangular) box with side length k > B k := { x ∈ Z d : 1 ≤ x i ≤ k i } .The 2 d external faces of B k are F i := { x ∈ ¯ B k : x i = 0 } , F d + i := { x ∈ ¯ B k : x i = k i + 1 } . Take a face, F i , and let w t : ¯ B k → N be the sequence of B ck -frozen parallel toppling odometerswith initial conditions w = 1 { x ∈ F i } and s (cid:48) = η . We say that B k ( η ) can be crossed indirection i if w t ( x ) ≥ { x ∈ B k } for t ≥ |B k | . Hypothesis 2 (Box-crossing) . For each δ > , there is k so that min j ∈ Z d P ( B ( j ) k ( η ) can be crossed in each direction ) > − δ, where (cid:91) j ∈ Z d B ( j ) k := (cid:91) j ∈ Z d ( B k + j · k ) = Z d is a tiling of the lattice by B k . If η were recurrent, Hypothesis 2 would imply η explodes with probability 1. In particular,no holes would develop in the support of the odometer. (If unfamiliar, see Section 7 belowfor the definition of recurrence, although this is not used here.) Our next hypothesis ensuresthis and more: any sufficiently long path of topplings fills its bounding rectangle. Hypothesis 3 (Path-filling) . There exists a constant γ > so that on an event of probability1, for all n ≥ n , and every path of distinct points, [ − n, n ] d ⊃ L m := { z , . . . , z m } , z i +1 ∼ z i ,of length m ≥ (log n ) γ the following holds. The L m -frozen parallel toppling odometer withinitial conditions w = 1 { x ∈ L m } and s (cid:48) = η quickly exceeds 1 on the bounding rectangle of L m : w t ≥ { br ( L m ) } for t ≥ m d . In order for η to have a limit shape in dimensions d ≥
3, we need to strengthen Hypothesis2. The next assumption prevents low-dimensional tendrils from burrowing through goodcubes (for a counterexample in three-dimensions take a large cube filled with 4 and connecteach of the faces with disjoint tunnels of 5). For a point z ∈ ¯ B k and direction 1 ≤ i ≤ d ,consider, as before, a line passing from one side of the box to the other(47) L ( i,z ) k := (cid:91) j =1 ,...,k i ( z , . . . , z i − , j, z i +1 , . . . , z d ) . We say B k ( η ) is strongly box-crossing if, for all 1 ≤ i ≤ d and z ∈ ¯ B k , w | k | ≥ { x ∈ B k } ,where w t is the odometer for {B ck ∪ L ( i,z ) k } -frozen parallel toppling with initial conditions w ( x ) = 1 { x ∈ L ( i,z ) k } , s (cid:48) = η . ypothesis 4 (Strongly box-crossing) . For each δ > , there is k so that, using the samenotation as Hypothesis 2, min j ∈ Z d P ( B ( j ) k ( η ) is strongly box-crossing ) > − δ. We now have made enough assumptions to prove convergence of the last-wave as in Section3.
Proposition 5.1 (Convergence of the last-wave) . Under Hypotheses 1, 3, and 4, on anevent of probability 1, η is explosive and the rescaled last-wave arrival times, n − ˆ T η ([ nx ]) := n − min { t > u t ([ nx ]) > } converge locally uniformly to N η , a continuous, convex, one-homogeneous function on R d .Proof. For δ > k from Hypotheses 4. For j ∈ Z d , let X j := 1 {B ( j ) k is strongly box-crossing } . By Theorem 0.0 in [LSS97], { X j } j ∈ Z d stochastically dominates a sequence of Bernoulli in-dependent random variables { Y j } j ∈ Z d with P ( Y j = 1) ≥ (1 − π ( δ )) for π : [0 , → [0 , π ( δ ) → δ →
0. Therefore, for δ > { X j } j ∈ Z d contains an infinite supercritical percolation cluster C ∞ .The rest of the argument follows almost exactly the proof in Section 3. The only minorchange is in the proof of Lemma 3.1. We use η ≥ η min rather than η ≥ (2 d −
2) and invokeTheorem 4.1 in [LP09] to get that the support of a ( C · R d )-wave contains [ − R, R ] d (wherethe constant C is larger than before). (cid:3) If additionally η ≥ (2 d − η ≥ (2 d −
2) isnot necessary, however, we have not found an alternative condition and are forced to assumethis:
Hypothesis 5 (Wave-approximation) . Suppose η is explosive, let u t denote the last-wave, v t the parallel toppling odometer for η + M η · δ , and ˆ T η , T η the respective arrival times. Onan event of probability 1, sup x ∈ [ − n,n ] d | ˆ T η ( x ) − T η ( x ) | = o ( n ) . Theorem 5.1 (Convergence of the exploding sandpile) . Under the assumptions in Proposi-tion 5.1 and Hypothesis 5, on an event of probability 1, η is explosive and the rescaled arrivaltimes, n − T η ([ nx ]) := n − min { t > v t ([ nx ]) > } converge locally uniformly to N η , acontinuous, convex, one-homogeneous function on R d .Proof. Immediate from Hypothesis 5 and Proposition 5.1. (cid:3)
Remark 3.
