Optimal local interventions in the two-dimensional Abelian sandpile model

O P T I M A L L O C A L I N T E R V E N T I O N S I N T H E T W O - D I M E N S I O NA L A B E L I A N S A N D P I L E M O D E L A P R E P R I N T Maike C. de Jongh a m.c.dejongh@utwente.nl Richard J. Boucherie a r.j.boucherie@utwente.nl a Department of Applied Mathematics Univ ersity of T wente P .O. Box 217, NL-7500 AE Enschede b Centrum W iskunde & Informatica P .O. Box 94079, NL-1090 GB Amsterdam M.N.M. van Lieshout b,a m.n.m.van.lieshout@cwi.nl March 26, 2026 A B S T R AC T The Abelian sandpile model serves as a canonical e xample of self-or ganized criticality . This critical behavior manifests itself through large cascading ev ents triggered by small perturbations. Such large-scale e v ents, kno wn as a v alanches, are often re garded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Moti v ated by this perspecti v e, we study strate gies to reduce av alanche sizes. W e provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. W e first de v elop and formalize an extended method to compute the expected size of an av alanche originating from a connected component of critical vertices, i.e., v ertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest a v alanche sizes and increasing the number of mitigated av alanches. Keyw ords: Abelian sandpile model, self-organized criticality , av alanche dynamics, a v alanche size, optimal control 1 Introduction The Abelian sandpile model was proposed by Bak et al. [2 , 3] to study systems dri v en by small external forces that organize themselv es by means of energy-dissipating a v alanches. In this model, grains of sand are successively dropped on randomly chosen vertices of a graph. Whenev er a verte x accumulates too many grains, it topples and transfers them to its neighbours, which may trigger a cascade in volving man y other v ertices. An interesting aspect of the model is that critical behaviour emer ges without the need to tune an y external parameter . This phenomenon, known as self-organized criticality , manifests itself in a range of real-world systems, including earthquakes [ 20 ], forest fires [ 12 ], financial markets [ 4 ], and the ev olution of species [ 1 ]. In many of these applications, lar ge a valanches may have destructiv e consequences. For this reason, a growing body of literature has focused on the use of intervention heuristics for controlling the size of av alanches. Sev eral studies pro vide an empirical in vestig ation of av alanche propagation in cases where an external controller can influence the landing position of the ne xt sand grain [ 5 , 13 , 16 , 17 , 21 ]. In contrast, Qi and Pfenninger [18] point out that in man y cases the origin of an a v alanche is uncontrollable and therefore focus on control strategies that intervene once an av alanche has started. In particular , they study a situation in which vertices can be prev ented from toppling, even when the number of sand grains e xceeds their capacity . Other intervention strategies Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T that ha ve been e xplored mitigate the risk of large-scale a v alanches by deliberately triggering smaller ones in a controlled way [6, 7, 14]. Another line of research explores the impact of sand grain absorption on a valanche sizes [19]. In contrast to the studies mentioned above, which provide numerical results on control strategies, we present a rigorous analysis of the impact of strate gic interventions. In particular , we consider a scenario in which an e xternal controller can remov e sand grains from selected vertices. In our analysis, we focus on connected components of vertices that hold the maximum amount of sand grains, called gener ators , and in vestigate ho w removing sand grains af fects av alanches originating from these generators. Our contribution is twofold. First, we formalize and extend the method proposed by Dorso and Dadamia [11] for computing the expected size of an a v alanche starting from a giv en generator . W e identify the cases in which their method does not apply and propose appropriate modifications. In addition, we provide a formal justification that the resulting algorithm indeed yields the correct result for all possible generators. Second, we consider generators in the shape of squares and provide rigorous results on the ef fect of removing sand grains, identifying optimal targets for interv entions. The motiv ation for analyzing such generators is that, when surrounded by noncritical vertices, they allo w for a local analysis. The boundary of noncritical vertices pre vents the propagation of av alanches beyond the square. Consequently , the internal av alanche dynamics are unaffected by the remainder of the configuration. This enables us to characterize the structure of av alanches originating from the square. Moreover , the size of av alanches in this setting provides a natural upper bound for a valanches in an y square-shaped region surrounded by noncritical vertices. The structure of this paper is as follows. In Section 2, we gi ve a definition of the Abelian sandpile model. Section 3 presents our adaptation of the method of Dorso and Dadamia [11] for computing the expected a valanche size correspond- ing to a particular generator . This method makes use of the decomposition of an a valanche into a sequence of w av es, introduced by Ivashk e vich et al. [15] . W e explain ho w our extension ensures that the method becomes applicable in all cases and we provide a proof of correctness. In Section 4 we explore the impact of strategic interv entions on expected av alanche sizes. W e deriv e rigorous results on the impact of removing sand grains from vertices in square-shaped generators by studying the structure of the wav es that constitute an av alanche. Also, we identify the set of vertices that are the optimal targets for the remo val of sand grains in this case. Some proofs are deferred to the Appendix. Finally , we reflect on our results in Section 5 and indicate some directions for future research. 2 Definition of the Abelian sandpile model W e consider the Abelian sandpile model on a graph that is constructed from a finite, two-dimensional box V = { 1 , . . . , L } 2 ∩ Z 2 in the following w ay . First, all vertices in the set V c = Z 2 \ V are identified with a single verte x s , called the sink . Next, all self-loops at s are remov ed. W e refer to the resulting graph as the wired lattice of size L . By construction, the corner vertices of the box are connected to the sink by two edges, whereas each of the remaining boundary vertices has exactly one edge connecting it to the sink. A sandpile , denoted by a map η : V → Z , is a configuration of indistinguishable particles, called sand grains , on this wired lattice. Let the set of all sandpiles on V be denoted by X . W e define the mass m ( η ) of a sandpile η ∈ X as the total number of sand grains in the sandpile η . Furthermore, we call a v ertex v ∈ V stable in a sandpile η ∈ X if 0 ≤ η ( v ) < 4 . If all vertices v ∈ V are stable in η , then η is called a stable sandpile . W e denote the set of stable sandpiles on V by Ω . A vertex v ∈ V is critical in η ∈ Ω if it contains three sand grains, i.e., if η ( v ) = 3 . A verte x v ∈ V that satisfies η ( v ) < 3 is called noncritical in η . The sandpile Markov c hain evolv es according to the following dynamics: each time step, a new sand grain is dropped on a verte x v ∈ V selected uniformly at random. If this ne w sand grain causes the vertex v to become unstable, then each of the sand grains located at v will be sent to one of its horizontal and vertical neighbours, an operation called toppling . Unstable vertices no w continue to topple until a stable sandpile is reached, a process called r elaxation . Sand grains that end up in the sink during the relaxation are discarded. T o formalize these dynamics, we label the vertices as V = { v 1 , v 2 , . . . , v L 2 } and write v i ∼ v j if vertices v i and v j are neighbours. Then, we define the L 2 × L 2 matrix ∆ , of which the i th row/column corresponds to v ertex v i , as ∆ ij =    4 , if i = j, − 1 , if i  = j, v i ∼ v j , 0 , otherwise. W e now define se veral operators on sandpiles η ∈ X . First, let α i : X → X denote an addition operator , which adds a sand grain to verte x v i ∈ V , i.e., ( α i η )( v j ) =  η ( v j ) + 1 , if i = j, η ( v j ) , otherwise . 