University of Groningen. Abelian Sandpiles and the Harmonic Model Schmidt, Klaus; Verbitskiy, Evgeny

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1 University of Groningen Abelian Sandpiles and the Harmonic Model Schmidt, Klaus; Verbitskiy, Evgeny Published in: Communications in Mathematical Physics DOI: /s IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2009 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Schmidt, K., & Verbitskiy, E. (2009). Abelian Sandpiles and the Harmonic Model. Communications in Mathematical Physics, 292(3), DOI: /s Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 Commun. Math. Phys. 292, (2009) Digital Object Identifier (DOI) /s Communications in Mathematical Physics Abelian Sandpiles and the Harmonic Model Klaus Schmidt 1,2, Evgeny Verbitskiy 3,4 1 Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria. klaus.schmidt@univie.ac.at 2 Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria 3 Philips Research, High Tech Campus 36 (M/S 2), 5656 AE, Eindhoven, The Netherlands. evgeny.verbitskiy@philips.com 4 Department of Mathematics, University of Groningen, PO Box 407, 9700 AK, Groningen, The Netherlands Received: 15 January 2009 / Accepted: 14 April 2009 Published online: 15 August 2009 The Author(s) This article is published with open access at Springerlink.com Abstract: We present a construction of an entropy-preserving equivariant surjective map from the d-dimensional critical sandpile model to a certain closed, shift-invariant subgroup of T Zd (the harmonic model ). A similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the Bernoulli property of the measure of maximal entropy for that model. Contents 1. Introduction Four models Outline of the paper A Potential Function and its l 1 -Multipliers The Harmonic Model Linearization Homoclinic points Symbolic covers of the harmonic model Kernels of covering maps The Abelian Sandpile Model The Critical Sandpile Model Surjectivity of the maps ξ g : R X f (d) Properties of the maps ξ g, g Ĩ d The Dissipative Sandpile Model The dissipative harmonic model The covering map ξ (γ ) : R (γ ) X f (d,γ ) Conclusions and Final Remarks References...759

3 722 K. Schmidt, E. Verbitskiy 1. Introduction For any integer d 2let h d = ( log 2d 2 ) d cos(2π x i ) dx 1 dx d, (1.1) i=1 h 2 = 1.166, h 3 = 1.673, etc. It turns out that for d 2, h d is the topological entropy of three different d-dimensional models in mathematical physics, probability theory, and dynamical systems. For d = 2, there is even a fourth model with the same entropy h d Four models. The d-dimensional abelian sandpile model was introduced by Bak, Tang and Wiesenfeld in [3,4] and attracted a lot of attention after the discovery of the Abelian property by Dhar in [8]. The set of infinite allowed configurations of the sandpile model is the shift-invariant subset R {0,...,2d 1} Zd defined in (4.4) and discussed in Sect In [10], Dhar showed that the topological entropy of the shift-action σ R on R is also given by (3.4), which implies that every shift-invariant measure µ of maximal entropy on R has entropy (1.1). Shift-invariant measures on R were studied in some detail by Athreya and Jarai in [1,2], Jarai and Redig in [13]; however, the question of uniqueness of the measure of maximal entropy is still unresolved. Spanning trees of finite graphs are classical objects in combinatorics and graph theory. In 1991, Pemantle in his seminal paper [17] addressed the question of constructing uniform probability measures on the set T d of infinite spanning trees on Z d i.e., on the set of spanning subgraphs of Z d without loops. This work was continued in 1993 by Burton and Pemantle [5], where the authors observed that the topological entropy of the set of all spanning trees in Z d is also given by the formula (1.1). Another problem discussed in [5] is the uniqueness of the shift-invariant measure of maximal entropy on T d (the proof in [5] is not complete, but Sheffield has recently completed the proof in [22]. This coincidence of entropies raised the question about the relation between these models. A partial answer to this question was given in 1998 by R. Solomyak in [24]: she constructed injective mappings from the set of rooted spanning trees on finite regions of Z d into X f (d) such that the images are sufficiently separated. In particular, this provided a direct proof of coincidence of the topological entropies of α f (d) and σ Td without making use of formula (1.1). In dimension 2, spanning trees are related not only to the sandpile models (cf. e.g., [19] for a detailed account) and, by [24], to the harmonic model, but also to a dimer model (more precisely, to the even shift-action on the two-dimensional dimer model) by [5]. However, the connections between the abelian sandpiles and spanning trees (as well as dimers in dimension 2), are non-local: they are obtained by restricting the models to finite regions in Z d (or Z 2 ) and constructing maps between these restrictions, but these maps are not consistent as the finite regions increase to Z d. In this paper we study the relation between the infinite abelian sandpile models and the algebraic dynamical systems called the harmonic models. The purpose of this paper is to define a shift-equivariant, surjective local mapping between these models: from the 1 In the physics literature it is more customary to view the sandpile model as a subset of {1,...,d} Zd by adding 1 to each coordinate.