Box-crossing with probability 1 implies η is recurrent (see Section 7 if unfamil-iar). However, not every recurrent sandpile is explosive - take η = (2 d − and use [FLP10] - and not every exploding sandpile has a recurrent initial condition - see Section 6. igure 4. Initial random backgrounds and computed limit shapes. The ran-dom backgrounds are built from Bernoulli clouds of the indicated point sets.Blue is 2 chips and red is 3 chips.5.2.
Examples satisfying the hypotheses.
The simplest way to ensure Hypotheses 3 and5 is to take η ≥ (2 d − Bernoulli cloud , see Figure 4. Take p >
0, fix a finite set of points,
S ⊂ Z d (say a triangle, circle, or a line), and independently sample a uniform random variable ateach site on the lattice, { U j } j ∈ Z d . Then, let(48) η ( x ) := (cid:40) (2 d −
1) if there exists j ∈ Z d such that U j < p and x ∈ {S + j } (2 d −
2) otherwise.Hypothesis 1 is satisfied as |S| < ∞ and Hypothesis 4 as p > random checkerboard . Fix a box B which tiles the lattice, Z d = (cid:83) j ∈ Z d B j . Take functions ζ , . . . , ζ m , defined on the box, ζ i : B → { (2 d − , (2 d − } .Suppose further that B ( ζ ) contains at least one site with (2 d −
1) chips along every straight (cid:2) (cid:3) (cid:20) (cid:21) Table 1.
Computed limited shapes of periodic, checkerboard backgrounds ofthe indicated box. ine. Let { Y j } j ∈ Z d be a field of i.i.d. random variables, where P ( Y j = i ) = p i , for i = 1 , . . . , m .Then, let(49) η ( x ) := ζ i ( z ( x )) if Y j ( x ) = i, where z ( x ) ∈ B is the position of x in its tiled box, B j ( x ) . Finite range of dependence isimmediate by construction. If we further assume p >
0, then Hypothesis 4 is satisfied bythe assumption on B ( ζ ).The random checkerboard includes the degenerate case p = 1, where η is a periodic copyof ζ . See Table 1 for pictures. In this case, if η ≥ (2 d −
2) but is not box-crossing, thenthe background is not explosive by Theorem 4.2 in [FLP10]. However, it is possible to buildrandom (and periodic) checkerboard, exploding sandpiles with η (cid:54)≥ (2 d − η (cid:54)≥ (2 d − Failure of convergence
In this section we construct a family of exploding sandpiles which fail to have a limitshape. As the construction indicates, the counterexample is stable: it can be random orperiodic.
Theorem 6.1.
For each d ≥ , there are explosive backgrounds η (cid:54)≥ (2 d − which fail tohave a limit shape; the first arrival times T ( n ) := min { t > v t ( ne ) > } do not converge, (50) lim sup n →∞ n − T ( n ) ≥ / and (51) lim inf n →∞ n − T ( n ) = 1 . Figure 5.
The counterexample from Theorem 6.1 for d = 2 and d = 3.We explicitly demonstrate a family of checkerboard backgrounds which are explosive butdo not have a limit shape. Our counterexample is essentially a two-dimensional one. Afterconstructing it in two dimensions, we embed it into higher-dimensions and show failure ofconvergence by comparison with the two-dimensional counterexample. .1. Proof of Theorem 6.1 for d = 2 . We use the notation of Section 5. Let B := { x ∈ Z : 0 ≤ x ≤ } denote a box of side length 4 and take ζ , ζ : B → { , } as ζ := ζ := , where the lower-left corner of the box is (0 ,
0) and left-to-right and down-to-up are increasing.Let η be an arbitrary tiling of ζ , ζ ; for example, η could be a sample from the randomcheckerboard measure. Fix coordinates so that η ( x , x ) = ζ i ( x mod 4 , x mod 4) . Let v t be the sequence of parallel toppling odometers for s = η + 3 · δ . We first verify (51). Step 1: Proof of (51) . We show for all n ≥ n + 1) ≤ T (4 n + 1) ≤ n + 4 . By inspection, v (0) = 1 and v ( e ) = v ( e ) = 1. Now, take n ≥ e to ( e + (4 n + 1) e ). Thus, v n +3 ( e + (4 n + 1) e ) = 1 and v n +4 ((4 n + 1) e ) = 1. The lower bound is immediate from η ≤ (2 d −
1) - a site can fire onlyif a neighbor has fired previously.
