Downloaded by guest on September 28, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1812015116 sandpile abelian the of domain. structure its fractal of the boundaries extend the to beyond identity possible be might coefficients it critical that different and showing subsets into the group split sandpile might fields harmonic that remarkable suggest results show Our dynamics robustness. stochastic corresponding the that processes Markov and simple by induced be can show fields harmonic we the Finally, that . natural a admits the group that configu- implies sandpile directly identifying which coordinates domains, different universal between of rations func- infinite set harmonic for a of space provides limits the tions scaling that show several we of Furthermore, size, domains. existence different of the domains conjecture on we dynamics extensive sandpile an these on of Based analysis rotations. and magnifications resemble ics transla- harmon- fourth-order by and and induced ones stretchings the second- while respectively, tions, smooth by resemble induced harmonics dynamics third-order The the identity. by patches the the characterized of constituting conservation group to apparent abelian and corresponds transformation the evolution smooth this through that cycles fields show harmonic periodic We under orders. identity different sandpile the of of ana- evolution we Here, the patches. lyze self-similar of whose composed group, fractal a abelian is an identity as form pro- configurations social serves recurrent phe- and Its physical, which a cesses. biological, criticality, automaton various self-organized in cellular study occurring nomenon to a 2018) 12, model July is archetypical review for sandpile the (received 2019 abelian 3, January approved The and WA, Redmond, Research, Microsoft Peres, Yuval by Edited a sandpile Lang abelian Moritz the of dynamics Harmonic eutn nsbeun opig napoesrfre oa an as unstable, to vertices referred other process a render in can topplings vertex subsequent a in due of particles resulting of toppling number redistribution the total The the the respectively. to at decreases two, domain vertices and the one of of by toppling corners whereas the sandpile, and number sides total the the in conserves of particles domain toppling the Thus, of of 1A). (Fig. interior one the particles by in of neighbors vertices number direct the its increasing of and each four of by num- particles the three, its decreasing of exceeds “topples,” ber vertex during and any unstable When of becomes particles vertex random. this of at number the chosen process are vertices this particles onto configuration, dropped initial slowly some from number Starting nonnegative sandpile. with a sand”), carries of (“grains vertex domain Each the lattice. square of standard the of domain angular below. explained abelian are the which of of this properties explained several 1), mathematical sandpile, be intriguing p. can various 3, community the (ref. scientific by SOC the study of to interest model the continued archetypical introduction, being and Despite initial research. first of its the field since active an passed remains y model sandpile 30 Even than char- (1–3). avalanches, more correlations as though spatiotemporal to scale-free referred become by processes acterized eventually in which relax and configurations automatically unstable critical certain to of forces into property fluctuating the converge by review), recent driven a systems for dissipative 2 ref. (see critical- (SOC) self-organized first ity of the concept was dropping the sandpile demonstrated abelian random which The model under (1). sand sandpile of grains idealized of (addition) an of evolution the T nttt fSineadTcnlg uti,30 lsenuug Austria Klosterneuburg, 3400 Austria, Technology and Science of Institute h adiemdli ellratmtndfie narect- a on defined automaton cellular a is model sandpile The a,Tn,adWeefl n18 1.Temdldescribes model The by (1). introduced 1987 in model Wiesenfeld mathematical and a Tang, Bak, is sandpile abelian he a,1 | identity n ihi Shkolnikov Mikhail and | C dynamics eerdt stecngrto fthe of configuration the as to referred | criticality a | C ij fparticles of (i , j ) 1073/pnas.1812015116/-/DCSupplemental. y at online information supporting contains article This 1 aadpsto:A pnsuc mlmnaino h loihst eeaethe generate to algorithms the at available of is dynamics implementation identity open-source sandpile An deposition: Data the under Published Submission.y Direct PNAS a is article This interest.y of conflict paper. y no the M.L. declare wrote research; authors M.S. The performed and M.L. M.S. and and data; M.L. analyzed M.S. research; and designed M.L. contributions: Author depth in it studied first who Creutz Michael identity—after (4). sandpile the of The relaxation (4). subsequent the group addition and vertex-wise sandpile configurations particles the of the recurrent to corresponds as all thereby known of operation group group, set abelian the an that forms fact the reach- configurations from transient be and stems can recurrent between it importance distinction if The the (4). only of configuration and stable carries minimally if the from recurrent that able vertex follows is It configuration drop (4). each given recurrent a to necessarily where is possible particles three configuration” always exactly is stable it “minimally relax- Since reached and (4). particles be C sandpile dropping can by the which configuration ing configurations other the Tran- any stable in (6). be from those nonzero often can as configurations is finitely recurrent defined vertex Equivalently, appear (6). given contrast, process any probability same in on the often configurations, where particle infinitely sient above a appears described a drop it process Thereby, to if Markov divided (6). the recurrent ones be in transient is can and configurations configuration recurrent stable stable classes, of two set into the that observed yet is law scale power (5). thus this unknown is for exponent and critical (1) the law a However, invariant. power after a topplings drop—follows of distri- particle number The random 13). total p. sizes—the 3, avalanche (ref. of topplings which of bution configuration order 1), the stable of a theorem independent reaches is 4, sandpile (ref. “relaxed” the terminates the of and eventually boundaries the process at this particles of domain, loss the to Due “avalanche.” owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence whom To hshl oepoesaiglmt o nntl i domains. big infinitely for limits scaling explore can to and help configuration thus sandpile possible universal every provide for directly coordinates dynamics harmonic on These depending wind. to merge, the and similar transform, fields, travel, sand- which harmonic dunes abelian under sand the dynamics smooth in that show arising demonstrate pile structures exten- we Here, fractal been y. self-similar 30 has the than more criticality” for “self-organized studied archetypical sively the such as for biological, sandpile abelian physical, model the many processes, in social of size. occur and any grain of phenomena avalanche additional an similar start each or Since all automatically where at nothing it state cause may sandpile, critical sand a a on to sand converges dropping slowly When Significance ij h dniyo hsaeingop h adieo Creutz or sandpile the group, abelian this of identity The was it (1), model sandpile the of introduction the after Soon atce noeeyvre fagvnconfiguration given a of vertex every onto particles NSlicense.y PNAS langmo.github.io/interpile/ www.pnas.org/lookup/suppl/doi:10. NSLts Articles Latest PNAS . y | C f10 of 1 the , 3 −