4 Abelian Sandpiles and the Harmonic Model 723 infinite critical sandpile model R to the harmonic model. Although we are not able to prove that this mapping is almost one-to-one it has the property that it sends every shift-invariant measure of maximal entropy on R to Haar measure on X f (d). Moreover, it sheds some light on the somewhat elusive group structure of R. Firstly, the dual group of X f (d) is the group G d = R d /( f (d) ), where R d = Z[u ± 1,...,u± d ] is the ring of Laurent polynomials with integer coefficients in the variables u 1,...,u d, and ( f (d) ) is the principal ideal in R d generated by f (d) = 2d d i=1 (u i +ui 1 ). The group G d is the correct infinite analogue of the groups of addition operators defined on finite volumes, see [9,19] (cf. Sect. 7). Secondly, the map ξ Id constructed in this paper gives rise to an equivalence relation on R with x y x y ker(ξ Id ), such that R / is a compact abelian group. Moreover, R /, viewed as a dynamical system under the natural shift-action of Z d, has the topological entropy (1.1). This extends the result of [16], obtained in the case of dissipative sandpile model, to the critical sandpile model. Finally, we also identify an algebraic dynamical system isomorphic to the dissipative sandpile model. This allows an easy extension of the results in [16]: namely, the uniqueness of the measure of maximal entropy on the set of infinite recurrent configurations in the dissipative case. Unfortunately, we are not yet able to establish the analogous uniqueness result in the critical case Outline of the paper. Sect. 2 investigates certain multipliers of the potential function (or Green s function) of the simple random walk on Z d. In Sect. 3 these results are used to describe the homoclinic points of the harmonic model. These points are then used to define shift-equivariant maps from the space l (Z d, Z) of all bounded d-parameter sequences of integers to X f (d). In Sect. 4 we introduce the critical and dissipative sandpile models. In Sect. 5 we show that the maps found in Sect. 3 send the critical sandpile model R onto X f (d), preserve topological entropy, and map every measure of maximal entropy on R to Haar measure on the harmonic model. After a brief discussion of further properties of these maps in Subsect. 5.2, we turn to dissipative sandpile models in Sect. 6 and define an analogous map to another closed, shift-invariant subgroup of T Zd. The main result in [16] shows that this map is almost one-to-one, which implies that the measure of maximal entropy on the dissipative sandpile model is unique and Bernoulli. 2. A Potential Function and its l 1 -Multipliers Let d 1. For every i = 1,...,d we write e (i) = (0,...,0, 1, 0,...,0) for the i th unit vector in Z d, and we set 0 = (0,...,0) Z d. We identify the cartesian product W d = R Zd with the set of formal real power series in the variables u ±1 1,...,u±1 d by viewing each w = (w n ) W d as the power series w n u n (2.1) n Z d

5 724 K. Schmidt, E. Verbitskiy with w n R and u n = u n 2 1 un d d w w on W d is defined by for every n = (n 1,...,n d ) Z d.theinvolution w n = w n, n Z d. (2.2) For E Z d we denote by π E : W d R E the projection onto the coordinates in E. For every p 1weregardl p (Z d ) as the set of all w W d with w p = 1/p <. n Z d w n p Similarly we view l (Z d ) as the set of all bounded elements in W d, equipped with the supremum norm. Finally we denote by R d = Z[u ±1 1,...,u±1 d ] l1 (Z d ) W d the ring of Laurent polynomials with integer coefficients. Every h in any of these spaces will be written as h = (h n ) = n Z d h nu n with h n R (resp. h n Z for h R d ). The map (m,w) u m w with (u m w) n = w n m is a Z d -action by automorphisms of the additive group W d which extends linearly to an R d -action on W d given by h w = h n u n w (2.3) n Z d for every h R d and w W d.ifw also lies in R d this definition is consistent with the usual product in R d. For the following discussion we assume that d 2 and consider the irreducible Laurent polynomial d f (d) = 2d (u i + ui 1 ) R d. (2.4) i=1 The equation f (d) w = 1 (2.5) with w W d admits a multitude of solutions. 2 However, there is a distinguished (or fundamental) solution w (d) of (2.5) which has a deep probabilistic meaning: it is a certain multiple of the lattice Green s function of the symmetric nearest-neighbour random walk on Z d (cf. [6,12,25,27]). Definition 2.1. For every n = (n 1,...,n d ) Z d and t = (t 1,...,t d ) T d we set n, t = d j=1 n j t j T. We denote by F (d) (t) = n Z d f (d) n e2πi n,t = 2d 2 d cos(2πt j ), t = (t 1,...,t d ) T d, (2.6) j=1 the Fourier transform of f (d). 2 Under the obvious embedding of R d l (Z d, Z), the constant polynomial 1 R d corresponds to the element δ (0) l (Z d, Z) given by { δ n (0) 1 if n = 0, = 0 otherwise.