Step 2: η is explosive. We construct a toppling procedure dominated by v t which transforms η into a configuration η (cid:48) which is not stabilizable. We may topple the origin, every 3, then2, and then the 2 × η (cid:48) := η + ∆ (1 { η ≥ } + δ + δ − e + δ − e + δ − e − e ) . The resulting configuration is (away from the origin) a tiling of ζ (cid:48) , ζ (cid:48) : B → { , } ,(53) ζ (cid:48) := ζ (cid:48) := . Remark 4.
The reason why convergence fails for this counterexample is that the limit shapeof the explosive background η (cid:48) is not a diamond. See Figure 5. When d = 2 the limit shapeis a regular octagon with boundary max( | x − y/ | , | x + y/ | , | x/ − y | , | x/ y | ) , but we willnot prove this. Both ζ (cid:48) and ζ (cid:48) are box-crossing, so we just check that we can construct a sequence offirings to the outer face of a box away from the origin dominated by v t . The box containingthe origin is at least η (cid:48) (0 : 3 , ≥ . From this we see that 3 · δ + η (cid:48) is not stabilizable - in a finite number of steps every site in(0 : 3 , tep 3: Reductions. Before proving (50), we make several reductions. We seek to lowerbound T , therefore, we are free to add to η as this will only decrease the arrival time. First,we may suppose all of the boxes are ζ rather then ζ .We then increment the background so as to reduce to a sandpile on a cylinder, C := { x ∈ Z : x ≥ , ≥ x ≥ } . Specifically, let ζ : C → { , , } , ζ := ζ ζ · · · with the origin, (0 , ζ : set for x ≥ η ( x , x ) := ζ ( x , x mod 4)and for x <
0, ˆ η ( x , x ) := ˆ η ( − ( x + 1) , x ) . Note that ˆ η ≥ η .The structure of ˆ η allows us to reduce to a symmetrized Laplacian on the cylinder C (see forexample Lemma 2.3 in [BR20]) with reflecting boundaries at x = 0: v t ( − , x ) = v t (0 , x )and torus boundary conditions for x ∈ { , } : v t ( x , −
1) = v t ( x ,
3) and v t ( x ,
4) = v t ( x , symmetrized Laplacian ∆ and nearest neighbors y ∼ x on C .Let ˜ u ( x ) : C → { , } be ˜ u ( x ) := 1 { ˆ η ( x ) ≥ } . Then,∆˜ u + ζ = ζ (cid:48) ζ (cid:48) · · · =: ζ (cid:48) . Let v t : C → Z + be the symmetrized parallel toppling odometer for ζ and w t : C → Z + thesame for ˆ η (cid:48) (defined with ζ (cid:48) as ˆ η was with ζ ). We claim that(54) T (cid:48) (3 + 8 n ) ≤ T (3 + 8 n ) , where T (cid:48) ( n ) := min { t > w t ( ne ) > } . In fact, we claim(55) v t ≤ w t + ˜ u for all t ≥
0. This includes (54) as ˜ u ((3 + 8 n ) e ) = 0. We observe (55) is a consequence ofinduction: the base case t = 0 is automatic and the inductive step is, v t +1 ( x ) = (cid:98) (cid:80) y ∼ x v t ( y ) + ζ ( x )4 (cid:99)≤ (cid:98) (cid:80) y ∼ x ( w t ( y ) + ˜ u ( y )) + ζ ( x )4 (cid:99) = ˜ u ( x ) + (cid:98) (cid:80) y ∼ x w t ( y ) + ∆˜ u ( x ) + ζ ( x )4 (cid:99) = ˜ u ( x ) + w t +1 ( x ) . tep 4: Proof of (50) . We show for all n ≥ T (cid:48) (3 + 8 n ) ≥ n. We do so by building a ‘pulsating front’ for w t in the horizontal direction. (Readers interestedin pulsating fronts in periodic media on R d may see Section 2.2 of [Xin09].)We first reduce to the last-wave for w t , ˆ w t , with initial conditions ˆ w = δ (0 , and ˆ s = ζ (cid:48) + ∆ ˆ w . The justification is identical to Step 1 of the proof of Proposition 4.1 and so isomitted. Using ˆ w t ≤