APPLIED MATHEMATICS A 0 3 0 B 3 42 Background Triangles 112 toppling Sierpinski 040 Triangles 403 Central 122 Square toppling Tropical 201 Curve 023 Tip Center Back 222

Fig. 1. Toppling of vertices and the sandpile identity. (A) If a vertex of the sandpile (here 3 × 3) carries four or more particles, it becomes unstable and topples, decreasing the number of its particles by four and increasing the number of particles carried by each of its (four or fewer) neighbors by one. The toppling of one vertex can render other, previously stable vertices unstable, resulting in an avalanche of subsequent topplings. (B) The sandpile identity on a 255 × 255 square domain. White, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively. Orange arrows denote the three different patch types in the identity: (i) the central square, (ii) background triangles, and (iii) Sierpinski triangles (ref. 3, p. 109ff). The latter are composed of three different patterns (yellow arrows), which we refer to as the tips, centers, and backs of the Sierpinski triangles. Additionally, thin 1D tropical curves are visible in the sandpile identity, slightly disturbing the patches they cross.

(4)—shows a remarkably complex self-similar fractal structure Results composed of patches covered with periodic patterns (Fig. 1B). Motivation. Recall that the stable configuration C = (C u )◦ Computational studies indicate that the structure of the iden- reached when relaxing an unstable configuration C u , with (·)◦ tity possesses a scaling limit for infinite domains. Indeed, for the relaxation operator corresponding to a series of topplings some configurations different from the sandpile identity, scal- resulting in a stable sandpile, can be expressed in terms of the ing limits for infinite domains were shown to exist, and the toppling function T [also referred to as the odometer function patches visible in these configurations as well as their robust- (16)], where Tij quantifies how often the vertex (i, j ) topples ness were analyzed (7–9). Nevertheless, corresponding results when relaxing C u (16, 17), for the sandpile identity—such as a closed formula for its construction—are still missing (ref. 3, p. 61), even though C = (C u )◦ = C u + ∆T , [1] recently a proof for the scaling limit of the sandpile identity was announced (10). At the time of writing, however, only rigorous with ∆ the discrete Laplace operator defined by proofs for some specific structural aspects of the sandpile iden- tity are available (11). For example, the thin (usually one pixel (∆T )ij = Ti+1,j + Ti−1,j + Ti,j +1 + Ti,j −1 − 4Ti,j . wide) “strings” or “curves” visible in the identity (Fig. 1B) were recently identified as tropical curves (12–15), structures from Note that we adopt the convention to set Tij = 0 for any (i, j ) tropical geometry arising, e.g., in string theory and statistical outside of the domain. physics. In general, it is nontrivial to find the toppling function T cor- In this article we study a yet unknown property of the sand- responding to the relaxation of an unstable configuration C u pile model: the evolution of the sandpile identity under harmonic without performing the relaxation itself. However, we can equiv- fields externally imposed by deterministically or stochastically alently state the inverse problem of how to construct an unstable dropping particles on boundary vertices of the domain. We show configuration C u such that the toppling function T takes some that such harmonic fields induce cyclic dynamics of the sand- predefined values. An interesting special case of this inverse pile identity through the , smoothly transforming problem arises when we require the toppling function to be har- individual patches and tropical curves, mapping them onto one monic, i.e., that (∆T )ij = 0 for all vertices (i, j ) in the interior another or merging them into different objects. The dynamics of the domain. Eq. 1 then implies that the relaxation of C u induced by the same harmonic field on domains of different size changes the particle numbers of vertices only at the boundary of u show remarkable similarities, strongly indicating that not only the domain and that Cij = Cij for all (i, j ) in the interior. Finally, the sandpile identity, but also sandpile dynamics possess scaling if we assume that C is recurrent, we can always require that C u limits. To mathematically interpret these observations, we intro- should be the result of adding some “potential” X to C itself, duce an extended analogue of the sandpile model where each i.e., that C u = C + X and, in consequence, that vertex at the domain boundary is allowed to carry a real number of particles. The set of recurrent configurations for this extended C = (C + X )◦ = C + X + ∆T . [2] sandpile model forms a connected , and we show that, on this group, the harmonic fields define closed and Clearly, the potential is given by X = −∆T and, since T is thus provide universal coordinates allowing us to map configu- harmonic, it has support only at the boundary; i.e., Xij = 0 for rations between the sandpile groups on different domains. Since every (i, j ) in the interior. In other words, we add particles there exists a natural inclusion of the usual sandpile group into to the boundary of a given recurrent configuration such that the extended one, the former can be interpreted as a discretiza- we arrive at the same configuration again after relaxing the tion of the latter, and the existence of universal coordinates thus sandpile. directly implies that the usual sandpile group admits a natural In the process described above, instead of adding all particles renormalization. at once, we can equivalently add one particle after the other and