6 Abelian Sandpiles and the Harmonic Model 725 (1) For d = 2, (2) For d 3, w n (2) e 2πi n,t 1 := T 2 F (2) dt for every n Z 2. (t) Td w n (d) e 2πi n,t := F (d) (t) dt for every n Zd. The difference in these definitions for d = 2 and d > 2 is a consequence of the fact that the simple random walk on Z 2 recurrent, while on higher dimensional lattices it is transient. Theorem 2.2. ([6,12,25,27]) We write for the Euclidean norm on Z d. (i) For every d 2, w (d) satisfies (2.5). (ii) For d = 2, w n (2) = 0 if n = 0, 1 8π log n κ n 2 c 4 (n4 1 +n4 2 ) O( n 4 ) if n = 0, n 2 1 (2.7) where κ 2 > 0 and c 2 > 0. In particular, w (2) 0 = 0 and w n (2) < 0 for all n = 0. Moreover, 4 w (2) n = (P(X k = n X 0 = 0) P(X k = 0 X 0 = 0)), k=1 where (X k ) is the symmetric nearest-neighbour random walk on Z 2. (iii) For d 3, n d 2 w (d) n = κ d + c d as n, where κ d > 0, c d > 0. Moreover, 2d w (d) n = 1 n 4 di=1 n 4 i 3 d+2 n 2 + O( n 4 ) (2.8) P(X k = n X 0 = 0) >0 for every n Z d, k=0 where (X k ) is again the symmetric nearest-neighbour random walk on Z d. Definition 2.3. Let w (d) W d be the point appearing in Definition 2.1. Weset I d = { } g R d : g w (d) l 1 (Z d ) ( f (d) ), (2.9) where ( f (d) ) = f (d) R d is the principal ideal generated by f (d). Since w n (d) for every n Z d it is clear that I d = Id ={g : g I d }. = w (d) n

7 726 K. Schmidt, E. Verbitskiy Theorem 2.4. The ideal I d is of the form where with 1 = (1,...,1). I d = ( f (d) ) + I 3 d, (2.10) I d = {h R d : h(1) = 0} = (1 u 1 ) R d + + (1 u d ) R d (2.11) For the proof of Theorem 2.4 we need several lemmas. We set J d = ( f (d) ) + I 3 d R d. (2.12) Lemma 2.5. Let g = k Z d g ku k R d. Then g J d if and only if it satisfies the following conditions (2.13) (2.16). g k = 0, (2.13) k Z d g k k i = 0 for i = 1,...,d, (2.14) k=(k 1,...,k d ) Z d g k k i k j = 0 for 1 i = j d, (2.15) k=(k 1,...,k d ) Z d g k (ki 2 k 2 j ) = 0 for 1 i = j d. (2.16) k=(k 1,...,k d ) Z d Proof. Condition (2.13) is equivalent to saying that g I d. In conjunction with (2.13), (2.14) is equivalent to saying that g I 2 d : indeed, if g I d, then it is of the form g = with a i R d for i = 1,...,d. Then g u j = d (1 u i ) a i (2.17) i=1 g k k j u k1 1 uk j 1 j k=(k 1,...,k d ) Z d u k d d = a j + and g u j (1) = 0 if and only if a j I d. If g I d is of the form (2.17) and satisfies (2.14) weset a j = d i=1 (1 u i ) a i u j, d (1 u i ) b i, j (2.18) i=1 with b i, j R d. Condition (2.15) is satisfied if and only if 2 g (1) = a i a j = b i, j (1) + b j,i (1) = 0 u i u j u j u i for 1 i = j d.

8 Abelian Sandpiles and the Harmonic Model 727 Finally, if g satisfies (2.13) (2.14) and is of the form (2.17) (2.18) with b i, j R d for all i, j, then (2.16) is equivalent to the existence of a constant c R with g k k 2 a i i = 2 (1) = 2b i,i (1) = c u i k=(k 1,...,k d ) Z d for i = 1,...,d. The last equation shows that b i,i b 1,1 I d for i = 2,...,d. By combining all these observations we have proved that g satisfies (2.13) (2.16) if and only if it is of the form g = h 1 d (1 u i ) 2 + h 2 (2.19) i=1 with c Z, h 1 R d and h 2 I 3 d. The set of all such g R d is an ideal which we denote by J. Clearly, I 3 d J and d i=1 (1 u i ) 2 J. Since (1 u i ) 2 (1 ui 1 ) I 3 d for i = 1,...,d as well, we conclude that f (d) = d (1 u i ) 2 i=1 d (1 ui 1 ) (1 u i ) 2 J. (2.20) i=1 This shows that J J d, and the reverse inclusion also follows from (2.20) and (2.19). Lemma 2.6. I d J d. Proof. We assume that g I d and set v = g w (d). In order to verify (2.13) we argue by contradiction and assume that k g k = 0. If d = 2 then k v n = g k log n + l.o.t., 2π for large n. Ifd 3, then v n = κ d k g k + l.o.t. n d 2 for large n. In both cases it is evident that v l 1 (Z d ). By taking (2.13) into account one gets that, for every d 2, v n = (g w (d) ) n = k = g k w (d) n k T d e 2πi n,t k g ke 2πi k,t 2d 2 d j=1 cos(2πt j ) dt. Hence v = (v n ) is the sequence of Fourier coefficients of the function H(t) = k g ke 2πi k,t 2d 2 d j=1 cos(2πt j ).