1, we make another reduction to initial condition ˆ w : C → { , } ,ˆ w = (cid:20) · · · (cid:21) . We now show, by manual computation, that the configuration of the odometer at the front,the rightmost 4 × C containing a site which has toppled, is 12-periodic in time. Fornotational ease, we denote sites which have toppled by ∗ ,ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) . ˆ s = (cid:20) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ · · · (cid:21) This shows that ˆ w = (cid:20) · · · (cid:21) , so the odometer at the front is identical to what it was at the start and the process ‘resets’.Hence, by induction on n ,(57) ˆ T (3 + 8 n ) = 12 n, where ˆ T ( n ) := min { t > w t ( ne ) > } . This implies (56), completing the proof by (54).6.2. Proof of Theorem 6.1 for d ≥ . Let η (2 D ) : Z → { , , } be the random back-ground defined in Section 6.1 and let d ≥ η : Z d → { d − , d − , d − } is built by stacking the two-dimensional one,(58) η ( x , x , . . . , x d ) = 2 · ( d −
2) + η (2 D ) ( x , x ) for all x ∈ Z d . Step 1: Proof of (51) . The argument is identical to d = 2. tep 2: η is explosive. Let η (cid:48) (2 D ) be the tiling of ζ (cid:48) , ζ (cid:48) , defined in (53). The higher-dimensional analogue, η (cid:48) : Z d → { d − , d − , d − } is also stacked,(59) η (cid:48) ( x , x , . . . , x d ) := η (cid:48) (2 D ) ( x , x ) + 2 · ( d − . The argument is as before: we construct a toppling procedure dominated by v t , the paralleltoppling odometer for η , that transforms η into η (cid:48) . Since 3 · δ + η (cid:48) ≥ (2 d −
2) is box-crossing,it is not stabilizable.Topple the origin, all sites with (2 d −
1) chips, then all sites with (2 d − d −
3) near the origin. Let ˜ u denote the odometer for this and ˜ u (2 D ) the two-dimensionalversion and observe that ˜ u ( x , x , . . . , x d ) = ˜ u (2 D ) ( x , x ). This implies, d (cid:88) i =1 (˜ u ( x − e i ) + ˜ u ( x + e i ) − u ( x )) + η = (cid:88) i =1 (˜ u (2 D ) ( x − e i ) + ˜ u (2 D ) ( x + e i ) − u (2 D ) ( x )) + η = (cid:88) i =1 (˜ u (2 D ) ( x − e i ) + ˜ u (2 D ) ( x + e i ) − u (2 D ) ( x )) + η (2 D ) + 2 · ( d − η (cid:48) (2 D ) + 2 · ( d − η (cid:48) . We conclude by observing η (cid:48) is box-crossing as 3 · δ + η (cid:48) ≥ (2 d −
2) and every layer in thebox B ( d ) := { x ∈ Z d : 0 ≤ x ≤ } , contains at least one site with (2 d −
1) chips. Indeed, forall (3 , , x d − ) ∈ B ( d ) , η (cid:48) (3 , , x d − ) = η (cid:48) (2 D ) (3 ,
3) + 2 · ( d −
2) = (2 d − , and every layer in η (cid:48) (2 D ) has at least one site with 3 chips. Step 3: Proof of (50) . Let v (2 D ) t be the parallel toppling odometer for η (2 D ) . By (56) itsuffices to show that(60) v t ( x , x , . . . , x d ) ≤ v (2 D ) t ( x , x ) . This is a consequence of the parabolic least action principle (Lemma 2.3 in [BR20]) but weprovide a self-contained proof here:Suppose (60) holds at time t and we want to show it holds at ( t + 1). Let x be given. If v t ( x ) < v (2 D ) t ( x , x ) we are done. Hence, we may suppose v t ( x ) = v (2 D ) t ( x , x ) and∆ v t ( x ) + η = d (cid:88) i =1 ( v t ( x + e i ) + v t ( x − e i ) − v t ( x )) + η ≥ d. By (60) at time t , this implies (cid:88) i =1 ( v (2 D ) t (( x , x ) + e i ) + v (2 D ) t (( x , x ) − e i ) − v (2 D ) t ( x , x )) + η ≥ d, nd by (58), ∆ v (2 D ) t ( x , x ) + η (2 D ) ( x , x ) ≥ d − · ( d −
2) = 4 , completing the proof. 7. A criteria for exploding
Let (Ω , F , P ) be a probability space of sandpile backgrounds defined in Section 5. For afinite domain V ⊂ Z d , we say η : V → Z is recurrent if the firing of ∂V causes every sitein V to eventually topple. Specifically, the V c -frozen parallel toppling odometer for initialconditions w = 1 { x ∈ ∂V } , s (cid:48) = η , is eventually is 1 on V : w t ( V ) = 1 for t ≥ | V | . Wesay η : Z d → Z is recurrent if its restriction to V is recurrent for every finite V ⊂ Z d .The measure P is recurrent if P ( η is recurrent) = 1. The arguments in Section 5 imply thefollowing criteria. Proposition 7.1.