2 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Second, we define a remov al operator γ i : X → X , which remov es all sand grains from verte x v i ∈ V , i.e., ( γ i η )( v j ) =  0 , if i = j, η ( v j ) , otherwise. Third, let τ i : X → X denote a toppling operator , which represents the procedure of toppling a vertex v i , i.e., ( τ i η )( v j ) = η ( v j ) − ∆ ij . W e call a toppling of a vertex v i ∈ V le gal with respect to a sandpile η ∈ X if η ( v i ) > 3 . Dhar [10] showed that toppling operators commute. It follows from similar reasoning that toppling operators commute with addition operators. Fourth, we define an identity operator I : X → X , which maps a sandpile configuration to itself, i.e., I η = η . Finally , let R denote a relaxation operator , which takes as input a sandpile η ∈ X and outputs the stable sandpile η ′ = Rη that results from toppling unstable v ertices until a stable sandpile is reached. As shown by Dhar [10] , each sequence of topplings of unstable vertices is finite and results in the same unique stable sandpile, which ensures that the operator R is well-defined. Hence, Rη = τ i k τ i k − 1 · · · τ i 1 η , where ( τ i 1 , τ i 2 , . . . , τ i k ) , k = 0 , 1 , . . . , is any sequence of legal topplings such that τ i k τ i k − 1 · · · τ i 1 η is a stable sandpile configuration. The dynamics of the sandpile Marko v chain can now be expressed in terms of the following transition probability kernel: P ( η , η ′ ) = 1 L 2 L 2 X i =1 1 [ η ′ = Rα i η ] , η , η ′ ∈ Ω . Our main variable of interest is the avalanche size . Dhar [10] showed that the number of times a vertex v j ∈ V topples during the av alanche induced by dropping a grain at verte x v i ∈ V onto a sandpile η is the same for each sequence of legal topplings leading from α i η to Rα i η . Let this number be denoted by n ( v i , v j ; η ) . W e define the size x ( η , v i ) of the avalanche caused by dropping a sand grain at v ertex v i ∈ V onto sandpile η as the total number of topplings that occur during the relaxation of the pile α i η , i.e., x ( η , v i ) := L 2 X j =1 n ( v i , v j ; η ) . Let the (random) size of the next a valanche that will occur gi ven a sandpile η ∈ Ω be denoted by X ( η ) . Note that P ( X ( η ) = k ) = 1 L 2 L 2 X i =1 1 [ x ( η , v i ) = k ] , k = 0 , 1 , . . . . In this work, we are particularly interested in the av alanche size conditional on the ev ent that the next sand grain lands in a subset A ⊆ V . 3 A valanche de velopment In order to analyze the de velopment of a valanches and in vestigate pre vention strategies, we b uild on the approach for computing the expected a valanche size proposed by Dorso and Dadamia [11] . Their method relies on the representation of an a valanche as a sequence of wa ves, a concept introduced by Iv ashkevich et al. [15 , p. 350 ] . Such a representation is established by choosing a specific type of sequence of legal topplings to relax an unstable sandpile, namely as follows. Suppose that η ∈ X is a sandpile that satisfies η ( v i ) = 3 for some v i ∈ V and η ( v j ) ≤ 3 for all v j ∈ V , v j  = v i . W e no w drop a ne w sand grain at vertex v i and choose any sequence of legal topplings that begins by toppling verte x v i and then proceeds to topple unstable vertices in the set V \ { v i } until all vertices are stable, e xcept possibly for v i . Hence, let ( τ i 1 , τ i 2 , . . . , τ i k ) denote a sequence of topplings such that i 1 = i and i j  = i for all j = 2 , . . . , k , the toppling τ i j , j = 2 , . . . , k , is leg al for the sandpile τ i j − 1 · · · τ i 1 α i η and τ i k · · · τ i 1 α i η ( v j ) ≤ 3 for all j  = i . Ivashk e vich et al. [15] show that { i 1 , i 2 , . . . , i k } is a set of k unique vertices. Hence, no vertex topples twice during the sequence of topplings ( τ i 1 , τ i 2 , . . . , τ i k ) . The set of vertices W i 1 ( η ) := { i 1 , i 2 , . . . , i k } is called the first wave of the a valanche generated by dropping a sand grain at vertex v i onto η . Let R η i 1 denote the operator that yields the sandpile that results from toppling the vertices in the first wa ve, i.e., R η i 1 α i η := τ i k · · · τ i 1 α i η . 3 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Figure 1: Example to which the method of Dorso and Dadamia [11] does not apply . The generator outlined in bold splits into two components after the first wa ve, each producing second wa ves of dif ferent sizes. If R η i 1 α i η ( v i ) > 3 , then relaxing this sandpile requires toppling vertex v i a second time. Repeating the same procedure yields the second wave W i 2 ( η ) and corresponding operator R η i 2 . Since any sandpile can be stabilized through a finite number of topplings [ 10 ], continuing this process yields a finite number of wa ves W i 1 ( η ) , . . . , W in ( η ) and corresponding operators R η i 1 , . . . , R η in , resulting in a stable sandpile R η in · · · R η i 1 α i η . W e proceed to discuss the key ideas of the method proposed by Dorso and Dadamia [11] , identify its limitations, and dev elop an extension that works in full generality , along with a formal justification. Suppose that dropping a sand grain at a critical verte x v i ∈ V onto a stable sandpile η ∈ Ω yields an av alanche that can be represented as a sequence of n ( η , v i ) wa ves. Let w ij ( η ) denote the size of the j th wa ve, i.e., w ij ( η ) := | W ij ( η ) | , j = 1 , . . . , n ( η , v i ) . Note that the size of the av alanche generated by dropping a sand grain at v i onto η is given by x ( η , v i ) = n ( η ,v i ) X j =1 w ij ( η ) . (1) Let A ⊆ V denote the connected component of critical vertices in η that contains v i . W e refer to such a connected component as a gener ator . Note that each verte x in A will topple during the first wa ve, i.e., A ⊆ W i 1 ( η ) . T o see this, suppose that some vertex v ∈ A does not topple during the first wav e. Since η ( v ) = 3 , this implies that none of the neighbours of v topples during the first wav e. Continuing this argument and using the fact that v and v i are in the same connected component of critical vertices, this implies that v i does not topple in the first wave, which yields a contradiction. Dorso and Dadamia [11] claim that the first wa ve of the a valanche caused by dropping a sand grain at verte x v ∈ A is the same for each v ∈ A , a statement we formally prove in Lemma 3.1. Let W A ( η ) denote the first wa ve of an av alanche arising from dropping a sand grain on some v ertex v ∈ A and let w A ( η ) denote its size. These are guaranteed to be well-defined by Lemma 3.1. Now , let ( x 1 , . . . , x k ) be a permutation of the elements of W A ( η ) . The key idea underlying the method of Dorso and Dadamia [11] is the fact that toppling operators and addition operators commute and therefore τ x k · · · τ x 1 α i η = α i τ x k · · · τ x 1 η . They argue that it suf fices to study the sandpile configuration η A , defined as η A = τ x k · · · τ x 1 η . By the fact that topplings commute, η A does not depend on the specific choice of permutation and is therefore well-defined. Note that the topplings in the sequence ( τ x 1 , . . . , τ x k ) are not necessarily legal with respect to η and it is possible that there e xists a v ∈ V such that η A ( v ) < 0 . An implicit assumption in the approach of Dorso and Dadamia [11] is that the sizes of the second and higher order waves are equal as well for all vertices v ∈ A at which the next sand grain may land, allo wing for expressions such as [ 11 , expression (20)]. This is the case if the set { v ∈ A | η A ( v ) = 3 } is connected. In general, howe ver , this is not guaranteed. Figure 1 displays a sandpile η and generator A (outlined in bold in the left panel) for which the set { v ∈ A | η A ( v ) = 3 } splits into two components (shown in the right panel). If a sand grain lands on a vertex in the upper component, it produces a second wa ve of size 2, whereas hitting the lo wer component generates a second wav e of size 1. W e now propose a modification of the method of Dorso and Dadamia [11] , which applies in full generality , along with a formal justification. W e start by proving that the first wa ve of the a valanche produced by dropping a sand grain at a verte x v ∈ A is the same for each v ∈ A . 