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Lemma lemma: following the state first us let fields, harmonic k sand- configuration the that recurrent of Note every domains. limits infinite scaling on the dynamics pile discussing when poten- formula nonnegative general with which fields for following, harmonic the tials on In asso- focus nonnegative. is potential we field the however, harmonic that given guaranteed a not to is ciated it field general, harmonic in given dynam- because, sandpile a the by the induced in ics appear configurations recurrent and S2). and S1 Figs. the Appendix, that (SI and fields dynamics the (Eq. func- in floor differences definition the negligible of to instead function leads round tion or ceiling the using that btX function monic with field harmonic ics of of let thermore, lattice square standard the H on function harmonic Identity. valued Sandpile the of Dynamics Harmonic harmonic higher-order for once at parti- all of adding our functions. instead when to one effect – by the one and analyzed cles dynamics, yet nontrivial nobody toppling any – harmonic in knowledge constant result this not will following, function the in clear function become toppling harmonic eventually potential constant the will adding, step, sandpile to each corresponds the repeated this in that Notably, are identity. out steps the turns to These convergence it relaxed. and is times, the sides sandpile several at the vertex the each at on and vertex particles corners, two each and on domain dropped rectangular the is of particle single a method, dniysatn rmteepyconfiguration sandpile empty the the pro- construct from to method Creutz starting Michael a identity by of ago y reminiscent 30 nearly is posed above described procedure the become article. dynamics this these of order that focus such the which is added in meaningful, be and to have “oscillations,” particles these the The it. of to properties from back converging dynamic dissimilar finally more before intermediate configuration, and initial more these the become Intuitively, first configurations will algorithm. stable configurations we intermediate, this However, of executing started. series while we a which observe with also arrive will configuration finally still same will particles the we of (4), at addition commute the operator toppling Since the step. and each after sandpile the relax agadShkolnikov and Lang (t Z | em 1 Lemma specific by induced dynamics sandpile the discussing Before term The o ie etnua domain rectangular given a For eoedsrbn u prahi oedti,w oethat note we detail, more in approach our describing Before H Γ. Γ t I ) Γ H ≥ H eut ntesm adiedynamics. sandpile same the in results b.c eisrsrcint an to restriction its be o h domain the for c 1) h ro olw iey() hc neddirectly indeed which (4), widely follows proof The (17). 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H for Z )X 1 = I (t Let ≥0 t H ) (C (i 0 ≥ (t oee,a will as However, . nue ya inte- an by induced C  , hsnsc that such chosen ◦ H 0 ij j )= 1) + 0 + , T t ) nue ythe by induced 0 = ≥ ea integer- an be nteinterior the in 0 X enote We 0. eoe the denotes 0 4.I this In (4). ) ◦ I = H Fur- Γ. Z ij (t C 2 ≥ ) Z and [3] for for Γ 0) / h oeta fec ai etrbcm ongtv,which nonnegative, became set the vector to that used basis ensure us We to each allows 1). basis of appropriate (Table potential an less of the har- or definition all the four of in space order freedom integer-valued vector of the nine functions of of basis monic the set forming fields feasible harmonic computationally and small a on dynamics for of 3) relaxation the to associated function 1. Remark harmonic constant the of case special function the for proof the provides oi field 1B. monic Fig. of in patches indicated 2D names various the the by to identity refer har- sandpile We with the one. discussion order the of start fields we monic dynamics, field nontrivial harmonic any constant one to by lead the induced Because dynamics the them. only of discuss we reflection diagonal, to the up along odd-order dynamics of equivalent pair induces each functions domain, harmonic square a on Since computations. that, assume to tempting be aeFormula four Name order of functions harmonic discrete less for or Basis 1. 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For stretch., left. of translation; values top respective the the from domains, center the constants to the Γ, closest of vertex center the the at in lie to assumed H H H H H 4b 4a 3b 3a 2b 2a 1b 1a 0 ntefloig efcsoraayi ntesnpl identity sandpile the on analysis our focus we following, the In h adieiett yaisidcdb h rtodrhar- first-order the by induced dynamics identity sandpile The h adieiett yaisidcdby induced dynamics identity sandpile The o l amnc n all and harmonics all For a ob harmonic, be to has t ∈ 6 1 1 6 −i j i  (i Z 3 3 i 2 i 3 − − 2 ≥0 4 j H − ij and j i − − − 3ji 3ij + + u otesmlrte ewe Eqs. between similarities the to Due ij + . j 0 j 2 6i ij 1 2 2 c 2 c 1 = c + 3 H  2 1b 1a 2a + + ) IAppendix, SI j + c 255 ij + 2 c 1a (.) c c 2b + . c 3b 3a c k 4a r hsnsc htmin( that such chosen are 4b = j (t 4 × c i (.) 0 = ) 255 + 352 17 o.I general, In too. o 255 for 8 8 0 0 c 16 32 0 290 290 0 220 1a qaedmi nue yareasonably a by induced domain square 332 5 etcltas.o TCs of transl. 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APPLIED MATHEMATICS A = +

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Fig. 2. Sandpile identity dynamics induced by harmonic fields of order four or less on a 255 × 255 square domain. For odd-order harmonics, only one of the two dynamics is shown since the other one is identical up to switching of axes. For each harmonic, the sandpile dynamics at five time points of specific interest are shown. The first frame (t = 0) displayed for the dynamics induced by H1a corresponds to the sandpile identity, where the dynamics induced by 1a 1a 2a 2a 2b 2 2 2b all harmonics start (t = 0) and end (t = 1). (A–F)The six subfigures correspond to he harmonics: (A) Hij = i + c ;(B) Hij = ij + c ;(C) Hij = i − j + c ; 3a 3 2 3a 4a 1 4 2 2 4 2 2 4a 4b 1 3 3 4b (D) Hij = i − 3ij + c ;(E) Hij = 6 (i − 6i j + j − i − j ) + c , and (F) Hij = 6 (i j − ij ) + c (F). In A–F, white, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively.

2b 2 2 2b Also the dynamics induced by Hij = i − j + c (Fig. 2C different positions. After t = 0.5, the dynamics go through a sim- and SI Appendix, Movie S3) resemble stretching actions, how- ilar cycle to that before and finally reach the sandpile identity at ever, along the horizontal and vertical axes instead of the diag- t = 1. During each of these two half cycles, self-similar patches onal ones. Different from the dynamics induced by H 2a , the are smoothly transformed onto each other, enter the domain at central square is “disassembled” in the horizontal direction into its left and right boundaries, or leave it at its top and bottom many tropical curves and thus becomes a rectangle of shrinking boundaries. 3a 3 2 3a width and growing height. At t = 0.25, the width of the rectan- The dynamics induced by Hij = i − 3ij + c (Fig. 2D and gle approaches zero, leading to the fusion of the two background SI Appendix, Movie S4) resemble a horizontal translation of the triangles which were originally to the left and the right of the sandpile identity, overlaid by stretching dynamics similar to the central square. Subsequently, a rectangle is reestablished in the ones induced by H 2b . New Sierpinski triangles enter the domain center of the domain by the “fusion” of many tropical curves at its right boundary, leading to regular repetitive configurations entering the domain from the top and the bottom, reaching the of more and more, smaller and smaller Sierpinski triangles filling shape of a square at t = 0.5. While the configuration at t = 0.5 the whole domain. When the size of each individual Sierpin- is very similar to the sandpile identity, the tropical curves are at ski triangle approaches one vertex, the dynamics enter extended