If Hypotheses 1 and 2 are satisfied and P is recurrent then P is explosive. In this section, we use Proposition 7.1 to prove Theorem 1.4. Let P denote a productmeasure with P ( η ≥ d ) = 1 and P ( η = 2 d − >
0. We show that P is recurrent andbox-crossing. Both arguments require a form of dimensional reduction, which we record inthe first subsection. Also, by monotonicity, we may assume η → { d, d − } .7.1. Dimensional reduction.
Let Q ( d ) n := { x ∈ Z d : 1 ≤ x ≤ n } and denote each of the(internal) 2 d faces of Q ( d ) n as(61) F i ( Q ( d ) n ) := { x ∈ Q ( d ) n : x i = 1 }F d + i ( Q ( d ) n ) := { x ∈ Q ( d ) n : x i = n } . We show that after firing the outer boundary of F i ( Q ( d ) n ), the sandpile dynamics on F i ( Q ( d ) n )can be coupled with ( d − Q ( d − n .Specifically, fix i = 1, and for each x ∈ F ( Q ( d ) n ) write x = (1 , x d − ). Let the initial d -dimensional background be given, η ( d ) : Q ( d ) n → Z . Let s ( d − t , u ( d − t , and s ( d ) t , u ( d ) t be thesequence of ( Q ( d ) n ) c , ( Q ( d − n ) c frozen toppling processes with initial conditions,(62) u ( d )0 (0 , x d − ) = 1 u ( d − ( x d − ) = u ( d )0 (1 , x d − ) s ( d )0 (1 , x d − ) = ∆ u ( d )0 (1 , x d − ) + η ( d ) (1 , x d − ) s ( d − ( x d − ) = η ( d ) ( x d − ) − η ( d − ( x d − )and constraints,(63) u ( d ) t ( x , x d − ) = 0 for all x > u ( d ) t (1 , x d − ) ≤ u ( d − t ( x d − ) ≤ , for all t ≥
0. We prove the following by an induction on time. Note that a symmetric resultholds for every face. roposition 7.2. For all t ≥ and (1 , x d − ) ∈ F ( Q ( d ) n ) , (64) u ( d ) t (1 , x d − ) = u ( d − t ( x d − ) and (65) s ( d ) t (1 , x d − ) = s ( d − t ( x d − ) + 2 if u ( d ) t (1 , x d − ) = 0 . Proof.
For all t ≥
0, if u ( d ) t (1 , x d − ) = 0 and u ( d ) t (1 , y d − ) = u ( d − t ( y d − ) for all y d − , s ( d ) t (1 , x d − )= η ( d ) (1 , x d − )+ (cid:16) − u ( d ) t (1 , x d − ) + u ( d ) t (0 , x d − ) + u ( d ) t (2 , x d − ) (cid:17) + d (cid:88) i =2 (cid:16) u ( d ) t ((1 , x d − ) + e i ) + u ( d ) t ((1 , x d − ) − e i ) − u ( d ) t (1 , x d − ) (cid:17) = η ( d − ( x d − ) + 2+ d − (cid:88) i =1 (cid:16) u ( d − t ( x d − + e i ) + u ( d − t ( x d − − e i ) − u ( d − t ( x d − ) (cid:17) = s ( d − t ( x d − ) + 2 . Therefore, we may begin the induction and suppose (64) and (65) hold at time t . If s ( d − t ( x d − ) ≥ · ( d −
1) = (2 d − s ( d ) t (1 , x d − ) = 2 d . The other direction is identical,showing u ( d ) t +1 (1 , x d − ) = u ( d − t +1 ( x d − ) = 1 in this case. (cid:3) P is recurrent.
By monotonicity of recurrence, it suffices to prove the following.
Proposition 7.3.