4 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Lemma 3.1. Given a stable sandpile configuration η ∈ Ω and a generator A = { v i 1 , v i 2 , . . . , v i k } ⊆ V , we have W i ℓ 1 ( η ) = W i m 1 ( η ) for all ℓ, m = 1 , . . . , k . Pr oof. Consider v i ℓ , v i m ∈ A . Let W i ℓ 1 ( η ) = A ∪ ˜ W i ℓ 1 ( η ) and W i m 1 ( η ) = A ∪ ˜ W i m 1 ( η ) , where ˜ W i ℓ 1 ( η ) , ˜ W i m 1 ( η ) ⊆ V \ A . Suppose first that ˜ W i ℓ 1 ( η ) = ∅ . W e show that this implies that ˜ W i m 1 ( η ) = ∅ . Since each verte x in A will topple exactly once during the first w ave and ˜ W i ℓ 1 ( η ) = ∅ [ 15 ], there exists a sequence of le gal topplings ( τ x 1 , τ x 2 , . . . , τ x k ) acting on α i ℓ η , where ( x 1 , . . . , x k ) is a permutation of the set { i 1 , . . . , i k } , such that η ′ = τ x k · · · τ x 1 α i ℓ η is a sandpile that satisfies η ′ ( v ) ≤ 3 for all v  = v i ℓ . Note that τ x k · · · τ x 1 α i ℓ η = α i ℓ τ x k · · · τ x 1 η , by the fact that toppling operators and addition operators commute. Since each vertex in A toppled exactly once, we ha ve η ′ ( v i ℓ ) ≤ 4 . Also, since A is a connected component there exists a sequence of legal topplings ( τ y 1 , τ y 2 , . . . , τ y k ) acting on α i m η , where ( y 1 , . . . , y k ) is again a permutation of the set { i 1 , . . . , i k } . Let ˜ η = τ y k · · · τ y 1 α i m η = α i m τ y k · · · τ y 1 η . Since topplings commute, we hav e τ x k · · · τ x 1 η = τ y k · · · τ y 1 η . It now follows from the fact that η ′ satisfies η ′ ( v ) ≤ 3 for all v  = v i ℓ and η ′ ( v i ℓ ) ≤ 4 that ˜ η ( v ) ≤ 3 for all v  = v i m . Thus, ˜ W i m 1 ( η ) = ∅ . Now , suppose that ˜ W i ℓ 1 ( η ) = { v j 1 , v j 2 , . . . , v j n } . Since A is a connected component of critical v ertices, there exists a sequence of legal topplings ( τ x 1 , τ x 2 , . . . , τ x k ) acting on α i ℓ η , where ( x 1 , . . . , x k ) is a permutation of the set { i 1 , . . . , i k } . Hence, there exists a sequence of leg al topplings ( τ x 1 , τ x 2 , . . . , τ x k , τ ˜ x 1 , τ ˜ x 2 , . . . , τ ˜ x n ) acting on α i ℓ η , where ( ˜ x 1 , ˜ x 2 , . . . , ˜ x n ) is a permutation of the set { j 1 , . . . , j n } , such that η ′ := τ ˜ x n · · · τ ˜ x 1 τ x k · · · τ x 1 α i ℓ η is a sandpile configuration that satisfies η ′ ( v ) ≤ 3 for all v  = v i ℓ . Since each verte x in W i ℓ 1 topples only once, we obtain η ′ ( v i ℓ ) ≤ 4 . Again using the fact that A is a connected component of critical vertices, we can construct a sequence of legal topplings ( τ y 1 , τ y 2 , . . . , τ y k ) acting on α i m η , where ( y 1 , . . . , y k ) is a permutation of the set { i 1 , . . . , i k } . Now , note that η ′ = τ ˜ x n · · · τ ˜ x 1 τ x k · · · τ x 1 α i ℓ η = τ ˜ x n · · · τ ˜ x 1 α i ℓ τ x k · · · τ x 1 η . By the f act that topplings commute, the sandpile configurations α i ℓ τ x k · · · τ x 1 η and α i m τ y k · · · τ y 1 η only dif fer at v ertices v i ℓ and v i m . Since { v j 1 , . . . , v j n } ⊆ V \ A and thus { v j 1 , . . . , v j n } ∩ { v i ℓ , v i m } = ∅ , it follo ws that ( τ ˜ x 1 , τ ˜ x 2 , . . . , τ ˜ x n ) is a sequence of legal topplings for the sandpile configuration α i m τ y k · · · τ y 1 η . Let ˜ η := τ ˜ x n · · · τ ˜ x 1 α i m τ y k · · · τ y 1 η = τ ˜ x n · · · τ ˜ x 1 τ y k · · · τ y 1 α i m η . Note that η ′ = α i ℓ τ ˜ x n · · · τ ˜ x 1 τ x k · · · τ x 1 η and ˜ η = α i m τ ˜ x n · · · τ ˜ x 1 τ y k · · · τ y 1 η . From the f acts that topplings commute, η ′ ( v ) ≤ 3 for all v  = v i ℓ and η ′ ( v i ℓ ) ≤ 4 , it now follows that ˜ η ( v ) ≤ 3 for all v  = v i m and ˜ η ( v i m ) ≤ 4 . Thus, ˜ W i m 1 ( η ) = ˜ W i ℓ 1 ( η ) . Giv en a generator A ⊆ V in a sandpile configuration η ∈ Ω , we proceed to establish a recursiv e expression for computing E [ X ( η ) | Y ∈ A ] , where Y denotes the random vertex at which the next sand grain lands. Let W A ( η ) denote the first wa ve of an av alanche arising from dropping a sand grain on some vertex v ∈ A and let w A ( η ) denote its size. These are guaranteed to be well-defined by Lemma 3.1. Now , let ( x 1 , . . . , x k ) be a permutation of the elements of W A ( η ) and let the sandpile configuration η A be defined as η A = τ x k · · · τ x 1 η . By the fact that topplings commute, η A does not depend on the specific choice of the permutation and is therefore well-defined. Note that the topplings in the sequence ( τ x 1 , . . . , τ x k ) are not necessarily legal with respect to η and it is possible that there exists a v ∈ V such that η A ( v ) < 0 . W e now define the set ˜ A as ˜ A = { v ∈ A | η A ( v ) < 3 } . Furthermore, if the set A ′ := { v ∈ A | η A ( v ) = 3 } is nonempty , let A 1 , . . . , A ℓ denote its connected components. Lemma 3.2 now pro vides a recursive e xpression for computing E [ X ( η ) | Y ∈ A ] . Lemma 3.2. Let η ∈ Ω denote a stable sandpile configur ation and let A ⊆ V denote a gener ator in η . Let Y denote the random verte x at which the ne xt sand grain lands. The expected value of the size of the ne xt avalanche corr esponding to A satisfies E [ X ( η ) | Y ∈ A ] =    w A ( η ) + ℓ P j =1 | A j | | A | E [ X ( η A ) | Y ∈ A j ] , if { v ∈ A | η A ( v ) = 3 }  = ∅ , w A ( η ) , otherwise . (2) Pr oof. Let ˜ H denote the subset of vertices in A that, when hit by a new sand grain, generate an avalanche consisting of a single wa ve. Let H ′ = A \ ˜ H . If H ′  = ∅ , let H 1 , . . . , H ℓ denote its connected components. By conditioning on Y , we obtain E [ X ( η ) | Y ∈ A ] = | ˜ H | | A | E [ X ( η ) | Y ∈ ˜ H ] + ℓ X j =1 | H j | | A | E [ X ( η ) | Y ∈ H j ] = | ˜ H | | A | w A ( η ) + ℓ X j =1 | H j | | A | E [ X ( η ) | Y ∈ H j ] , 5 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T if H ′  = ∅ and E [ X ( η ) | Y ∈ A ] = w A ( η ) , otherwise. Hence, to establish expression (2), it suf fices to sho w that ˜ H = ˜ A , H j = A j for all j = 1 , . . . , ℓ and E [ X ( η ) | Y ∈ A j ] = w A ( η ) + E [ X ( η A ) | Y ∈ A j ] . (3) W e start by showing that ˜ H = ˜ A . First, consider a vertex v i ∈ ˜ H . Let ( x 1 , . . . , x k ) be a permutation of W A ( η ) . Since an av alanche caused by dropping a sand grain at v i consists of only one wa ve, we hav e τ x k · · · τ x 1 α i η ( v i ) ≤ 3 . Since τ x k · · · τ x 1 α i η = α i τ x k · · · τ x 1 η , this implies that τ x k · · · τ x 1 η ( v i ) = η A ( v i ) < 3 . Hence, v i ∈ ˜ A . Now , suppose that v i ∈ ˜ A . This implies that η A ( v i ) < 3 . Therefore, we have τ x k · · · τ x 1 α i η ( v i ) ≤ 3 for each permutation ( x 1 , . . . , x k ) of W A ( η ) . It follows that an a valanche generated by dropping a sand grain at verte x v i consists of only a single wa ve, and thus v i ∈ ˜ H . Thus, we hav e ˜ H = ˜ A . W e proceed to prove that H j = A j for all j = 1 , . . . , ℓ . Let A ′ = S ℓ j =1 A j . Note that it suffices to show that H ′ = A ′ . First, consider v i ∈ H ′ . Let ( x 1 , . . . , x k ) be a permutation of W A ( η ) . Since the av alanche that is generated after a new sand grain drops at v i can be decomposed into more than one wa ve, we hav e τ x k · · · τ x 1 α i η ( v i ) > 3 . Since each verte x in W A topples only once, this implies τ x k · · · τ x 1 α i η ( v i ) = 4 . Hence, τ x k · · · τ x 1 η ( v i ) = η A ( v i ) = 3 . It follows that v i ∈ A ′ . Now , suppose that v i ∈ A ′ . Hence, η A ( v i ) = 3 . This implies that τ x k · · · τ x 1 α i η ( v i ) = 4 for each permutation ( x 1 , . . . , x k ) of W A ( η ) . Therefore, if a new sand grain lands at v i , the av alanche that is generated consists of multiple wa ves. Hence, v i ∈ H ′ . Thus, we obtain H ′ = A ′ . Finally , we sho w the validity of expression (3). Through further conditioning on Y , we obtain E [ X ( η ) | Y ∈ A j ] = 1 | A j | X v i ∈ A j x ( η , v i ) . Suppose that the av alanche caused by dropping a sand grain at vertex v i consists of k i wa ves. In voking expression (1), we obtain E [ X ( η ) | Y ∈ A j ] = 1 | A j | X v i ∈ A j " w A ( η ) + k i X n =2 w in ( η ) # . Let ( x 1 , . . . , x k ) denote a permutation of W A ( η ) . Since τ x k · · · τ x 1 α i η = α i τ x k · · · τ x 1 η = α i η A , it follows that w in ( η ) = w i ( n − 1) ( η A ) for all n = 2 , . . . , k