9 728 K. Schmidt, E. Verbitskiy If v l 1 (Z d ), then H must be a continuous function on T d. Since t = 0 is the only zero of F (d) on T d (cf. (2.6)), the numerator G = k g ke 2πi k, must compensate for this singularity. Consider the Taylor series expansion of G at t = 0: G(t) = g k +2πi k +h.o.t. d t j g k k j 2π 2 j=1 k d j=1 The Taylor series expansion of F (d) at t = 0 is given by t 2 j g k k 2 j 4π 2 t i t j g k k i k j i = j k k Suppose that d F (d) (t) = 4π 2 t 2 j + h.o.t. j=1 h(t) = a 0 + d j=1 b j t j + d j=1 c j t 2 j + i = j d i, j t i t j + h.o.t t t2 d + h.o.t is continuous at t = 0. Then a 0 = 0, b j = 0 for all j, c j = c for all j, d ij = 0 for all i = j, and for some constant c. If any of these conditions is violated, then one easily produces examples of sequences t (m) 0 as m with distinct limits lim m h(t (m) ).By applying this to H we obtain (2.13) (2.16), so that g J d by Lemma 2.5. To establish the inclusion J d I d,wehavetoshowthatforanyg J d, g u l 1 (Z d ), where u W d of the form di=1 ni 4 ω n = n d+4, or ω n = 1 n γ with γ d 2. For d = 2, we also have to treat the case ω n = log n. These results are obtained in the following three lemmas. Lemma 2.7. Suppose that d 2 and that ω W d is given by { 0 if n = 0, ω n = di=1 ni 4 n d+4 if n = 0. If g R d satisfies (2.13), then g ω l 1 (Z d ). Proof. Let M = max{ k :g k = 0}, and suppose that n > M. Then (g ω) n = di=1 (n i k i ) 4 g k n k d+4 = di=1 ni 4 + O( n 3 ) g k n d+4 (1+O( n 1 )) k k ( ) di=1 ni 4 ( ) ( ) 1 1 = n d+4 g k + O n d+1 = O n d+1. Therefore, n (g ω) n <. k

10 Abelian Sandpiles and the Harmonic Model 729 For the reverse inclusion J d I d we need different arguments for d = 2 and for d 3. We start with the case d = 2. Lemma 2.8. Suppose that g = k Z 2 g ku k R 2 satisfies (2.13). WesetS + = {k : g k > 0} and S ={k : g k < 0}. Put M g = 2 k S + g k = 2 k S g k and define two polynomials in the variables (n 1, n 2 ): P + (n 1, n 2 ) = ((n 1 k 1 ) 2 + (n 2 k 2 ) 2) g k = n k 2gk, k S + k S + P (n 1, n 2 ) = ((n 1 k 1 ) 2 + (n 2 k 2 ) 2) g k = n k 2 gk. k S k S Let m g bethedegreeofp= P + P.If then g ω l 1 (Z 2 ), where ω n = (2.21) M g m g 3, (2.22) { 0 if n = (0, 0), log n if n = (0, 0). Proof. Since k Z 2 g k = 0by(2.13), M g = deg P + = deg P and m g = deg P < max(deg P +, deg P ) = M g. Let v = g ω. Hence, for all n with n > max{ k :k S + S }, one has (g ω) n = 1 2 log P +(n 1, n 2 ) P (n 1, n 2 ) = 1 ( 2 log 1+ P ) +(n 1, n 2 ) P (n 1, n 2 ). P (n 1, n 2 ) There exist constants C, N such that P + (n 1, n 2 ) P (n 1, n 2 ) P (n 1, n 2 ) C n m g n M = C g n M < 1 g m g 2 for n N. Hence we can find another constant C such that (g ω) n C n M g m g for all sufficiently large n. Since M g m g 3, we finally conclude that g ω l 1 (Z 2 ). Lemma 2.9. Suppose that g J d (cf. (2.13) (2.16)), and that ω W d is given by { 0 if n = 0, ω n = 1 n γ if n = 0, for some integer γ d 2. Then g ω l 1 (Z d ).