For every d ≥ , η : Z d → { d } is recurrent.Proof. By consistency of recurrence, it suffices to show this for domains which are cubes (seefor example Remark 3.2.1 in [Red05]). Write Q ( d ) n for a cube of side length n in Z d . Weinduct on dimension, then cube side length. The base case for dimension d = 1 is immediate.Moreover, the base case n = 2 is also immediate for every dimension. It remains to check η : Q ( d ) n → { d } is recurrent given η : Q ( d ) n − → { d } is recurrent.We decompose the cube into its faces and an inner cube,(66) Q ( d ) n = Q ( d ) n − ∪ d (cid:91) i =1 F i ( Q ( d ) n ) . By the inductive hypotheses on n , once every external face of Q ( d ) n − is toppled, every sitein Q ( d ) n − eventually fires. Therefore, by (66), it suffices to check that every site in F i ( Q ( d ) n )fires after the boundary of Q ( d ) n fires. This, however, is a consequence of the inductivehypothesis on dimension and Proposition 7.2, any site in F i ( Q ( d ) n ) which topples for the( d − d -dimensions. (cid:3) Remark 5.
The argument given here is similar to the proof of Lemma 3.1 in [Sch92] . .3. P is box-crossing.
A coupling between the sandpile and bootstrap percolation hasbeen observed before [FLP10]. Bootstrap percolation is a cellular automata on Z d witha random initial state and a deterministic update rule. Every site x ∈ Z d starts off as infected independently at random with probability p . Infected sites remain infected and ifan uninfected site contains at least d neighbors which are infected, it becomes infected.These dynamics exactly match parallel toppling for a background η (cid:48) : Z d → { d, d } wheresites are constrained to topple at most once. Infected sites are those which have toppled,and sites with 2 d chips start as infected. Indeed, any site beginning with d chips topples ifand only if it has at least d neighbors which have toppled.Our proof that P is box-crossing uses this coupling together with a large deviation resultof Schonmann. Borrowing the terminology of Schonmann, we say a cube Q ( d ) n ⊂ Z d is η (cid:48) − internally spanned if the ( Q ( d ) n ) c -frozen parallel toppling procedure with u = 0, s (cid:48) = η (cid:48) ,and sites constrained to topple at most once, concludes with every site in Q ( d ) n toppling. Proposition 7.4 ([Sch92]) . Let η (cid:48) ( x ) := (cid:40) d with probability pd otherwise.There are constants c, C depending only on dimension and p > so that (67) P ( Q ( d ) n is η (cid:48) -internally spanned ) ≥ − c exp( − Cn ) . We use Proposition 7.4 to show P is box-crossing. Proposition 7.5.
In all dimensions, P is box-crossingProof. The claim is immediate in dimension one. Let ( d + 1) ≥ n ≥ Q ( d +1) n = n (cid:91) i =1 L i , where L i := { x ∈ Q ( d +1) n : x = ( i, x d ) } . The projection of each layer to a d -dimensional boxis L ( d ) i . Let Ω (cid:48) i := { η : L ( d ) i is η (cid:48) -internally spanned where η (cid:48) ( x d ) := η ( i, x d ) − } and let Ω (cid:48) := ∩ ni =1 Ω i . By definition of being internally spanned and Proposition 7.2, if η ∈ Ω (cid:48) , then Q ( d +1) n ( η ) canbe crossed in direction e . We conclude by symmetry and Proposition 7.4. (cid:3) References [AC18] Scott Armstrong and Pierre Cardaliaguet,
Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions , Journal of the European Mathematical Society (2018), no. 4, 797–864.[AM20] Ian Alevy and Sevak Mkrtchyan, The limit shape of the leaky Abelian sandpile model , arXivpreprint arXiv:2010.01946 (2020). AMP02] Oswaldo SM Alves, Fabio P Machado, and S Yu Popov,
The shape theorem for the frog model ,The Annals of Applied Probability (2002), no. 2, 533–546.[AP96] Peter Antal and Agoston Pisztora, On the chemical distance for supercritical Bernoulli percola-tion , The Annals of Probability (1996), 1036–1048.[AS19] Hayk Aleksanyan and Henrik Shahgholian,
Discrete balayage and boundary sandpile , Journald’Analyse Mathematique (2019), no. 1, 361–403.[BG07] L´aszl´o Babai and Igor Gorodezky,