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A 4b t fe some After . + r o com- not are 0 = and H H = j 0 = 2b 2a . 4 5 1 6 H . .Fur- B). 1/12 − (i or scom- is .99995 2a 3 i H H 2 j (SI or 2a 2b − − ecietetasaino rpclcre.We ecompared we different When between curves. dynamics tropical these of translation the describe For fields. harmonic other in discussed lines effects time different all In of situations. incorporating 3C). superposition such model a (Fig. proposes ones a which stronger depict above, of we vicinity 3D, configurations temporal Fig. fractal the same weak in the seemingly some emerge approximately of Interestingly, at periods caused times. start that distortion” absolute always ensuring “space seems size, configurations the thus domain random of fractals the effect strong scaling local the by of The for proximity sizes. compensate the had domain to in period different distortion this for time observable, lengths by was entrainment absolute configuration the similar which fractal during strong period when a time contrast, In the dynamics. estimated identity we sandpile extend- whole patches the small over of ing composed configurations con- domains, regular enough fractal see large strong only to vertex. for of expect, one vicinity thus would approach the we in figurations, patches distortion individual time the any Without of enter only areas can configurations the dynamics random when seemingly sandpile by the characterized that periods recall configurations, tal match. to patches run to a on had of time factor the a example, strong to by the faster For proportional of coincided. vicinity inversely the configurations factor in fractal dynamics a the size, by domain time respective However, configura- scaled different. clearly the we were Here, times when absolute 3B). same the (Fig. at configurations tions fractal strong sizes of domain different for dynamics dynamics sizes identity further sandpile the To the by of compared induced structure. boundaries we fractal effect, the this at the analyze emerge extending patches seemingly fractals, new these domain, that by sense “entrained” config- the be fractal to in these seem of of dynamics multiples vicinity the temporal urations, to the corresponding In fractions. times simple at by configurations induced dynamics random the in emerge rations section. this in eevsal eetbeol o ag nuhdmi sizes domain enough large for (t only detectable visually were tions (t became the fractals simple” these “less 1/12 pronounced the visually and less size the domain fraction, the smaller of the However, multiples correspond- to times absolute ing same the at appeared configurations dynamics the on analysis our focus by We induced size. when domain change field the harmonic analyze given scaling we a section, by induced this dynamics In the orders. how different induced of domain fields given harmonic a by on dynamics identity sandpile Size. the cussed Domain the Scaling of Effect identity struc- sandpile domain. fractal the square local sense of a structure on spaced the the regularly in resembling domains S6), emerged, tures tested Fig. all Appendix, for (SI that domains by nonconvex induced a effect, on dynamics this on lar the analyzed identity that further we sandpile observed When we the 2E). square of (Fig. central domain a typical square of patches emergence the surrounding show and domain circular a or lar S5 9 = iia fet otoefor those to effects Similar frac- strong of vicinity the in distortion” “time this interpret To sandpile the compared we when emerged picture different A configu- fractal accentuated section, previous the in shown As .Itrsigy h yaisidcdby induced dynamics the Interestingly, ). 255 N or i.3A). Fig. /10, o ifrn oanszs iia cetae fractal accentuated similar sizes, domain different For . t × 3 = 255 H H /12 3a 3a qaedmi o h oiin n hpso the of shapes and positions the for domain square u ics te amncfilsa h n of end the at fields harmonic other discuss but , on ,adsm ekfatlconfigura- fractal weak some and 3A), Fig. in N 255/63 1/12 × N rohrsml rcin Fg 3A). (Fig. fractions simple other or qaedmisfrdfeetdomain different for domains square ≈ H 4 1a na on h adieiett dynamics identity sandpile the , H ntepeiu eto,w dis- we section, previous the In 3a N 63 × N a lob bevdfor observed be also can × NSLts Articles Latest PNAS N ntetmoa vicinity temporal the in 63 oan with domains H qaedmi than domain square H H 3a 4a 4a rmseemingly from narectangu- a on r vnsimi- even are | N f10 of 5 odd, =

APPLIED MATHEMATICS A = 3 − 2 + 63 × 63 255 × 255 1/12 2/12 3/12 1/3 9/10 B = 3 − 2 + 63 × 63 4/1200 8/1200 12/1200 16/1200 20/1200 255 × 255 1/1200 2/1200 3/1200 4/1200 5/1200

C = 3 − 2 + D

1/30 1/8

Fig. 3. Effect of scaling the domain size on the sandpile identity dynamics. (A) For different domain sizes (Top row, 63 × 63; Bottom row, 255 × 255), 3a 3 2 3a pronounced fractal configurations occur at the same times in the dynamics induced by Hij = i − 3ij + c corresponding to multiples of simple fractions (e.g., of 1/12). These fractals are easier to visually detect at larger domain sizes, and some are visually detectable only for sufficiently large domains (t = 0.9). (B) Fractal structures seem to “entrain” the dynamics of patches in their temporal vicinity. To map the positions of such entrained patches onto one another for different domain sizes, time has to be scaled by a factor proportional to the domain size (here, by 255/63 ≈ 4), which is in contrast to the absolute timescale at which the fractal configurations themselves appear. (C) Weak fractal structures can appear in the vicinity of stronger ones (here, at t = 1/30 and t = 1/8, i.e., close to the fractal structures at t = 0 and t = 1/6, respectively). (D) Qualitative model for the interplay between the different effects described in A–C. A global timescale determines when fractal configurations occur. These fractals impose a time distortion proportional to the domain size in their temporal vicinity. The absolute time periods during which these time distortions affect the dynamics are approximately constant and independent of the domain size. The entrainment by weak fractals in the vicinity of stronger ones might lead to a superposition of different local timescales. In A–C, white, green, blue, and black pixels represent vertices carrying zero, one, two, or three particles, respectively.

the tropical curves seemed to have similar relative positions A harmonic field of order o acts on and transforms an object at the same absolute times (SI Appendix, Fig. S7A). Note that of dimension d only if o ≥ d, and the speed of the object in the already in the identity the positions of the tropical curves are sandpile dynamics then becomes proportional to N o−d , with N completely different for odd- and even-sized domains and that the domain size. they “fluctuate” when increasing the domain size. Similarly, the 2a 2b In the following, we formalize several aspects of our obser- patches in the dynamics induced by H and H had the same vations in terms of mathematical conjectures about the scal- relative position and shapes at the same absolute times (SI ing limits of configurations appearing in the sandpile identity Appendix, Fig. S7B). However, only when we scaled time by a fac- dynamics. These limits are defined with respect to sequences tor inversely proportional to the domain size, did the positions of 1 2 {ΓN }N >0 of discrete domains ΓN = Ω ∩ N Z approximating a the tropical curves become comparable (SI Appendix, Fig. S7C). 2 given continuous and compact convex domain Ω ⊂ R , where Finally, for fourth-order harmonic fields, hyperfractal configura- 1 2 1 tions appeared at the same absolute times, while the speed of the N Z denotes the standard square lattice scaled by N . For exam- 2 regularly spaced local fractal structures scaled with the domain ple, when setting Ω = [0, 1] , ΓN corresponds to an (N + 1) × size, and the speed of individual patches scaled with the square of (N + 1) square domain and {ΓN }N >0 to the sequence of square the domain size (SI Appendix, Fig. S8). Given these observations, domains with increasing size. For every domain ΓN , we denote by we propose to assign the dimensions dc = 1 to tropical curves, GN the corresponding sandpile group and by IN ∈ GN its iden- dp = 2 to patches, df = 3 to fractals, and de = 4 to hyperfractals. tity. To compare configurations CN ∈ GN belonging to different