11 730 K. Schmidt, E. Verbitskiy Proof. Let S g ={k Z d : g k = 0}, M = max{ k :k S g }, and note that where is the Euclidean norm on Z d R d. We fix n Z d with n > M and set S g B d ={y R d : y M}, (2.23) ( d ) γ/2 h (n) (k) = n k γ = (n i k i ) 2. (2.24) In calculating the Taylor expansion of h (n) as a function of the variables k 1,...,k d we use the notation I!=i 1! i d!, I =i i d and I h (n) i1+ +id h (n) = i=1 k I k i 1 1 k i n n, (2.25) for I = (i 1,...,i d ) Z d +, k = (k 1,...,k d ) Z d, where Z + ={n Z : n 0}. Then the Taylor expansion of h (n) for k M is given by where h (n) (k) = I 2 1 I! I h (n) k I (0) k I + I =3 R (n) 1 I h (n) I sup y B d I! k I (y) R (n) I k I, (cf. (2.23)). The first and second order derivatives of h (n) have the following form. h (n) (k) = γ (n i k i ) n k γ 2 for i = 1,...,d, k i 2 h (n) γ 4 (k) = γ (γ +2) (n i k i ) (n j k j ) n k k i k j 2 h (n) k 2 i It follows that for i, j = 1,...,d, i = j, (k) = γ (γ +2) (n i k i ) 2 n k γ 4 γ 2 γ n k for i = 1,...,d. h (n) (0) = n γ, h (n) (0) = γ n i n γ 2, k i 2 h (n) (0) = γ (γ +2) n i n j n γ 4, i = j, k i k j 2 h (n) (0) = γ (γ +2) ni 2 n γ 4 γ n γ 2. k 2 i

12 Abelian Sandpiles and the Harmonic Model 731 For I = (i 1,...,i d ) Z d + and y Rd, I h (n) k I (y) = P I (n 1,...,n d ) n y γ 2 I, where P I is a polynomial of degree at most I in the variables n 1,...,n d. Therefore, for every I Z d + with I =3, R (n) I O( n γ 3 ). (2.26) By using the Taylor series expansion of h (n) above we obtain that, for all n with sufficiently large norm, (g ω) n = g k h (n) (k) k S g h(n) (0) g k k S g + d h (n) (0) g k k i k i=1 i k Sg + 2 h (n) (0) g k k i k j k i = j i k j k Sg + 1 d 2 h (n) (0) 2 k 2 g k ki 2 i=1 i k Sg + O( n (γ +3) ). (2.27) The first three terms on the right-hand side of the above inequality vanish because of (2.13), (2.14), and (2.15). The fourth term is estimated as follows: (2.16) impliesthat g k ki 2 = const for all i = 1,...,d, k S g and we denote by C this common value. Then d 2 h (n) (0) k 2 g k k 2 i i=1 i k Sg d ( = γ(γ +2) ni 2 n γ 4 γ n γ 2) C = [ γ(γ +2) γ d ] C n γ 2. i=1 Therefore, if γ = d 2, then the fourth term vanishes. If γ>d 2, i.e., if γ d 1, then the fourth term is of the order O( n (d+1) ), and is thus summable over Z d.the remainder term in (2.27) is always summable since γ +3 d +1. Proof of Theorem 2.4. We start with the case d 3. Recall that for n = 0, w n (d) = κ di=1 d n d 2 +c ni 4 d n d+4 3c d 1 d +2 n d +O( n (d+2) ) =: ω n (1) + ω(2) n + ω(3) n + r n.

13 732 K. Schmidt, E. Verbitskiy Applying g, we conclude that g w l 1 (Z d ), because g ω (1), g ω (3) l 1 (Z d ) by Lemma 2.9 for γ = d 2 and γ = d, respectively; g ω (2) l 1 (Z d ) by Lemma 2.7; (g r) n = O( n (d+2) ), and hence g r l 1 (Z d ) as well. Now consider the case d = 2. Then w n (2) = 1 8π log n κ n c + n4 2 2 n For any g J 2, 1 n 2 +O( n 4 )=ω n (1) + ω(2) n + ω(3) n + r n. g ω (2), g ω (3), g r l 1 (Z 2 ) (2.28) by the results of Lemmas 2.7 and 2.9. The remaining term g ω (1) has to be treated slightly differently. First of all, note that since J 2 = ( f ) + (u 1 1) 3 R 2 + (u 1 1) 2 (u 2 1) R 2 + (u 1 1)(u 2 1) 2 R 2 + (u 2 1) 3 R 2, it is sufficient to check that g ω (1) l 1 (Z 2 ) only for the set of generators, i.e., for g = f (2),(u 1 1) 3,(u 1 1) 2 (u 2 1), (u 1 1)(u 2 1) 2,(u 2 1) 3. For g = f (2), f (2) w (2) = δ (0) l 1 (Z 2 ) (cf. (2.5) and Footnote 2 on page 724), and hence, given (2.28), f ω (1) l 1 (Z 2 ) as well. For g = (u 1 1) 3 R 2 we apply Lemma 2.8. Note that S + ={(1, 0), (3, 0)}, S ={(0, 0), (2, 0)}, and P + = ((n 1 3) 2 + n 2 2 )((n 1 1) 2 + n 2 2 )3, P = ((n 1 2) 2 + n 2 2 )3 (n n2 2 ) P + P i = 9 60n n n n4 1 4n5 1 36n n 1n n 2 1 n2 2 +8n3 1 n2 2 18n n 1n 4 2. Hence M g = deg P + = deg P = 8, m g = deg P = 5, M g m g = 3. Therefore, by Lemma 2.8, (g ω (1) ) n =O( n 3 ), and hence g ω (1) l 1 (Z 2 ), which is equivalent to g I 2. The same calculation shows that (u 2 1) 3 I 2. Furthermore, since f (2) I 2 and u 1 1 (u 1 1) 3 + f (2) = u 1 2 (u 1 1)(u 2 1) 2, we obtain that (u 1 1)(u 2 1) 2 I 2 and, by symmetry, that (u 1 1) 2 (u 2 1) I 2. This proves that J 2 I 2, and Lemma 2.6 yields that J 2 = I 2.