i . Hence, we obtain E [ X ( η ) | Y ∈ A j ] = 1 | A j | X v i ∈ A j " w A ( η ) + k i − 1 X n =1 w in ( η A ) # = w A ( η ) + 1 | A j | X v i ∈ A j k i − 1 X n =1 w in ( η A ) = w A ( η ) + 1 | A j | X v i ∈ A j x ( η A , v i ) = w A ( η ) + E [ X ( η A ) | Y ∈ A j ] . This concludes the proof. Based on expression (2), we recursi vely define the depth ρ ( η , A ) of a generator A in η as ρ ( η , A ) = ( 1 + max j =1 ,...,ℓ { ρ ( η A , A j ) } , if { v ∈ A | η A ( v ) = 3 }  = ∅ , 1 , otherwise. (4) Here, recall that the sets A j , j = 1 , . . . , ℓ , are the connected components of { v ∈ A | η A ( v ) = 3 }  = ∅ . The depth of a generator A corresponds to the maximum number of topplings at a single vertex v ∈ A during an av alanche originating from this generator . Algorithm 1 now pro vides a method to compute the e xpected av alanche size based on the recursi ve e xpression presented in Lemma 3.2. Theorem 3.3 offers theoretical justification for the algorithm. Theorem 3.3. Given a stable sandpile configuration η ∈ Ω and a generator A = { v i 1 , v i 2 , . . . , v i k } ⊆ V , Algorithm 1 terminates and yields E [ X ( η ) | Y ∈ A ] . Pr oof. First, we show that the quantity w ′ in Algorithm 1 equals w A ( η ) , the size of the first wa ve. T o this end, we start by sho wing that each vertex v ∈ V is selected as the tail of ne w outgoing edges in steps 2-4 at most once. This 6 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Algorithm 1: Analysis of av alanche dev elopment through decomposition into wa ves. Input: A sandpile configuration η ∈ Ω on the L × L wired lattice V and a generator A ⊆ V . 1 . Initialize a directed graph ( V , E ) with E = ∅ . Let indeg ( v ) and outdeg ( v ) , v ∈ V , denote the indegree and the outdegree of v ertex v in this graph. 2 . F or each v ∈ A , add an edge to E from v to each of the neighbours (w .r .t. the lattice) of v ; 3 . Set w ′ = | A | ; 4 . while ther e is a v / ∈ A that satisfies η ( v ) + inde g ( v ) − outde g ( v ) ≥ 4 do Select an arbitrary vertex v / ∈ A that satisfies η ( v ) + indeg ( v ) − outdeg ( v ) ≥ 4 , add edges to E from v to each of the neighbours (w .r .t. the lattice) of v and increment w ′ by 1; end 5 . F or each v ∈ V , set η ′ ( v ) = η ( v ) + indeg ( v ) − outdeg ( v ) ; 6 . a . if the set { v ∈ A | η ′ ( v ) = 3 } is nonempty , then Let a 1 , a 2 , . . . , a ℓ denote its connected components (w .r .t. the lattice). Let z i be the output of Algorithm 1 for the sandpile configuration η ′ and the generator a i , i = 1 , . . . , ℓ . Set w = w ′ + ℓ P i =1 | a i | | A | z i . end 6 . b else Set w = w ′ . end 7 . Output : W ave size w . is obviously true for each v ertex v ∈ A . Suppose that some v ertex v / ∈ A is selected twice in step 4. W ithout loss of generality , let v be the first vertex that is selected for the second time. At this point, we have outdeg ( v ) = 4 . Since η ( v ) < 4 , this implies that indeg ( v ) > 4 . Hence, some neighbour of v was selected more than once as well, which is a contradiction. This immediately yields that step 4 terminates, since the number of vertices is finite. Now , we prov e that a verte x v ∈ V has outgoing edges if and only if it is part of the first wave W A ( η ) . This naturally holds for all v ertices v ∈ A . No w , let v j 1 , v j 2 , . . . , v j m denote an arbitrary total sequence of vertices selected in step 4. Also, let ( x 1 , . . . , x k ) be a permutation of { i 1 , . . . , i k } . Observ e that at the start of step 4, the quantity η ( v ) + indeg ( v ) − outde g ( v ) for v / ∈ A is equiv alent to τ x k · · · τ x 1 η ( v ) . Hence, the fact that v j 1 is selected first in step 4 implies that τ j 1 is a legal toppling in the sandpile τ x k · · · τ x 1 η . Similarly , at the point that verte x v j i , i = 2 , . . . , m is selected in step 4, the quantity η ( v j i ) + indeg ( v j i ) − outdeg ( v j i ) is equi valent to the number of sand grains that vertex v j i holds in the sandpile configuration τ j i − 1 · · · τ j 1 τ x k · · · τ x 1 η . Hence, τ j i is a legal toppling in the configuration τ j i − 1 · · · τ j 1 τ x k · · · τ x 1 η . At the end of step 4, the quantity η ( v ) + indeg ( v ) − outdeg ( v ) is equal to τ j m · · · τ j 1 τ x k · · · τ x 1 η ( v ) for all v / ∈ A . Since at this point η ( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v / ∈ A , there are no legal topplings in the sandpile configuration τ j m · · · τ j 1 τ x k · · · τ x 1 η . Hence, we ha ve W A ( η ) = A ∪ { v j 1 , v j 2 , . . . , v j m } . Since this argument can be applied to each sequence of vertices selected in step 4, it follows immediately that any such sequence is a permutation of the set { v j 1 , v j 2 , . . . , v j m } . Hence, the quantity w ′ in Algorithm 1 equals w A ( η ) . W e proceed to sho w that η ′ = η A . Let { v j 1 , . . . , v j m } = W A ( η ) \ A and let ( x 1 , . . . , x k ) and ( y 1 , . . . , y m ) be permutations of { i 1 , . . . , i k } and { j 1 , . . . , j m } . It follo ws from the arguments above that at the start of step 5, the indegree of a v ertex v ∈ V equals the number of neighbours of v that topple in the first w av e and the outdegree of a verte x v ∈ V equals 0 if v / ∈ W A ( η ) and 4 otherwise. This implies that η ′ ( v ) = τ y m · · · τ y 1 τ x k · · · τ x 1 η ( v ) = η A ( v ) . It now follo ws that a i = A i for all i = 1 , . . . , ℓ . By the facts that w ′ = w A ( η ) and a i = A i for all i = 1 , . . . , ℓ , it follo ws that step 6 simply computes the recursion in expression (2). The f acts that each sandpile can be stabilized by means of a finite number of topplings [ 10 ] and w A ( η ) is al ways positi ve ensures that the recursion in expression (2) has finite depth. This implies that Algorithm 1 terminates in a finite number of steps. 7 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T 4 Reducing a valanche sizes thr ough strategic interventions This section explores the impact of removing sand grains from critical vertices in a sandpile η ∈ Ω . W e define the stability level λ ( η, A ) of a set A ⊆ V in η as the quantity λ ( η , A ) = min v i ∈ A E [ X ( γ i η ) | Y ∈ A ] E [ X ( η ) | Y ∈ A ] , (5) where Y denotes the vertex at which the next sand grain lands. The stability level is a measure of the impact of the remov al of sand grains from a vertex in A on the expected av alanche size. W e call the set of vertices in A for which the minimum in expression (5) is attained the set of cornerstone vertices of A , denoted by B ∗ A ( η ) . W e consider square-shaped generators of size N × N . For such generators, av alanches exhibit a tractable structure, which enables an analytical computation of the expected avalanche size. Moreover , a valanches originating from the square do not propagate beyond its boundary , which allows for a local analysis of av alanche dynamics and control strategies. W e explicitly compute the expected av alanche size after interventions at different locations within the square and characterize the optimal targets for sand grain remo val. Giv en a set A ⊂ V that has the shape of an N × N square, where N ≥ 3 , let δ in ( A ) , C ( A ) , int ( A ) and δ out ( A ) denote the inner boundary , the corners, the interior and the outer boundary of A , i.e., δ in ( A ) = { v ∈ A | ∃ exactly one w / ∈ A such that v ∼ w } , C ( A ) = { v ∈ A | ∃ exactly tw o w / ∈ A such that v ∼ w } , int ( A ) = { v ∈ A | w ∈ A for all w ∈ V such that v ∼ w } , δ out ( A ) = { v / ∈ A | ∃ w ∈ A such that v ∼ w } . Additionally , we define the rings R 1 ( A ) , R 2 ( A ) , . . . , R ⌈ N/ 2 ⌉ ( A ) of A as R ⌈ N/ 2 ⌉ ( A ) = δ in ( A ) ∪ C ( A ) , R j ( A ) = δ in ( A \ ( ∪ ⌈ N/ 2 ⌉ k = j +1 R k ( A ))) ∪ C ( A \ ( ∪ ⌈ N/ 2 ⌉ k = j +1 R k ( A ))) , for j = 2 , . . . , ⌈ N / 2 ⌉ − 1 , R 1 ( A ) = A \ ( ∪ ⌈ N/ 2 ⌉ k =2 R k ( A )) . Let A ( N ) denote a generator that has the shape of an N × N square. Theorem 4.1 provides the expected a v alanche size corresponding to this type of generator . Theorem 4.2 giv es the depth of such a generator . Theorem 4.1. Consider a gener ator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e. The expected avalanche size corr esponding to this generator is given by E [ X ( η ) | Y ∈ A ( N ) ] = (3 N 4 + 15 N 3 + 20 N 2 − 8) / (30 N ) . (6) Pr oof. W e compute the