6 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1812015116 Lang and Shkolnikov Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 agadShkolnikov and Lang Ω to verges 3 Fig. b·c in images” pixel “scaled ojcue4. Conjecture limit. 2 smooth piecewise (Fig. a harmonics second-order by induced let conjectures, these For it. following been from the I of yet extend all not because conjectures has here original it limit included a we such folklore, sidered of existence though Even the (Introduction). shown field, the in by tion denoted limit, scaling a function smooth piecewise 1. follow- Conjecture the in tilde configurations the of drop to sequence converges we a that notation, say of and abuse ing weak a By C domains edo oe re.Frhroe ic iceeplnma har- polynomial discrete harmonic since a Furthermore, field, is order. inducing limit lower its of conjectured to field the adding, terms diffu- to that the lower-order respect follows with on its directly invariant thus It all and limit. potential field, scaling the sive harmonic on Due effects the (19). diminishing of of processes have type physical scaling a of the limit, models to many scaling by diffusive obtained so-called limit a to corresponds this har- the scale alternatively can we field monic fields, (Eq. harmonic dynamics valued identity sandpile real the to of definition proportional the approximately extend because increases occurs vertices dynamics ary harmonic number the average the of of speedup factor the a itively, by scaled size domain be the become increasing dynamics when sandpile the faster fields, such for Since, fields. monic Ω, function smooth piecewise a to every on smooth piecewise are function this of I {I 3. Conjecture be should them limit. by 2A), the induced (Fig. in dynamics constant curves the first-order tropical that conjecture of that thus positions indicate we the simulations only our affect Since when harmonics zero limit. to the converges influence taking whose “defects” 1D represent {I 2. Conjecture field harmonic oapeeiesot ucinfrall for function smooth piecewise a to sequence the that stating converges. domain the e N ∞ H H N N N → ie hs entos sequence a definitions, these Given ojcue3 Conjecture 1 Conjecture h rpclcre nteiett r omnycniee to considered commonly are identity the in curves tropical The ojcue4 Conjecture H H h auso hsfnto r icws mohon smooth piecewise are function this of values the (t (t prtn coordinate-wise. operating (p (t (t R ) ) t = ) )} )} o every for eoetesnpl yais(Eq. dynamics sandpile the denote ∈ N N [0, C Γ N >0 >0 N lim N →∞ C H ∞), (bNp weak-∗ C ∞ o vr oyoilhroi field harmonic polynomial every For ntesm eune ednt by denote we sequence, same the in weak-∗ o udai amncfields harmonic quadratic For h sequence The tefby itself o ierhroi fields harmonic linear For t ∞ H Ω : Z ttsta h adieiett tefpossesses itself identity sandpile the that states h sequence the ∈ generalizes Ω omlzsoritiinta h dynamics the that intuition our formalizes whenever ntedomain the on h ahmtcleuvlneo the of equivalence mathematical the c), [0, → ψ (p | ovre to converges ∞). X R ovre oapeeiesot function smooth piecewise a to converges N ) N ojcue4 Conjecture H C e −o ffraysot etfunction test smooth any for if |/∂ N Moreover, I N {I ∞ (p I { uhta t restriction its that such 2−o N ∞ N ojcue3 Conjecture {I C Γ e on {I )dp −o hl en omnassump- common a being While . I N N N ∞ H A H N } | Ω. H oaheesaiglmt.Intu- limits. scaling achieve to I Γ (tN (t } N = fprilsdopda bound- at dropped particles of ∞ and (tN N N >0 ). ojcue1 Conjecture o vr on in point every for Z . >0 hnbcmseuvln to equivalent becomes then o every for 2−o Ω Moreover, 2 does. ihteflo function floor the with B, t ψ )} N t . weak-∗ { )} (p ∈ N C e nue yagiven a by induced 3) ,tm a to has time 3B), (Fig. N ohge-re har- higher-order to >0 )C [0, N >0 B t } ∞ H Interestingly, ∞). {C N weak-∗ ∈ H and weak-∗ (p o vr on in point every for >0 , H a hsb con- be thus can [0, ovre oa to converges , N )dp h sequence the forder of } h sequence the N approach C) ∞). weak-∗ Ω, C t N e . −o . N >0 N converges h values the converges Ω : H o weak-∗ fwe If . | o Γ → to 3) con- N and ψ R, to : iceo egh1 hsipista h utetgroup quotient the that by decimal implies denote the to this we the isomorphic 1, If after domain. length positions the of the of circle boundary in the only at differ point and interior the osdrta,frec configuration each group, for that, consider group—of sandpile extended the group sandpile as original denote the the we which that of group group configurations abelian Lie topological recurrent a a for forms of as model the space same sandpile top- the extended at The are The one. particles model interior. original sandpile of the extended the numbers the in for real numbers rules integer drop pling Accord- only to particles. but allow of boundaries also number still can integer we domain nonnegative ingly, the a of interior only the carry in vertex each whereas ticles, boundary the at vertex domain each where model pile underlying its existence. its of explains homomorphism which group, group sandpile Lie the a of renormalization approximates natural How- the group trivial. show, be will to we as has ever, them between homomorphism group any (|G of vertices consisting adjacent domain a two exam- (|G on vertex groups For one sandpile sandpile. the just the abelian of respect orders that, the the see cannot of ple, to structure renormalization easy group is nontrivial it underlying any since glance general, first renormalization in at natural surprising a seem natu- of 3A. might a existence Fig. admits The group in renormalization. sandpile indicated ral the that mapping implies the immediately to This size—similar domains between different configurations universal of map provide to har- directly us the allowing dynamics that coordinates these show inducing we sand- fields Furthermore, group. monic usual continuous Lie approximate this the dynamics of harmonic geodesics that the that showing and group by intuition group pile this the following, formalize domains, the In we that finite trajectories. suggests continuous some dynamics but approximate harmonic they big the limits. of sufficiently scaling “smoothness” size their apparent on different concerning already of conjectures However, domains between several on relationship stated dynamics the and analyzed sandpile Group. we harmonic Sandpile section, the Abelian previous the the of In Renormalization and Topology The exist. the might limits scaling scaling when 2 more fourth-order positions (Fig. of and size dynamics times domain the predictable in at appear harmonics that for structures consider limits we fractal scaling when local different However, two dynamics. identity least sandpile at the exist there that with at expect Together appear we should times. configurations rational fractal regular all such to 3A). limit, (Fig. corresponding the fractions in times simple at less occur and more less configurations and regular more size, such 2 configurations domain of (Fig. the fractions increasing fractal simple when to Furthermore, regular corresponding times at fields, appear harmonic higher-order function smooth piecewise t 5. Conjecture continuous by limit. induced in diffusive the are only in dynamics fields counterparts the harmonic continuous (20), their terms from low-order differ fields monic ∈ oudrtn h oooyo h xeddsnpl group, sandpile extended the of topology the understand To sand- extended an introducing first by analysis our start We by induced dynamics sandpile the in above, described As [0, ∞) G ˜ Γ ∩ salwdt ar ongtv elnme fpar- of number real nonnegative a carry to allowed is otisalcngrtoswihaeeulto equal are which configurations all contains G Q, o vr oyoilhroi field harmonic polynomial every For a eitrrtda iceiaino Lie a of discretization a as interpreted be can h sequence the | 4 = E and n nadmi ossigof consisting domain a on and 4A) Fig. , (R/Z) | I r orm,adthus and coprime, are 4B) Fig. 15, = ∞ H F {I (t ∂ and ). N Γ H . (t npriua,teei natural a is there particular, In G hsmasthat means this 4, Conjecture ,even S8B), Fig. Appendix, SI )} sadsrt subgroup. discrete a is N >0 C NSLts Articles Latest PNAS ojcue5 Conjecture fteuulsandpile usual the of weak-∗ ∂ Γ ovre oa to converges fteconvex the of H ttsthat, states n every and R/Z G | ˜ D–F /G f10 of 7 C G ˜ the — in is ).