14 Abelian Sandpiles and the Harmonic Model The Harmonic Model Let d > 1. We define the shift-action α of Z d on T Zd by (α m x) n = x m+n (3.1) for every m, n Z d and x = (x n ) T Zd and consider, for every h R d, the group homomorphism h(α) = m Z d h m α m : T Zd T Zd. (3.2) Since R d is an integral domain, Pontryagin duality implies that h(α) is surjective for every nonzero h R d (it is dual to the injective homomorphism from R d = TZ d to itself consisting of multiplication by h). Let f (d) R d be given by (2.4) and let X f (d) T Zd be the closed, connected, shift-invariant subgroup d X f (d) = ker f (d) (α) = x = (x n) T Zd : 2dx n (x n+e ( j) + x n e ( j)) = 0 for every n Z d. j=1 (3.3) We denote by α f (d) the restriction of α to X f (d). Since every α m, m Z d, is a continuous automorphism of X f (d),thez d -action α f (d) preserves the normalized Haar measure f (d) λ X of X f (d) f (d). The Laurent polynomial f (d) can be viewed as a Laplacian on Z d and every x = (x n ) X f (d) is harmonic (mod 1) in the sense that, for every n Z d,2d x n is the sum of its 2d neighbouring coordinates (mod 1). This is the reason for calling (X f (d),α f (d)) the d-dimensional harmonic model. According to [21, Theorem 18.1] and [21, Theorem 19.5], the metric entropy of α f (d) with respect to λ X coincides with the topological entropy of α f (d) f (d) and is given by h λx f (d) (α f (d)) = h top(α f (d)) = log f (d) (2πit 1,...,2πit d ) dt 1 dt d <. (3.4) Furthermore, α f (d) is Bernoulli with respect to λ X f (d) (cf. [21]). Since every constant element of T Zd lies in X f (d), α f (d) has uncountably many fixed points and is therefore nonexpansive: for every ε > 0 there exists a nonzero point x = (x n ) in X f (d) with where x n <ε for every n Z d, t (mod 1) =min { t n :n Z}, t R. (3.5)

15 734 K. Schmidt, E. Verbitskiy 3.1. Linearization. Consider the surjective map ρ : W d = R Zd T Zd given by ρ(w) n = w n (mod 1) (3.6) for every n Z d and w = (w n ) W d. We write σ for the shift action (σ m w) n = (u m w) n = w m+n (3.7) of Z d on W d (cf. (2.3)). As in (3.2) we set, for every g = n Z d g nu n R d, h = n Z d h nu n l 1 (Z d ), h(σ ) = h n σ n : W d W d. (3.8) n Z d Then h(σ )(w) = h w, g(α)(ρ(w)) = ρ(g (3.9) w) for every w W d (cf. (2.2) and (2.3)). We set W d (Z) = Z Zd W d. According to (3.3), W f (d) For later use we denote by := ρ 1 (X f (d)) ={w W d : ρ(w) X f (d)} = f (d) (σ ) 1 (W d (Z)) ={w W d : f (d) w W d (Z)}. (3.10) R W d, Z W d (Z), T T Zd (3.11) the set of constant elements. If c is an element of R, Z or T we denote by c the corresponding constant element of R, Z or T. Equation (3.10) allows us to view W f (d) as the linearization of X f (d) Homoclinic points. Let β be an algebraic Z d -action on a compact abelian group Y, i.e., a Z d -action by continuous group automorphisms of Y. An element y Y is homoclinic for β (or β-homoclinic to 0) if lim n β n y = 0. The set of all homoclinic points of β is a subgroup of Y, denoted by β (Y ). If β is an expansive algebraic Z d -action on a compact abelian group Y then β (Y ) is countable, and β (Y ) = {0} if and only if β has positive entropy with respect to the Haar measure λ Y (or, equivalently, positive topological entropy). Furthermore, β (Y ) is dense in Y if and only if β has completely positive entropy w.r.t. λ Y. Finally, if β is expansive, then β n x 0 exponentially fast (in an appropriate metric) as n. All these results can be found in [14]. If β is nonexpansive on Y, then there is no guarantee that β (Y ) = {0} even if β has completely positive entropy. Furthermore, β-homoclinic points y may have the property that β n y 0veryslowlyas n. The Z d -action α f (d) on X f (d) is nonexpansive and the investigation of its homoclinic points therefore requires a little more care. In particular we shall have to restrict our attention to α f (d)-homoclinic points x for which α n f (d) x 0 sufficiently fast as n. For this reason we set