Sandpile transience on the grid is polynomially bounded , Pro-ceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, 2007, pp. 627–636.[BR19] Ahmed Bou-Rabee,
Convergence of the random Abelian sandpile , arXiv preprintarXiv:1909.07849 (2019).[BR20] ,
Dynamic dimensional reduction in the Abelian sandpile , arXiv preprint arXiv:2009.05968(2020).[BS98] Guy Barles and Panagiotis E Souganidis,
A new approach to front propagation problems: theoryand applications , Archive for Rational Mechanics and Analysis (1998), no. 3, 237–296.[CD81] J Theodore Cox and Richard Durrett,
Some limit theorems for percolation processes with neces-sary and sufficient conditions , The Annals of Probability (1981), 583–603.[CKF20] Joe Chen and Jonah Kudler-Flam,
Laplacian growth and sandpiles on the sierpi´nski gasket: limitshape universality and exact solutions , Annales de l’Institut Henri Poincar´e D (2020), no. 4.[CM14] LA Caffarelli and R Monneau, Counter-example in three dimension and homogenization of geo-metric motions in two dimension , Archive for Rational Mechanics and Analysis (2014), no. 2,503–574.[DF91] Persi Diaconis and William Fulton,
A growth model, a game, an algebra, Lagrange inversion,and characteristic classes , Rend. Sem. Mat. Univ. Pol. Torino (1991), no. 1, 95–119.[Dha99] Deepak Dhar, The Abelian sandpile and related models , Physica A: Statistical Mechanics and itsApplications (1999), no. 1-4, 4–25.[DL81] Richard Durrett and Thomas M Liggett,
The shape of the limit set in Richardson’s growth model ,The Annals of Probability (1981), no. 2, 186–193.[DS13] Deepak Dhar and Tridib Sadhu, A sandpile model for proportionate growth , Journal of StatisticalMechanics: Theory and Experiment (2013), no. 11, P11006.[FdBR08] Anne Fey-den Boer and Frank Redig,
Limiting shapes for deterministic centrally seeded growthmodels , Journal of Statistical Physics (2008), no. 3, 579.[Fel19] William M Feldman,
Mean curvature flow with positive random forcing in 2-d , arXiv preprintarXiv:1911.00488 (2019).[FL11] Anne Fey and Haiyan Liu,
Limiting shapes for a non-abelian sandpile growth model and relatedcellular automata. , Journal of Cellular Automata (2011).[FLP10] Anne Fey, Lionel Levine, and Yuval Peres, Growth rates and explosions in sandpiles , Journal ofstatistical physics (2010), no. 1-3, 143–159.[FMR +
09] Anne Fey, Ronald Meester, Frank Redig, et al.,
Stabilizability and percolation in the infinitevolume sandpile model , The Annals of Probability (2009), no. 2, 654–675.[FR05] Anne Fey and Frank Redig, Organized versus self-organized criticality in the Abelian sandpilemodel , Markov Processes Relat. Fields (2005), 425–442.[GG93] Janko Gravner and David Griffeath, Threshold growth dynamics , Transactions of the AmericanMathematical society (1993), 837–870.[GG96] ,
First passage times for threshold growth dynamics on Z2 , The Annals of Probability(1996), 1752–1778.[GG98] ,
Cellular automaton growth on Z2: theorems, examples, and problems , Advances in Ap-plied Mathematics (1998), no. 2, 241–304.[GG06] , Random growth models with polygonal shapes , The Annals of Probability (2006), 181–218.[GM04] Olivier Garet and R´egine Marchand,
Asymptotic shape for the chemical distance and first-passagepercolation on the infinite bernoulli cluster , ESAIM: Probability and Statistics (2004), 169–199. GM07] ,
Large deviations for the chemical distance in supercritical Bernoulli percolation , TheAnnals of Probability (2007), no. 3, 833–866.[GM12] , Asymptotic shape for the contact process in random environment , The Annals of AppliedProbability (2012), no. 4, 1362–1410.[GMP21] Eric Goles, Pedro Montealegre, and K´evin Perrot, Freezing sandpiles and Boolean threshold net-works: Equivalence and complexity , Advances in Applied Mathematics (2021), 102161.[Gri13] Geoffrey R Grimmett,
Percolation , vol. 321, Springer Science & Business Media, 2013.[HJL19] Robert D Hough, Daniel C Jerison, and Lionel Levine,
Sandpiles on the square lattice , Commu-nications in Mathematical Physics (2019), no. 1, 33–87.[HLM +