APPLIED MATHEMATICS A 1 vertex 1 B 2 vertices

0,4 1,4 2,4 3,4 4,4 3,3 01234 0,3 1,3 2,3 3,3 2 0 0,3 1,3 2,3 3,3 4,3 0,2 1,2 2,2 3,2 0,2 1,2 2,2 3,2 4,2 0,1 1,1 2,1 3,1 0,1 1,1 2,1 3,1 4,1 3 3,3 1,02,0 3,0 0,0 1,0 2,0 3,0 4,0 3,3 C D

1 2 0 3 3 3

E Stochastic, 2 = +

0.2 0.6 1.0 10.0 100.0 F G

H Stochastic, = 3 − 2 +

0.0 0.03 0.075 1/31.0 Fig. 4. (A) Topology of the usual and extended sandpile groups on a domain consisting of a single vertex: space of nonnegative configurations (Left) ˜ and the corresponding sandpile groups G = Z/4Z ⊂ G = R/4Z (Right). (B) Space of legal configurations (Left) and the sandpile group (Right) on a domain consisting of two adjacent vertices. The extended sandpile group is a 2D torus obtained by gluing the opposite sides of the rhombus. The lattice points on G˜ form G. Moving along horizontal or vertical directions corresponds to increasing the amount of sand carried by the left or right vertex. (C) Har- monic functions H serve as universal coordinates for configurations C˜ of the extended sandpile group and thus define natural renormalization maps between the extended sandpile groups on different domains [here from the two-vertices domain (B) to the one-vertex domain (A)]. A harmonic func- ˜ P i i tion H identifying a given configuration C can be determined by expressing it as a linear combination H = i tiH of harmonic basis functions H , which ˜ = −∆ | + + ◦ results in a set of linear equations when inserted into the surjective homomorphism CΓ2 ( (H Γ k) I) which can be solved for the coefficients ti. The renormalization of the usual sandpile group corresponds to the floor of the renormalization of the extended one, restricted to integer-valued con- figurations. (D) Visual depiction of the result of the renormalization described in C.(E) Stochastic realization of the sandpile identity dynamics induced 2a I(I,I(t)) by Hij = ij.(F) Evolution of the normalized variation of information VI(I; I(t)) = 1 − H(I,I(t)) between the stochastic identity dynamics in D and the sand- pile identity over four periods, with I the mutual information and H the joint entropy. F, Inset shows the evolution at t = 1, 2, ... for 100 periods. (G) Avalanche size distribution over one full period of the stochastic identity dynamics induced by H1a (blue) and H2a (red). The dashed black lines show power-law distributions with critical coefficients −1.371 (H1a) and −1.481 (H2a), respectively. (H) Encoding of information (here the string “PNAS”) into the stochastic sandpile dynamics induced by H3a. The encoded information is visually not detectable in intermediate configurations, while becom- 1 ing clearly visible at multiples of t = 3 . In E and H, white, green, blue, and black pixels represent vertices carrying zero, one, two, and three particles, respectively.

choice of the Riemannian metric on the extended sandpile group directly follows that each connected component of G˜ is a torus. ∂Γ coming as a lift of the standard flat metric on (R/Z) , which To determine the number of such connected components, recall ∂Γ allows us to discuss the volume of G˜ . Since (R/Z) is a torus that G is generated by dropping particles only onto vertices at ˜ of volume 1 and G˜ is its extension by G, the volume of G˜ is the domain boundaries (4), which implies that G is the quotient ∂Γ equal to |G|, the number of elements in the usual sandpile group. of R . Since the continuous image of a connected space is con- Furthermore, the extended sandpile group can be viewed to be nected, G˜ contains only one connected component, and thus the constructed of cubes [0, 1]∂Γ, one for each element of G, which extended sandpile group is a single torus. We summarize these are glued along their boundary faces. From this construction, it results in the following proposition.