16 Abelian Sandpiles and the Harmonic Model 735 (1) α (X f (d)) = x α(x f (d)) : x n <, (3.12) n Z where is defined in (3.5). d In order to describe the homoclinic groups α (X f (d)) and (1) α (X f (d)) we set x = ρ(w (d) ) X f (d). (3.13) The fact that x X f (d) is a consequence of Theorem 2.2 (1) and (3.10). Proposition 3.1. Let α f (d) be the algebraic Z d -action on the compact abelian group X f (d) defined in (3.3). Then every homoclinic point z α (X f (d)) is of the form z = ρ(h w (d) ) for some h R d. Furthermore, ) (1) α (X f ({h (d)) = ρ w (d) : h I d } (3.14) (cf. Theorem 2.2, (2.9) and (3.12)). Proof. If z α (X f (d)), then we choose w l (Z d ) with lim n w n = 0 and ρ(w) = z.from(3.10) we know that f (d) w W d (Z), and the smallness of (most of) the coordinates of w guarantees that h = f (d) w R d = l 1 (Z d ) l (Z d, Z), where l (Z d, Z) ={w = (w n ) l (Z d ) : w n Z for every n Z d }. If we multiply the last identity by w (d) we get that w (d) f (d) w = w = w (d) h = h w (d) for some h R d. If z (1) α (X f (d)) then w l 1 (Z d ) and hence, by definition, h I d. Conversely, if h I d, then z = ρ(h w (d) ) (1) α (X f (d)). Remark 3.2. A homoclinic point z of an algebraic Z d -action β on a compact abelian group Y is fundamental if its homoclinic group β (Y ) is the countable group generated by the orbit {β n z : n Z d } (cf. [14]). Proposition 3.1 shows that x = ρ(w (d) ) also has the property that its orbit under α f (d) generates the homoclinic groups α (X f (d)) and (1) α (X f (d)), although x itself may not be homoclinic (e.g., when d = 2) Symbolic covers of the harmonic model. We construct, for every homoclinic point z (1) α (X f (d)), a shift-equivariant group homomorphism from l (Z d, Z) to X f (d) which we subsequently use to find symbolic covers of α f (d). According to Proposition 3.1, every homoclinic point z (1) α (X f d ) is of the form z = g(α)(x ) = ρ(g w (d) ) for some g I d. We define group homomorphisms ξ g : l (Z d ) l (Z d ) and ξ g : l (Z d ) T Zd by ξ g (w) = (g w (d) )(σ )(w) = (g w (d) ) w and ξ g (w) = (ρ ξ g )(w). (3.15)

17 736 K. Schmidt, E. Verbitskiy These maps are well-defined, since ξ g (w) n = k Z d w n k (g w (d) ) k converges for every n, and equivariant in the sense that ξ g σ n = σ n ξ g, ξ g σ n = α n ξ g, ξ g h(σ ) = h(σ ) ξ g, ξ g h(σ ) = h(α) ξ g, for every n Z d, g I d and h R d. We also note that ξ g (v) = v n α n ( g(α)(x ) ) for every v = (v n Z d n ) l (Z d, Z). Proposition 3.3. For every g I d, { ξ g (l (Z d {0} if g ( f (d) ),, Z)) = X f (d) if g Ĩ d := I d ( f (d) ), (3.16) (3.17) (cf. (2.9) and (3.15) (3.16)). We begin the proof of Proposition 3.3 with two lemmas. Lemma 3.4. For every w l (Z d ) and g I d, ( f (d) (σ ) ξ g )(w)= f (d) (g w (d) ) w = g ( f (d) w (d) ) w = g w = g(σ )(w). (3.18) Furthermore, ξ g (l (Z d, Z)) X f (d). Proof. For every h,v R d, Theorem 2.2 (1) implies that f (d) h w (d) v = h f (d) w (d) v = h v. (3.19) Fix g I d and let K 1 and V K ={ K +1,...,K 1} Zd l (Z d, Z). Then V K is shift-invariant and compact in the topology of pointwise convergence, and the set V K V K of points with only finitely many nonzero coordinates is dense in V K.For v V K R d, ξ g (v) = (g w (d) ) v (3.20) and ( f (d) (σ ) ξ g )(v) = f (d) g w (d) v = g f (d) w (d) v = g v (3.21) by (3.15) and (3.19). Since both ξ g and multiplication by g are continuous on V K,(3.21) holds for every v V K. By letting K we obtain (3.21) for every v l (Z d, Z), hence for every v M 1 l (Z d, Z) with M 1, and finally, again by coordinatewise convergence, for every w l (Z d ), as claimed in (3.18). For the last assertion of the lemma we note that ξ g (v) = ρ((g w (d) ) v) = (g v )(α)(x ) X f (d) (3.22) for every v V K (cf.(3.13)). The continuity argument above yields that ξ g(v) X f (d) for every v l (Z d, Z).