expected av alanche size corresponding to the generator A ( N ) by induction over N . W e start by showing the v alidity of expression (6) for N = 1 and N = 2 . First, consider N = 1 . Let the unique verte x in A (1) be denoted by v i . T o find the expected a valanche size corresponding to A (1) , we use Algorithm 1. After initializing the directed graph ( V , E ) with E = ∅ , we add an edge to E from v i to each of its neighbours and set w ′ = 1 . Now , consider v ∈ V , v  = v i . If v is not a neighbour of v i (w .r .t. the lattice), we have indeg ( v ) = 0 and thus η ( v ) + indeg ( v ) < 4 . If, on the other hand, v is a neighbour of v i (w .r .t. the lattice), we hav e indeg ( v ) = 1 . Also, the fact that v / ∈ A (1) implies η ( v ) < 3 . Hence, η ( v ) + indeg ( v ) < 4 . Therefore, step 4 of the algorithm terminates and we obtain w A (1) ( η ) = 1 and A (1) 1 := { v ∈ A (1) | η A (1) ( v ) = 3 } = ∅ . Thus, we hav e E [ X ( η ) | Y ∈ A (1) ] = 1 , which is consistent with expression (6). W e proceed to consider N = 2 . Again, we initialize the directed graph ( V , E ) with E = ∅ and we augment E with an edge from v to each of its neighbours for each v ∈ A (2) . Note that, at this point, we have η ( v ) + indeg ( v ) − outde g ( v ) < 4 for each v / ∈ A (2) . Hence, step 4 of Algorithm 1 terminates and we obtain w A (2) ( η ) = 4 and A (2) 1 := { v ∈ A (2) | η A (2) ( v ) = 3 } = ∅ . The directed graph and the sandpile configuration η A (2) are shown in Figure 2. W e obtain E [ X ( η ) | Y ∈ A (2) ] = 4 , which is consistent with expression (6). 8 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Figure 2: Directed graph constructed in Algorithm 1 applied to a square generator (left) with N = 2 and sandpile configuration η A (2) (right). Now , suppose that expression (6) holds for N = k − 2 for some k = 3 , 4 , . . . . W e show that it continues to hold for N = k . W e again use Algorithm 1 to find the size w A ( k ) ( η ) of the first wa ve, the sets A ( k ) 1 , . . . , A ( k ) ℓ and the sandpile η A ( k ) . Again, we initialize a directed graph ( V , E ) with E = ∅ , add edges to E from v to each of the neighbours (w .r .t. the lattice) of v for each v ∈ A ( k ) and set w ′ = | A ( k ) | = k 2 . Note that for each v / ∈ A ( k ) ∪ δ out ( A ( k ) ) , we hav e indeg ( v ) = outdeg ( v ) = 0 and that for v ∈ δ out ( A ( k ) ) , we hav e indeg ( v ) = 1 and outdeg ( v ) = 0 . Furthermore, since v ∈ δ out ( A ( k ) ) is not part of A ( k ) , we have η ( v ) < 3 . This implies that η ( v ) + inde g ( v ) < 4 for all v / ∈ A ( k ) and step 4 of the algorithm terminates at this point, resulting in w A ( k ) ( η ) = k 2 . Now , note that η A ( k ) ( v ) = 3 if and only if indeg ( v ) = 4 . This implies that η A ( k ) ( v ) = 3 if and only if v ∈ int ( A ( k ) ) . Hence, the set { v ∈ A ( k ) | η A ( k ) ( v ) = 3 } consists of a single connected component A ( k ) 1 , which has the form of a square of size k − 2 × k − 2 . The directed graph constructed in Algorithm 1 and the sandpile configuration η A ( k ) are depicted in Figure 3. Figure 3: Directed graph constructed in Algorithm 1 applied to a square generator (left) and sandpile configuration η A ( k ) (right). Inserting the obtained size of the first wa ve w A ( k ) ( η ) into expression (2), using the structure of A ( k ) 1 and in voking the induction hypothesis no w yields E [ X ( η ) | Y ∈ A ( k ) ] = k 2 + ( k − 2) 2 k 2 3( k − 2) 4 + 15( k − 2) 3 + 20( k − 2) 2 − 8 30( k − 2) = 3 k 4 + 15 k 3 + 20 k 2 − 8 30 k , establishing expression (6) for N = k . It follo ws that expression (6) holds for all positi ve integers N . 9 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Remark 1 (Local confinement of av alanches and upper bound) . Observe from the ar guments above that an a valanche emanating from a square generator is confined to this re gion, since the surrounding boundary of noncritical v ertices acts as an insurmountable barrier . Moreover , giv en two sandpile configurations η , η ′ ∈ X with η ( v ) ≥ η ′ ( v ) for all v ∈ V , note that x ( η , v ) ≥ x ( η ′ , v ) for all v ∈ V . Consequently , expression (6) provides an upper bound for the expected av alanche size associated with any N × N square-shaped region enclosed by noncritical v ertices. Theorem 4.2. Consider a gener ator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e. The depth of this generator is given by ρ ( η , A ( N ) ) = ⌈ N / 2 ⌉ . (7) Pr oof. W e prove the statement by induction o ver N , using the definition of depth gi ven in e xpression (4). The result is obvious for N = 1 . Consider N = 2 . Recall from the proof of Theorem 4.1 that { v ∈ A (2) | η A (2) ( v ) = 3 } = ∅ (see Figure 2). Hence, it follows from e xpression (4) that ρ ( η , A (2) ) = 1 , which is consistent with expression (7). W e proceed to assume that expression (4) is v alid for N = k − 2 for some k = 3 , 4 , . . . and show that this implies its correctness for N = k . It follows from the proof of Theorem 4.1 that the set { v ∈ A ( k ) | η A ( k ) ( v ) = 3 } is a square connected component of size k − 2 × k − 2 (see Figure 3). Hence, by expression (4) and the induction hypothesis, we obtain ρ ( η , A ( k ) ) = 1 + ρ ( η , A ( k − 2) ) = 1 + ⌈ ( k − 2) / 2 ⌉ = ⌈ k / 2 ⌉ , (8) which establishes expression (7) for N = k . W e proceed to characterize the set of cornerstone v ertices for square-shaped generators. T o this end, we first analyze the effect of removing the sand grains from critical v ertices at different locations within such a generator . The results are collected in Lemmas 4.3 – 4.5. W e include the proof of Lemma 4.3. The proofs of the remaining lemmas follow similar lines and are deferred to the Appendix. Lemma 4.3. Consider a g enerator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e, where N ≥ 2 , and let v i ∈ R 1 ( A ( N ) ) . Then, E [ X ( γ i η ) | Y ∈ A ( N ) ] =        ( N 2 − 1)( N 2 + 5 N + 6) 10 N , if N is odd , N 5 + 5 N 4 + 5 N 3 − 5 N 2 − 6 N − 30 10 N 2 , if N is e ven. (9) Pr oof. Let ˜ A N , 1 R denote the remainder of the generator A ( N ) in the sandpile configuration γ i η , i.e., ˜ A N , 1 R = A ( N ) \ { v i } . Note that conditioning on the verte x Y that the next sand grain lands upon yields E [ X ( γ i η ) | Y ∈ A ( N ) ] = N 2 − 1 N 2 E [ X ( γ i η ) | Y ∈ ˜ A N , 1 R ] . Hence, it suffices to pro ve that E [ X ( γ i η ) | Y ∈ ˜ A N , 1 R ] =          N ( N 2 + 5 N + 6) 10 , if N is odd , N 5 + 5 N 4 + 5 N 3 − 5 N 2 − 6 N − 30 10( N 2 − 1) , if N is e ven. (10) W e prove the v alidity of expression (10) by applying Algorithm 1 to the generator ˜ A N , 1 R in the configuration γ i η and performing induction over N . First, we show that expression (10) holds for N = 2 and N = 3 . Consider the case that N = 2 . W e initialize a directed graph ( V , E ) with E = ∅ and augment E with edges from v to each of its neighbours for each v ∈ ˜ A 2 , 1 R . Now , observe that ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v / ∈ ˜ A 2 , 1 R . Thus, we obtain w ˜ A 2 , 1 R ( γ i η ) = 3 . The directed graph and the sandpile configuration ( γ i η ) ˜ A 2 , 1 R are shown in Figure 4. 10 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Figure 4: Directed graph constructed in Algorithm 1 applied to generator ˜ A 2 , 1 R in configuration γ i η (left) and sandpile configuration ( γ i η ) ˜ A 2 , 1 R (right). It follows that E [ X ( γ i η ) | Y ∈ ˜ A 2 , 1 R ] = 3 , which agrees with expression (10). Now , consider the case N = 3 . W e start by initializing the directed graph ( V , E ) with E = ∅ and add edges from v to each of its neighbours for each v ∈ ˜ A 3 , 1 R . At this point, the only v ertex v / ∈ ˜ A 3 , 1 R that satisfies ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) ≥ 4 is the verte x v i . Thus, we add edges to E from v i to each of its neighbours. Since each neighbour of v i ∈ R 1 ( A ( N ) ) is part of ˜ A 3 , 1 R , step 4 of Algorithm 1 terminates at this point. W e obtain w ˜ A 3 , 1 R ( γ i η ) = 9 . Observe that for each v ∈ δ in ( A ( N ) ) , we hav e indeg ( v ) = 3 . Hence, ( γ i η ) ˜ A 3 , 1 R ( v ) = 2 . Also, for each v ∈ C ( A ( N ) ) , we hav e indeg ( v ) = 2 . Therefore, ( γ i η ) ˜ A 3 , 1 R ( v ) = 1 . Finally , we have indeg ( v i ) = outdeg ( v i ) = 4 , and thus, η ˜ A 3 , 1 R ( v i ) = 0 . The directed graph constructed in the algorithm and the sandpile configuration ( γ i η ) ˜ A 3 , 1 R after the first wa ve are depicted in Figure 5. It follows that applying Algorithm 1 in the case N = 3 yields E [ X ( γ i η ) | Y ∈ ˜ A 3 , 1 R ] = 9 , which is consistent with expression (10). Figure 5: Directed graph constructed in Algorithm 1 applied to generator ˜ A 3 , 1 R in configuration γ i η (left) and sandpile configuration ( γ i η ) ˜ A 3 , 1 R (right). Now , we assume that expression (10) holds for N = k − 2 for some k = 3 , 4 , . . . . Consider the generator ˜ A k, 1 R in the configuration γ i η . The directed graph and the sandpile ( γ i η ) ˜ A k, 1 R that remains after the first w av e are provided in Figure 6 for k = 5 . The algorithm ev olves in a similar way for e ven k , although in this case, there are four options for the verte x v i . 