8 of 10 | www.pnas.org/cgi/doi/10.1073/pnas.1812015116 Lang and Shkolnikov Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 h xsec fteeuieslcodntsdrcl mle that implies directly coordinates universal these of existence The of inclusion to Since restricted natural group. a sandpile exists extended there the for coordinates universal domains of finite to corresponding family the of inclusion choice specific the Z Z 2. Lemma on problem Dirichlet the to solution of a outside of existence the for of holds also which (4), vertices G boundary same to the only the particles in that by ping recall indistinguishable Again, induced are domain. the dynamics when of interior their time and same field, the harmonic at topple sandpiles on dynamics sandpile monic G h nescino all of intersection the to isomorphic canonically values integer taking quotients functions harmonic of on space the to equal G 1. Theorem projection The closed. is on geodesics by continuous group sandpile extended the I of dynamics (Eq. corresponding group the sandpile as usual the written of by dynamics homomorphism identity the define (−∆(H then We nonnegative. number function minimal harmonic the given on group a sandpile For extended lows: the of configurations onto homomorphism the define on functions let harmonic this, by discrete For real-valued functions. of harmonic and groups R group sandpile yteiaeo h Laplacian the of image the by configuration the by group sandpile configuration The on the being configurations configuration stable nonrecurrent only 16 the If of 4A). 15 (Fig. 4 length of circle sandpile is usual group the Therefore, recurrent. are when configurations case stable the in example, For 1. configuration from configurations maps agadShkolnikov and Lang of sion subgroup discrete its dimension 1. Proposition ˜ ˜ ˜ H 2 2 Γ N hs h olwn em sasrihfradconsequence straightforward a is lemma following the Thus, . Proof: domains of family injective exhausting an consider Now Since sandpile two the between relationship the analyze we Next, h eainhpbtenteetne adiegroup sandpile extended the between relationship The . } i h orfunction floor the via /∆ (t H Γ +1 N Let N = ) Z ≥1 Z → . H ⊂ Γ Therefore, G G | i.e., ; ˜ Z/4 η Γ H G H ˜ N ,where 4B), (Fig. (tH h homomorphism The H + Let Γ |∂ into /H N /H I h usaeo nee-audhrois ethen We harmonics. integer-valued of subspace the eteetne adiegopof group sandpile extended the be /K G k (21): Γ| hr r aoia ujciehomomorphisms surjective canonical are There , H Γ h rjcoiso hs yaiscrepn to correspond dynamics these of trajectories The ). C + ) Z with K (t r louieslfrteoiia adiegroup. sandpile original the for universal also are , Z n h xeddsnpl ru is group sandpile extended the and h xeddsnpl group sandpile extended The N N n volume and G G ˜ → ∈ rjcsot l xeddsnpl groups sandpile extended all onto projects ˜ N = ) +1 ⊂ I nteopst ieto,teflo function floor the direction, opposite the In . G ) etekre of kernel the be sa2 ou fae 5 ti smrhcto isomorphic is It 15. area of torus 2D a is G K Γ ˜ H → G ◦ η ˜ G N ie hshmmrhs,tesandpile the homomorphism, this Given . ne h orfnto sacb fvolume of cube a is function floor the under ∞ G N C (btH ∞ k K +1 +1 ij . 3 ∈ scci ihtesnpl dniygiven identity sandpile the with cyclic is sa olw:Teeeit aua inclu- natural a exists There follows: as is N : b.c /K 3 = hr h no fall of union the where , rjlim proj = G Z ⊂ ˜ is n ti hnntrlt define to natural then is it and c), G G ˜ R ≥0 ˜ η |G N ie h ylctaetr ftehar- the of trajectory cyclic the gives K hog h dniy If identity. the through H Since . G Γ Γ N to By . Γ N /∆Z |. ∆Z Z : +1 η hs h sa n h extended the and usual the Thus, . uhthat such → H . hc iepoetosbetween projections give which , ossso w daetvertices, adjacent two of consists G ssurjective. is Γ and Γ {Γ hs utet are quotients these 2, Lemma h riaeo recurrent a of preimage The . Γ, I Γ G ossso utoevre,all vertex, one just of consists N ⊂ H η |∂ G N eoe h utetof quotient the denotes →∞ ˜ N hoe 1 Theorem into H Z = Γ| G } ˜ Γ : apn amncfields harmonic mapping Γ N N b → H H ∈ = . −∆( G r eurn,wt the with recurrent, are ≥1 I ˜ G ial,w oethat note we Finally, . ˜ G H ˜ |Γ N sgnrtdb drop- by generated is . c efis determine first we , hs coordinates, these , , H hr sanatural a is There G ˜ fti edscto geodesic this of ˜ 2, = G needn from independent Γ ˜ N | H N Γ Γ Z hskre is kernel This . rvdsaset a provides . .. the i.e., R/4Z, + N satrsof torus a is 2 etespace the be h extended the H k is n denote and ) a be can 3) H ∈ S becomes Γ C N η {Γ G (H sfol- as ˜ ij Γ Z 0 Γ the , 0 = N N and and = ) R . G = ⊂ ˜ Γ . rlrnraiain.I oedti,frapi fdomains of pair a for nat- detail, admits group more sandpile Γ In extended the renormalizations. as ural well as original the ehave we η of jection for coordinate tion configuration given G tv.Atrec atcedo,w a hnrlxtesandpile the relax then time can the we associate drop, and particle each After ative. potential prob- field harmonic the a integer to to proportional according distribution stochastically ability them (Eq. drop vertices alternatively boundary can at particles Potentials. dropping Harmonic ministically of Realizations Stochastic 1. Question research: future our in answer to hope we which question, group. map renormalization natural this the combining with inclusion group By the sandpile extended defined. the for uniquely projection renormalization is projection malization i adieiett yaisidcdb amncfilsof fields harmonic by stochas- induced the for dynamics distributions identity size sandpile critical avalanche the tic stochastic compare the the and (Eq. of of determine step to dynamics exponents time us sandpile every allows at This the dropped algorithm. is generate particle to one only algorithm our of their of neighborhood the 4G). to (Fig. returning configurations extended initial before show noise dynamics of sandpile periods higher-order that configurations random utilizing seemingly As by possi- into becomes studies. information furthermore encode different it to ble in robustness, this distribution coeffi- of size consequence critical a avalanche the the for discrepancy of values the cient the obtained explains explore partially experimentally to sufficient which between be group, billions not sandpile of might whole hundreds drops even particle that random indicates of result This fields ergodic. harmonic most are particles by that induced implies dynamics dropping which sandpile (4), stochastic domain by the the of reached vertices boundary be onto only can become configuration identity recurrent the of structure the (t disturbed does significantly time longer sandpile deterministic (t icantly the of stochas- reproduced their details the are to all identity nearly domain close period, the very one After of remain counterparts. interior dynamics the sandpile particles in tic where while domain the dropped, of noise are boundaries This the noise. toward by stronger overlaid is are but counterparts, deterministic har- the for process field Markov monic this of realization a to corresponding physical a sandpile. in possible abelian experimentally be deter- the process might resembling its system Markov it of this simplicity, interval the implement its time onto to to (fixed) Due drops the counterpart. particle as ministic expected successive vertex same boundary in two the same resulting between with process interval dynamics Markov time identity a sandpile to stochastic corresponds algorithm This with ˜ 1 2 ial,nt that note Finally, ial,w oeta ifrn rmtedtriitcversion deterministic the from different that note we Finally, every that considering when remarkable is robustness This dynamics identity sandpile stochastic the depict we 4D, Fig. In sntijcie o any for injective, not is ⊂ to H Γ k 2 H ∈ ,1 2, 1, 0, = G ˜ hr sacnnclrnraiainpoeto from projection renormalization canonical a is there , 1 η C ˜ 1 Fg 4 (Fig. steinclusion the Is satisfying H 2 (H sgvnby given is G 2a a C ˜ 2 = ) h tcatcdnmc lsl eebetheir resemble closely dynamics stochastic The . . . . 2 → Fg 4 (Fig. C η C G ˜ h ubro atce led dropped. already particles of number the ˜ 1 2 C (H 2 ˜ and hoe 1 Theorem H ∈ 2 n h orfunction floor the and t b = 10 = C rvddta hsptnili nonneg- is potential this that provided , G .Given C). ˜ ˜ = ) 1 H 2 .T banti rjcinfra for projection this obtain To D). η 1 = since efis eemn amncfunc- harmonic a determine first we , |X = 2 H /H (H k and H η a 1 , | .Ol fe signif- a after Only 4D). Fig. in Z ) (H H G otersligconfiguration, resulting the to → Γ n hssriga universal a as serving thus and t b ietypsstefollowing the poses directly 2 1 ) H 100 = H ∈ → G ⊂ ˜ .Nt ht while that, Note 4D). (Fig. h eomlzto pro- renormalization the , ∞ Γ G NSLts Articles Latest PNAS 2 nisomorphism? an 1 with n hsterenor- the thus and , nFg 4 Fig. in o h sa sandpile usual the for nta fdeter- of Instead η G ˜ 2 1 (H X → D H a = ) G and fagiven a of 1 eget we , η | E). 2 ,we 3), f10 of 9 (H 3), b ),