08] Alexander E Holroyd, Lionel Levine, Karola M´esz´aros, Yuyal Peres, James Propp, and David BWilson,
Chip-firing and rotor-routing on directed graphs , In and out of equilibrium 2, Springer,2008, pp. 331–364.[HS19] Robert Hough and Hyojeong Son,
Cut-off for sandpiles on tiling graphs , arXiv preprintarXiv:1902.04174 (2019).[IPS99] Hitoshi Ishii, Gabriel E Pires, and Panagiotis E Souganidis,
Threshold dynamics type approxi-mation schemes for propagating fronts , Journal of the Mathematical Society of Japan (1999),no. 2, 267–308.[J´ar18] Antal A J´arai, Sandpile models , Probability Surveys (2018), 243–306.[Kli18] Caroline J Klivans, The mathematics of chip-firing , CRC Press, 2018.[KS08] Harry Kesten and Vladas Sidoravicius,
A shape theorem for the spread of an infection , Annals ofmathematics (2008), 701–766.[LBR02] Yvan Le Borgne and Dominique Rossin,
On the identity of the sandpile group , Discrete mathe-matics (2002), no. 3, 775–790.[LKG90] SH Liu, Theodore Kaplan, and LJ Gray,
Geometry and dynamics of deterministic sand piles ,Physical Review A (1990), no. 6, 3207.[LP09] Lionel Levine and Yuval Peres, Strong spherical asymptotics for rotor-router aggregation and thedivisible sandpile , Potential Analysis (2009), no. 1, 1.[LP10] Lionel Levine and James Propp, What is... a sandpile , Notices Amer. Math. Soc, 2010.[LP17] Lionel Levine and Yuval Peres,
Laplacian growth, sandpiles, and scaling limits , Bulletin of theAmerican Mathematical Society (2017), no. 3, 355–382.[LPS16] Lionel Levine, Wesley Pegden, and Charles K Smart, Apollonian structure in the Abelian sandpile ,Geometric and functional analysis (2016), no. 1, 306–336.[LPS17] , The Apollonian structure of integer superharmonic matrices , Annals of Mathematics(2017), 1–67.[LS19] Moritz Lang and Mikhail Shkolnikov,
Harmonic dynamics of the Abelian sandpile , Proceedingsof the National Academy of Sciences (2019), no. 8, 2821–2830.[LSS97] Thomas M Liggett, Roberto H Schonmann, and Alan M Stacey,
Domination by product measures ,The Annals of Probability (1997), no. 1, 71–95.[LZ19] Jessica Lin and Andrej Zlatoˇs, Stochastic homogenization for reaction–diffusion equations ,Archive for Rational Mechanics and Analysis (2019), no. 2, 813–871.[Mar02] R´egine Marchand,
Strict inequalities for the time constant in first passage percolation , The Annalsof Applied Probability (2002), no. 3, 1001–1038.[Mat08] P Mathieu, Quenched invariance principles for random walks with random conductances , Journalof Statistical Physics (2008), no. 5, 1025–1046.[Mel20] Andrew Melchionna,
The sandpile identity element on an ellipse , arXiv preprint arXiv:2007.05792(2020).[Ost03] Srdjan Ostojic,
Patterns formed by addition of grains to only one site of an Abelian sandpile ,Physica A: Statistical Mechanics and its Applications (2003), no. 1-2, 187–199.[Pao13] Guglielmo Paoletti,
Deterministic Abelian sandpile models and patterns , Springer Science & Busi-ness Media, 2013.[PS13] Wesley Pegden and Charles K Smart,
Convergence of the Abelian sandpile , Duke mathematicaljournal (2013), no. 4, 627–642. PS20] ,
Stability of patterns in the Abelian sandpile , Annales Henri Poincar´e, vol. 21, Springer,2020, pp. 1383–1399.[PW85] Norman H Packard and Stephen Wolfram,
Two-dimensional cellular automata , Journal of Sta-tistical physics (1985), no. 5-6, 901–946.[Red05] Frank Redig, Mathematical aspects of the Abelian sandpile model , Les Houches lecture notes (2005), 657–659.[Sch92] Roberto H Schonmann, On the behavior of some cellular automata related to bootstrap percolation ,The Annals of Probability (1992), 174–193.[Wil78] Stephen J Willson,
On convergence of configurations , Discrete Mathematics (1978), no. 3,279–300.[Xin09] Jack Xin, An introduction to fronts in random media , vol. 5, Springer Science & Business Media,2009.[ZZ20] Yuming Paul Zhang and Andrej Zlatos,
Long time dynamics for combustion in random media ,arXiv preprint arXiv:2008.02391 (2020).,arXiv preprint arXiv:2008.02391 (2020).