11 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Figure 6: Directed graph constructed in Algorithm 1 applied to generator ˜ A k, 1 R in configuration γ i η (left) and sandpile configuration ( γ i η ) ˜ A k, 1 R (right) for k = 5 . It follows that Algorithm 1 yields w ˜ A k, 1 R ( γ i η ) = k 2 and that the remainder of the generator ˜ A k, 1 R after the first wav e is a generator ˜ A k − 2 , 1 R in the sandpile configuration ( γ i η ) ˜ A k, 1 R . Inserting this in expression (2) and using the induction hypothesis no w yields E [ X ( γ i η ) | Y ∈ ˜ A k, 1 R ] = k 2 + ( k − 2) 2 − 1 k 2 − 1 ( k − 2)(( k − 2) 2 + 5( k − 2) + 6) 10 = k ( k 2 + 5 k + 6) 10 for odd k and E [ X ( γ i η ) | Y ∈ ˜ A k, 1 R ] = k 2 + ( k − 2) 2 − 1 k 2 − 1 ( k − 2) 5 + 5( k − 2) 4 + 5( k − 2) 3 − 5( k − 2) 2 − 6( k − 2) − 30 10(( k − 2) 2 − 1) = k 5 + 5 k 4 + 5 k 3 − 5 k 2 − 6 k − 30 10( k 2 − 1) for ev en k . This establishes the validity of e xpression (10) for all positive inte gers N . Lemma 4.4. Consider a g enerator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e, where N ≥ 3 , and let v i ∈ R k ( A ( N ) ) \ C  S k j =1 R j ( A ( N ) )  , k = 2 , . . . , ⌈ N / 2 ⌉ . Then, E [ X ( γ i η ) | Y ∈ A ( N ) ] =        3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 24 k 2 + 23 k − 6) 30 N 2 , if N is odd, 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 20 k (2 k 2 − 9 k + 1) 30 N 2 , if N is e ven. (11) Pr oof. See Appendix A. Lemma 4.5. Consider a g enerator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e, where N ≥ 3 , and let v i ∈ C  S k j =1 R j ( A ( N ) )  , k = 2 , . . . , ⌈ N / 2 ⌉ . Then, E [ X ( γ i η ) | Y ∈ A ( N ) ] =        3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 24 k 2 + 23 k − 3) 30 N 2 , if N is odd, 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 18 k 2 + 2 k + 3) 30 N 2 , if N is e ven. (12) Pr oof. See Appendix A. 12 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Theorem 4.6 no w provides a characterization of the set of cornerstone vertices for s quare-shaped generators, i.e., the vertices for which the minimum in e xpression (5) is attained. Theorem 4.6. Consider a generator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e. The set of cornerstone vertices corr esponding to this generator is given by B ∗ A ( N ) ( η ) =    R 1 ( A ( N ) ) , if N = 1 , 2 , R 2 ( A ( N ) ) \ C ( A ( N ) ) , if N = 3 , 4 , R 3 ( A ( N ) ) \ C ( S 3 i =1 R i ( A ( N ) )) , if N ≥ 5 . Pr oof. The statement is ob vious for N = 1 , 2 . Now , consider N = 3 . Let v i ∈ R 2 ( A (3) ) \ C ( A (3) ) , v j ∈ C ( A (3) ) and v k ∈ R 1 ( A (3) ) . Comparing expressions (9), (11) and (12), we obtain E [ X ( γ i η ) | Y ∈ A (3) ] < E [ X ( γ j η ) | Y ∈ A (3) ] < E [ X ( γ k η )) | Y ∈ A (3) ] . Hence, we hav e B ∗ A (3) ( η ) = R 2 ( A (3) ) \ C ( A (3) ) . W e proceed to consider N = 4 . Let v i ∈ R 2 ( A (4) ) \ C ( A (4) ) , v j ∈ C ( A (4) ) and v k ∈ R 1 ( A (4) ) . Again comparing expressions (9), (11) and (12) yields E [ X ( γ i η ) | Y ∈ A (4) ] < E [ X ( γ j η ) | Y ∈ A (4) ] < E [ X ( γ k η )) | Y ∈ A (4) ] . It follows that B ∗ A (4) ( η ) = R 2 ( A (4) ) \ C ( A (4) ) . Now , consider N ≥ 5 . Let v i k ∈ R k ( A ( N ) ) \ C ( S k j =1 R j ( A ( N ) )) , k = 2 , . . . , ⌈ N / 2 ⌉ . Minimizing expression (11) with respect to k , we obtain arg min k =2 ,..., ⌈ N / 2 ⌉ E [ X ( γ i k η ) | Y ∈ A ( N ) ] = 3 for both odd and ev en N . Now , let v i k ∈ C  S k j =1 R j ( A ( N ) )  , k = 2 , . . . , ⌈ N / 2 ⌉ . Minimizing expression (12) with respect to k yields arg min k =2 ,..., ⌈ N / 2 ⌉ E [ X ( γ i k η ) | Y ∈ A ( N ) ] = 3 for both odd and ev en N . Consider v i ∈ R 3 ( A ( N ) ) \ C  S 3 j =1 R j ( A ( N ) )  , v j ∈ C  S 3 j =1 R j ( A ( N ) )  and v k ∈ R 1 ( A ( N ) ) . Comparing expressions (9) and e xpressions (11) and (12) for k = 3 no w yields E [ X ( γ i η ) | Y ∈ A ( N ) ] < E [ X ( γ j η ) | Y ∈ A ( N ) ] < E [ X ( γ k η ) | Y ∈ A ( N ) ] , which completes the proof. A surprising aspect of the result in Theorem 4.6 is that the location of the set of cornerstone vertices does not scale with the size of the square. Observe that the result strikes a balance between the impact of the interv ention on the largest possible av alanches and the amount of a v alanches that are af fected by the intervention. After all, emptying a verte x near the center of the square may prevent a lar ge avalanche, b ut it has no effect on the size of av alanches triggered when the new sand grain lands f arther from the center . On the other hand, emptying a more peripheral vertex mitigates man y of the potential av alanches, but the reduction in their size is smaller . 13 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Corollary 4.7. Consider a generator A ( N ) in a sandpile η ∈ Ω that has the form of an N × N squar e. The stability level of this gener ator is given by λ ( η , A ( N ) ) =                                            0 , if N = 1 , 9 / 16 , if N = 2 , 32 / 41 , if N = 3 , 15 / 17 , if N = 4 , 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N − 450 N (3 N 4 + 15 N 3 + 20 N 2 − 8) , if N ≥ 5 , N odd, 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N − 480 N (3 N 4 + 15 N 3 + 20 N 2 − 8) , if N ≥ 5 , N even. Pr oof. The case N = 1 is e vident from e xpression (5). The remaining cases follo w immediately from Theorem 4.1, Lemmas 4.3 – 4.5 and Theorem 4.6. 5 Conclusion As a paradigmatic model of self-organized criticality , the Abelian sandpile model provides a theoretical framework for large-scale cascading phenomena, including forest fires, earthquakes or financial mark et crashes. Since lar ge a v alanches in such settings correspond to catastrophic ev ents, understanding how to mitigate them is of considerable practical interest. This work took a first step to wards a theoretical understanding of the impact of interventions in the sandpile model. Specifically , we in vestigated the effect of remo ving sand grains from connected components of critical vertices. T o study this ef fect, we first e xtended the method proposed by Dorso and Dadamia [11] for computing the e xpected size of an av alanche emanating from a giv en generator , and we provided a formal justification for the resulting scheme. Using this algorithm, we performed a detailed analysis of the impact of remo ving sand grains at different locations in square-shaped generators. This class of generators admits an explicit local analysis and a characterization of av alanche structures. Our results rev eal a class of optimal target vertices, which we refer to as cornerstone vertices, that balance the tradeoff between reducing the maximal a valanche size and increasing the number of a v alanches that are mitigated when a new sand grain is added to the square. Interestingly , the locations of these optimal targets do not scale with the size of the square. This work opens se veral promising directions for future research. Our analysis of square-shaped generators provides insight into the impact of interventions on a valanche structure and highlights the tradeoff between restricting maximal av alanche sizes and increasing the number of mitigated a valanches. It would be interesting to e xtend this analysis to generators dra wn from the stationary distri bution of the sandpile Mark ov chain. Such generators may initiate av alanches that cov er a large part of the lattice, precluding a purely local analysis and potentially leading to fundamentally dif ferent optimal intervention strate gies. Furthermore, rather than focusing solely on the immediate ef fect of interventions, it would be valuable to in vestigate long-term mitigation strategies, for example within a Markov decision process framework. Such an approach has recently been explored for de veloping control strate gies in the Ising model [ 8 , 9 ]. In addition, it may be worthwhile to explore alternati ve intervention mechanisms beyond sand grain remo val, possibly inspired by specific applications. Examples include artificially triggering small av alanches to release stress and reduce the likelihood of larger e vents, or increasing the toppling thresholds of selected vertices. 