APPLIED MATHEMATICS different order (Fig. 4F). Our results show that the critical expo- be composed of fractals. These two interpretations do not nec- nent depends on the specific harmonic field, indicating that there essarily have to contradict each other, e.g., if we allowed tropical might exist different regions of the sandpile group where the crit- curves to be composed of fine strings of fractals. We note that ical coefficient has different values or that the critical coefficient such interpretations are somewhat reminiscent of discussions in might depend on probability distribution determining on which string theory, where strings (tropical curves) and branes of dif- vertices particles are dropped. ferent dimensions (patches, fractals, hyperfractals, and so on) occur. Discussion The close relationship between the usual and the extended We expect that our results showing the existence of smooth sand- sandpile group (Fig. 4 A–D) indicates that we can extend our pile identity dynamics induced by harmonic fields will provide knowledge of the former by studying the latter. For example, important guidelines for future studies of the abelian sand- Question 1 proposes a concrete scaling limit for the extended pile. For example, it might be possible to prove the conjec- sandpile group. Based on the observation that, when induced by tured limits for the sandpile identity dynamics (Conjectures 2–5) the same harmonic field, both sandpile models topple at identi- using well-established techniques from statistical mechanics (19). cal times while the time interval between subsequent topplings Specifically, if one can determine the universal coordinates of quickly decreases when increasing the domain size, one might particularly simple configurations, like the minimally stable one speculate whether the limit of the usual sandpile group might where each vertex carries three particles (SI Appendix, Fig. S9), actually be the same as the limit of the extended one. Fur- it might become possible to use these configurations as initial thermore, given that the harmonic fields themselves generate conditions for a coarse-grained model describing the limiting the sandpile group, our observation that regular fractal config- dynamics and, by simulating back in time, indirectly prove the urations occur at times corresponding to multiples of simple existence of a scaling limit for the identity. As another example, fractions has an interesting explanation: These simple fractions the emergence of new patches at boundary positions seemingly correspond to simple roots of the identity with respect to the consistent between the dynamics induced by different harmonic respective generator/harmonic. fields (Fig. 2) suggests that it might be possible to extend the frac- Finally, our results indicate that harmonic fields might divide tal structure of the sandpile identity beyond the boundaries of the the sandpile group into different regions, each showing scale-free domain, similar to the continuation of analytic functions in the spatiotemporal relationships, but with different critical expo- complex plane. Provided that the dynamics induced by the har- nents (Fig. 4G). If one could determine the critical exponents monic H 4a (Fig. 2E) are indeed similar to zooming actions, this corresponding to a basis for the harmonic fields with high con- extension might result in a (potentially infinite) wallpaper-like fidence using the periodicity of the sandpile identity dynamics, structure composed of regularly spaced, locally similar fractal it might be possible to reconstruct the critical exponent of the structures. The similarities between the sandpile identity dynam- whole sandpile group by taking an adequately weighted mean. ics induced by H 4a on different domains (SI Appendix, Fig. S6) furthermore suggest that there might exist only one fundamental Materials and Methods extension of the identity for each domain topology. An open-source implementation of the algorithms to generate the sandpile That second-order harmonics transform the central square of identity dynamics is available at langmo.github.io/interpile/ (22, 23). This the identity either into the tips of two Sierpinski triangles (H 2a , website also contains additional movies for other domains, harmonics, and Fig. 2B) or into a large set of tropical curves (H 2b , Fig. 2C) sug- initial configurations. gests that all patches constituting the sandpile identity might be ACKNOWLEDGMENTS. M.L. is grateful to the members of the C Guet and G composed of tropical curves. In contrast, the dynamics induced Tkacikˇ groups for valuable comments and support. M.S. is grateful to Nikita 4a by H (Fig. 2C) suggest that the sandpile identity itself might Kalinin for inspiring communications.

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