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Mitigating cascades in sandpile models: an immunization strategy for systemic risk? Eur . Phys. J. Special T opics , 225(10):2017–2023, 2016. [20] C.H. Scholz. The Mechanics of Earthquakes and F aulting . Cambridge Uni versity Press, 3 edition, 2019. [21] M. T uralska and A. Swami. Greedy control of cascading failures in interdependent networks. Sci. Rep. , 11(3276), 2021. 15 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T A Remaining proofs Proof of Lemma 4.4 Let ˜ A N ,k R denote the remainder of A ( N ) in the sandpile configuration γ i η , i.e., ˜ A N ,k R = A ( N ) \ { v i } . Conditioning on the verte x Y at which the next sand grain lands, we obtain E [ X ( γ i η ) | Y ∈ A ( N ) ] = N 2 − 1 N 2 E [ X ( γ i η ) | Y ∈ ˜ A N ,k R ] . It follows that it is suf ficient to prove that E [ X ( γ i η ) | Y ∈ ˜ A N ,k R ] =          3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 24 k 2 + 23 k − 6) 30( N 2 − 1) , if N is odd, 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 20 k (2 k 2 − 9 k + 1) 30( N 2 − 1) , if N is e ven. (13) W e again prove this statement by induction over N . W e start by showing that expression (13) holds for N = 3 and k = 2 and for N = 4 and k = 2 . First, suppose that N = 3 and k = 2 . Figure 7 shows the directed graph obtained by applying Algorithm 1 to the generator ˜ A 3 , 2 R in the sandpile configuration γ i η . Figure 7: Directed graph constructed in Algorithm 1 applied to generator ˜ A 3 , 2 R in configuration γ i η (left) and sandpile configuration ( γ i η ) ˜ A 3 , 2 R (right). Observe that w ˜ A 3 , 2 R ( γ i η ) = 8 and { v ∈ ˜ A 3 , 2 R | ( γ i η ) ˜ A 3 , 2 R ( v ) = 3 } = ∅ . Thus, by expression (2), we ha ve E [ X ( γ i η ) | Y ∈ ˜ A 3 , 2 R ] = 8 , which agrees with expression (13). Now , consider N = 4 and k = 2 . The e volution of Algorithm 1 for this case is depicted in Figure 8. Figure 8: Evolution of Algorithm 1 applied to generator ˜ A 4 , 2 R in sandpile configuration γ i η . 16 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Let v i 1 , v i 2 and v i 3 denote the neighbours of v i that are part of the corners, the inner boundary and the interior of A (4) respectiv ely . W e initialize a directed graph ( V , E ) with E = ∅ . For each v ∈ ˜ A 4 , 2 R , we no w add edges to E from v to each of its neighbours. Now , observe that ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v / ∈ ˜ A 4 , 2 R . Also, we obtain ( γ i η ) ˜ A 4 , 2 R ( v ) =        0 , if v = v i 1 , 1 , if v ∈ ( C ( A (4) ) \ { v i 1 } ) ∪ { v i 2 } , 2 , if v ∈ ( δ in ( A (4) ) \ { v i , v i 2 } ) ∪ { v i 3 } , 3 , otherwise. Hence, we have w ˜ A 4 , 2 R = 15 and ˜ A 4 , 2 R, 1 := { v ∈ ˜ A 4 , 2 R | ( γ i η ) ˜ A 4 , 2 R ( v ) = 3 } = int ( A (4) ) \ { v i 3 } . Applying Al- gorithm 1 to the generator ˜ A 4 , 2 R, 1 in the sandpile configuration ( γ i η ) ˜ A 4 , 2 R now yields w ˜ A 4 , 2 R, 1 (( γ i η ) ˜ A 4 , 2 R ) = 5 and { v ∈ ˜ A 4 , 2 R, 1 | (( γ i η ) ˜ A 4 , 2 R ) ˜ A 4 , 2 R, 1 ( v ) = 3 } = ∅ . Thus, we obtain E [ X ( γ i η ) | Y ∈ ˜ A 4 , 2 R ] = 16 , which is consistent with expression (13). Suppose now that expression (13) holds for N = n − 2 and all k = 2 , . . . , ⌈ ( n − 2) / 2 ⌉ for some n = 5 , 6 , . . . . W e show that this implies its v alidity for N = n and all k = 2 , . . . , ⌈ n/ 2 ⌉ by means of an embedded induction argument. W e first consider the case k = ⌈ n/ 2 ⌉ . The ev olution of Algorithm 1 for N = 7 and k = 4 is depicted in Figure 9. Figure 9: Evolution of Algorithm 1 applied to generator ˜ A 7 , 4 in sandpile configuration γ i η . Let v i 1 and v i 2 denote the neighbours of v i that are part of δ in ( A ( N ) ) ∪ C ( A ( N ) ) and let v i 3 denote the neighbour of v i that is part of int ( A ( N ) ) . Although Figure 9 sho ws a case in which both v i 1 and v i 2 are part of δ in ( A ( N ) ) , observe that either v i 1 or v i 2 may be part of C ( A ( N ) ) . Follo wing Algorithm 1, we start by initializing the directed graph ( V , E ) with E = ∅ and add edges to E from v to each of its neighbours for each v ∈ ˜ A n, ⌈ n/ 2 ⌉ R . At this point, note that ( γ i η )( v ) + inde g ( v ) − outde g ( v ) < 4 for all v ∈ V . The obtained directed graph for the case N = 7 , k = 4 is provided in Figure 9 (left). Hence, we obtain ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v ) = 3 for all v ∈ { v i } ∪ ( int ( A ( N ) ) \ { v i 3 } ) and ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v i 3 ) = 2 . In addition, ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v ) < 3 for all v ∈ ( δ in ( A ( N ) ) ∪ C ( A ( N ) )) \ { v i } . This implies that w ˜ A n, ⌈ n/ 2 ⌉ R ( γ i η ) = n 2 − 1 and the set ˜ A n, ⌈ n/ 2 ⌉ R, 1 := { v ∈ ˜ A n, ⌈ n/ 2 ⌉ R | ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v ) = 3 } = int ( A ( N ) ) \ { v i 3 } . W e no w apply Algorithm 1 to the generator ˜ A n, ⌈ n/ 2 ⌉ R, 1 in the sandpile configuration ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R . Again, we start by initializing the directed graph ( V , E ) with E = ∅ and add edges to E from v to each of its neighbours for each v ∈ ˜ A n, ⌈ n/ 2 ⌉ R, 1 . At this point, note that the only verte x v / ∈ ˜ A n, ⌈ n/ 2 ⌉ R, 1 that satisfies ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v ) + indeg ( v ) − outdeg ( v ) ≥ 4 is v = v i 3 . Hence, we add edges to E from v i 3 to each of its neighbours. No w , the only vertex v / ∈ ˜ A n, ⌈ n/ 2 ⌉ R, 1 for which ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ( v )+ indeg ( v ) − outdeg ( v ) ≥ 4 is v = v i . After adding edges from v i to each of its neigbours, step 4 of Algorithm 1 terminates. The directed graph resulting from this iteration is illustrated in Figure 9 (mid- dle). Note that w ˜ A n, ⌈ n/ 2 ⌉ R, 1 (( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ) = ( n − 2) 2 + 1 and the set { v ∈ ˜ A n, ⌈ n/ 2 ⌉ R, 1 | (( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ) ˜ A n, ⌈ n/ 2 ⌉ R, 1 ( v ) = 3 } 17 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T is exactly a square of size ( n − 4) × ( n − 4) , which we will denote by A ( n − 4) . Evaluating e xpression (2) and using expression (6) no w yields E [ X ( γ i η ) | Y ∈ ˜ A n ⌈ n/ 2 ⌉ R ] = n 2 − 1 + ( n − 2) 2 − 1 n 2 − 1 E [ X (( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ R ) | Y ∈ ˜ A n, ⌈ n/ 2 ⌉ R, 1 ] = n 2 − 1 + ( n − 2) 2 − 1 n 2 − 1  ( n − 2) 2 + 1 + ( n − 4) 2 ( n − 2) 2 − 1 3( n − 4) 4 + 15( n − 4) 3 + 20( n − 4) 2 − 8 30( n − 4)  = n (3 n 4 + 15 n 3 + 20 n 2 − 60 n − 8) 30( n 2 − 1) , which is consistent with expression (13) for k = ⌈ n/ 2 ⌉ . W ithin the embedded induction argument, we now assume that expression (13) holds for N = n and k = ℓ + 1 for some ℓ = 2 , . . . , ⌈ N / 2 ⌉ − 1 . W e proceed to sho w that the statement holds for k = ℓ . The directed graph resulting from Algorithm 1 and the sandpile configuration ( γ i η ) ˜ A n,k R for the case n = 7 and k = 3 are provided in Figure 10. Figure 10: Evolution of Algorithm 1 applied to generator ˜ A 7 , 3 R in sandpile configuration γ i η . Let v i 1 and v i 3 denote the neighbours of v i that are part of R ℓ ( A ( N ) ) , let v i 2 denote the neighbour of v i that is part of R ℓ +1 ( A ( N ) ) and let v i 4 denote the neighbour of v i that is part of R ℓ − 1 ( A ( N ) ) . Again, we start by initializing the directed graph ( V , E ) with E = ∅ . For each v ∈ ˜ A n,ℓ R , we add edges to E from v to each of its neighbours. Note that at this point, we hav e ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) = 4 if and only if v = v i . Hence, we continue to add edges to E from v i to each of its neighbours. Now , we obtain ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v ∈ V . The directed graph resulting from Algorithm 1 is depicted in Figure 10 (left). Observe no w that ( γ i η ) ˜ A n,ℓ R ( v ) =        0 , if v = v i , 1 , if v ∈ C ( A ( N ) ) , 2 , if v ∈ R ⌈ n/ 2 ⌉ ( A ( N ) ) \ C ( A ( N ) ) , 3 , otherwise . It follo ws that w ˜ A n,ℓ R ( γ i η ) = n 2 and that the set ˜ A n,ℓ R, 1 := { v ∈ ˜ A n,ℓ R | ( γ i η ) ˜ A n,ℓ R ( v ) = 3 } = int ( A ( N ) ) \ { v i } . W e now use the induction hypothesis to e v aluate expression (2): E [ X ( γ i η ) | Y ∈ ˜ A n,ℓ R ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 E [ X (( γ i η ) ˜ A n,ℓ R ) | Y ∈ ˜ A n,ℓ R, 1 ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 3( n − 2) 5 + 15( n − 2) 4 + 15( n − 2) 3 − 15( n − 2) 2 − 18( n − 2) + 10(4 ℓ 3 − 24 ℓ 2 + 23 ℓ − 6) 30(( n − 2) 2 − 1) = 3 n 5 + 15 n 4 + 15 n 3 − 15 n 2 − 18 n + 10(4 ℓ 3 − 24 ℓ 2 + 23 ℓ − 6) 30( n 2 − 1) if n is odd and E [ X ( γ i η ) | Y ∈ ˜ A n,ℓ R ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 E [ X (( γ i η ) ˜ A n,ℓ R | Y ∈ ˜ A n,ℓ R, 1 ] 18 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T = n 2 + ( n − 2) 2 − 1 n 2 − 1 3( n − 2) 5 + 15( n − 2) 4 + 15( n − 2) 3 − 15( n − 2) 2 − 18( n − 2) + 20 ℓ (2 ℓ 2 − 9 ℓ + 1) 30(( n − 2) 2 − 1) = 3 n 5 + 15 n 4 + 15 n 3 − 15 n 2 − 18 n + 20 ℓ (2 ℓ 2 − 9 ℓ + 1) 30( n 2 − 1) if n is e ven. Thus, we established the validity of e xpression (13) for N = n and all k = 2 , . . . , ⌈ n/ 2 ⌉ . The complete induction argument no w yields the statement for all N ≥ 3 and all k = 2 , . . . , ⌈ n/ 2 ⌉ . Proof of Lemma 4.5 Let ˜ A N ,k C denote the remainder of A ( N ) in the sandpile configuration γ i η , i.e., ˜ A N ,k C = A ( N ) \ { v i } . Note that E [ X ( γ i η ) | Y ∈ A ( N ) ] = N 2 − 1 N 2 E [ X ( γ i η ) | Y ∈ ˜ A N ,k C ] . Hence, it suffices to sho w that E [ X ( γ i η ) | Y ∈ ˜ A N ,k C ] =          3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 24 k 2 + 23 k − 3) 30( N 2 − 1) , if N is odd, 3 N 5 + 15 N 4 + 15 N 3 − 15 N 2 − 18 N + 10(4 k 3 − 18 k 2 + 2 k + 3) 30( N 2 − 1) , if N is e ven. (14) W e prove the statement by induction ov er N . W e first sho w that expression (14) holds for N = 3 and k = 2 and for N = 4 and k = 2 . Consider the case N = 3 and k = 2 . The result of applying Algorithm 1 to the generator ˜ A 3 , 2 C in the sandpile configuration γ i η is depicted in Figure 11. Figure 11: Evolution of Algorithm 1 applied to generator ˜ A 3 , 2 C in sandpile configuration γ i η . Let the vertex in the center of the generator A (3) be denoted by ˆ v . Observe that w ˜ A 3 , 2 C ( γ i η ) = 8 and ˜ A 3 , 2 C, 1 = { v ∈ ˜ A 3 , 2 C | ( γ i η ) ˜ A 3 , 2 C ( v ) = 3 } = { ˆ v } . Also, note that w ˜ A 3 , 2 C, 1 (( γ i η ) ˜ A 3 , 2 C ) = 1 and { v ∈ ˜ A 3 , 2 C, 1 | (( γ i η ) ˜ A 3 , 2 C ) ˜ A 3 , 2 C, 1 ( v ) = 3 } = ∅ . Using expression (2), we obtain E [ X ( γ i η ) | Y ∈ ˜ A 3 , 2 C ] = 65 8 , which is consistent with expression (14). Now , consider the case N = 4 and k = 2 . The e volution of Algorithm 1 is sho wn in Figure 12. 19 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T Figure 12: Evolution of Algorithm 1 applied to generator ˜ A 4 , 2 C in sandpile configuration γ i η . It follo ws that w ˜ A 4 , 2 C ( γ i η ) = 15 and ˜ A 4 , 2 C, 1 = R 1 ( A (4) ) . From the second iteration of Algorithm 1, we obtain w ˜ A 4 , 2 C, 1 (( γ i η ) ˜ A 4 , 2 C ) = 4 and { v ∈ ˜ A 4 , 2 C, 1 | (( γ i η ) ˜ A 4 , 2 C ) ˜ A 4 , 2 C, 1 ( v ) = 3 } = ∅ . Inserting this into expression (2) no w yields E [ X ( γ i η ) | Y ∈ ˜ A 4 , 2 C ] = 241 15 , which aligns with expression (14). W e proceed to assume that e xpression (14) is v alid for N = n − 2 and all k = 2 , . . . , ⌈ ( n − 2) / 2 ⌉ for some n = 5 , 6 , . . . . W e show that this implies its correctness for N = n and all k = 2 , . . . , ⌈ n/ 2 ⌉ by means of an embedded induction argument o ver k . W e start by considering the case k = ⌈ n/ 2 ⌉ . The result of applying Algorithm 1 to the case N = 7 and k = 4 is illustrated in Figure 13. Figure 13: Evolution of Algorithm 1 applied to generator ˜ A 7 , 4 C in sandpile configuration γ i η . Let the neighbours of v i be denoted by v i 1 and v i 2 . In accordance with Algorithm 1, we initialize the directed graph ( V , E ) with E = ∅ and add edges to E from v to each of its neighbours for each v ∈ ˜ A n, ⌈ n/ 2 ⌉ C . Observe that at this point, we hav e ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v ∈ V . The resulting directed graph for the case N = 7 , k = 4 is depicted in Figure 13 (left). It follows that ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ C ( v ) =    3 , if v ∈ int ( A ( n ) ) , 2 , if v ∈ ( δ in ( A ( n ) ) \ { v i 1 , v i 2 } ) ∪ { v i } , 1 , if v ∈ ( C ( A ( n ) ) \ { v i } ) ∪ { v i 1 , v i 2 } . This implies that w ˜ A n, ⌈ n/ 2 ⌉ C ( γ i η ) = n 2 − 1 and ˜ A n, ⌈ n/ 2 ⌉ C, 1 := { v ∈ ˜ A n, ⌈ n/ 2 ⌉ C | ( γ i η ) ˜ A n, ⌈ n/ 2 ⌉ C ( v ) = 3 } = int ( A ( N ) ) . Using expression (2), the fact that int ( A ( N ) ) is a square generator of size ( n − 2) × ( n − 2) and expression (6), we now obtain E [ X ( γ i η ) | Y ∈ ˜ A n, ⌈ n/ 2 ⌉ C ] = n 2 − 1 + ( n − 2) 2 n 2 − 1 3( n − 2) 4 + 15( n − 2) 3 + 20( n − 2) 2 − 8 30( n − 2) 20 Optimal local interventions in the two-dimensional Abelian sandpile model A P R E P RI N T = 3 n 4 + 18 n 3 + 38 n 2 − 22 n − 30 30( n + 1) . (15) Note that for odd n , expression (15) is equi valent to the first case in expression (14) with k = ( n + 1) / 2 and that for ev en n , it is equiv alent to the second case in expression (14) with k = n/ 2 . Hence, expression (6) agrees with expression (14) for k = ⌈ n/ 2 ⌉ . W e now make the additional assumption that expression (14) holds for N = n and k = ℓ + 1 for some ℓ = 2 , . . . , ⌈ N/ 2 ⌉ − 1 and sho w that this implies its validity for k = ℓ . T ogether with the fact that (14) holds for N = n and k = ⌈ n/ 2 ⌉ , this finalizes the embedded induction argument, establishing the statement for N = n and all k = 2 , . . . , ⌈ n/ 2 ⌉ . The directed graph obtained from Algorithm 1 and the sandpile configuration ( γ i η ) ˜ A n,ℓ C for the case n = 7 , ℓ = 3 are depicted in Figure 14. Figure 14: Evolution of Algorithm 1 applied to generator ˜ A 7 , 3 C in sandpile configuration γ i η . After initializing the directed graph ( V , E ) with E = ∅ , we again add edges to E from v to each of its neighbours for each v ∈ ˜ A n,ℓ C . Note that after this procedure, we have ( γ i η )( v i ) + indeg ( v i ) − outdeg ( v i ) = 4 . Hence, we append edges to E from v i to each of its neighbours. Now , we obtain ( γ i η )( v ) + indeg ( v ) − outdeg ( v ) < 4 for all v ∈ V . The directed graph resulting from Algorithm 1 is shown in Figure 14 (left). W e deri ve that ( γ i η ) ˜ A n,ℓ C ( v ) =        0 , if v = v i , 1 , if v ∈ C ( A ( N ) ) , 2 , if v ∈ δ in ( A ( N ) ) , 3 , if v ∈ int ( A ( N ) ) \ { v i } . It follo ws that w ˜ A n,ℓ C ( γ i η ) = n 2 and ˜ A n,ℓ C, 1 := { v ∈ ˜ A n,ℓ C | ( γ i η ) ˜ A n,ℓ C ( v ) = 3 } = int ( A ( N ) ) \ { v i } . Inserting this in expression (2) and using the induction hypothesis, we no w obtain E [ X ( γ i η ) | Y ∈ ˜ A n,ℓ C ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 E [ X (( γ i η ) ˜ A n,ℓ C ) | Y ∈ ˜ A n,ℓ C, 1 ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 3( n − 2) 5 + 15( n − 2) 4 + 15( n − 2) 3 − 15( n − 2) 2 − 18( n − 2) + 10(4 ℓ 3 − 24 ℓ 2 + 23 ℓ − 3) 30(( n − 2) 2 − 1) = 3 n 5 + 15 n 4 + 15 n 3 − 15 n 2 − 18 n + 10(4 ℓ 3 − 24 ℓ 2 + 23 ℓ − 3) 30( n 2 − 1) if n is odd and E [ X ( γ i η ) | Y ∈ ˜ A n,ℓ C ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 E [ X (( γ i η ) ˜ A n,ℓ C ) | Y ∈ ˜ A n,ℓ C, 1 ] = n 2 + ( n − 2) 2 − 1 n 2 − 1 3( n − 2) 5 + 15( n − 2) 4 + 15( n − 2) 3 − 15( n − 2) 2 − 18( n − 2) + 10(4 ℓ 3 − 18 ℓ 2 + 2 ℓ + 3) 30(( n − 2) 2 − 1) = 3 n 5 + 15 n 4 + 15 n 3 − 15 n 2 − 18 n + 10(4 ℓ 3 − 18 ℓ 2 + 2 ℓ + 3) 30( n 2 − 1) if n is even. This yields the correctness of expression (14) for N = n and all ℓ = 2 , . . . , ⌈ n/ 2 ⌉ . The full induc- tion argument now establishes the validity of expression (14), and thus of expression (12), for all N ≥ 3 and all k = 2 , . . . , ⌈ n/ 2 ⌉ . 21