18 Abelian Sandpiles and the Harmonic Model 737 Lemma 3.5. If g Ĩ d then ξ g (l (Z d, Z)) = X f (d). In fact, ξ g ( 2d ) = X f (d), where m ={0,...,m 1} Zd l (Z d, Z) for every m 1. Furthermore, the restriction of ξ g to 2d (or to any other closed, bounded, shift-invariant subset of l (Z d, Z)) is continuous in the product topology on that space. Proof. We fix x X f (d) and define w W f (d) by demanding that ρ(w) = x and 0 w n < 1 for every n Z d.ifv = f (d) (σ )(w) then 2d +1 v n 2d 1for every n Z d. Since ξ g commutes with f (d) (σ ) by (3.16), (3.21) shows that Hence ξ g (v) = (ρ ξ g )(v) = g(α)(x). (3.23) X f (d) ξ g (l (Z d, Z)) ξ g (V 2d ) g(α)(x f (d)), (3.24) where V K ={ K +1,...,K 1} Zd l (Z d, Z). We claim that Indeed, consider the exact sequence g(α)(x f (d)) = X f (d). (3.25) {0} ker g(α) X f (d) X f (d) g(α) X f (d) {0}, set Y = ker g(α) X f (d), Z = g(α)(x f (d)) X f (d), write α Y and α Z for the restrictions of α to Y and Z, and denote by α the Z d -action induced by α on X f (d)/z. Yuzvinskii s addition formula ([21, (14.1)]) implies that h top (α f (d)) = h top (α Y ) + h top (α Z ) = h top (α ) + h top (α Z ), where we are using the fact that the topological entropies of these actions coincide with their metric entropies with respect to Haar measure. Since the polynomials f (d) and g have no common factors, h top (α Y ) = 0by[21, Corollary 18.5], hence h top (α f (d)) = h top (α Z ) is given by (3.4) and 0 < h top (α f (d)) <. Since the Haar measure λ X f (d) of X f (d) is the unique measure of maximal entropy for α f (d) we conclude that λ X f (d) (g(α) (X f (d))) = 1 and g(α)(x f (d)) = X f (d), as claimed in (3.25). We have proved that ξ g (V 2d ) = X f (d). Ifv l (Z d, Z) satisfies that v n = 2d 1 for every n Z d, then v + V 2d = 4d 1, and (3.24) implies that ξ g ( 4d 1 ) = ξ g (V 2d ) + ξ g (v ) = X f (d) + ξ g (v ) = X f (d). We still have to show that ξ g ( 2d ) = X f (d). FixM 1 for the moment and put Let Q M ={ M,...,M} d Z d. (3.26) l (Z d, Z + ) ={v l (Z d, Z) : v n 0 for every n Z d }.

19 738 K. Schmidt, E. Verbitskiy For every v l (Z d, Z + ) and n Z d we set { h (v,n) u n f (d) if v n 2d = 0 otherwise, and we put H (v,m) = h (v,n), T (v) = v H (v,m). n Q If M D M (v) = n Q M v n n 2 max, (3.27) where max is the maximum norm on R d, then T (v) = v if and only if v n < 2d for every n Q M, and D M (T (v)) D M (v) + 2 (3.28) otherwise. We define inductively T n (v) = T (T n 1 (v)), n 2, and conclude from (3.28) that there exists, for every v l (Z d, Z + ), an integer K M (v) 0 with ṽ (M) = T k (v) for every k K M (v). (3.29) For v 4d 1 and any M 1, the corresponding ṽ (M) satisfies 0 ṽ n (M) 2d 1ifn Q M, ṽ n (M) v n if n max = M +1, ṽ n (M) v n (2d 1) (2M +1) d, {n: n max =M+1} ṽ (M) n = v n if n max > M +1, (3.30) where max is the maximum norm on R d. Let Ṽ (M) ={ṽ (M) : v 4d 1 }. Since v ṽ (M) ( f (d) ) it is clear that ξ g (v) = ξ g (ṽ (M) ) for every v 4d 1 and g Ĩ d. Since g Ĩ d, Theorem 2.4 implies that there exists a constant C > 0 with (g w (d) ) n C n max d 1 for every nonzero n Z d. Hence ξ g (ṽ (M) ) 0 ξ g ( v (M) ) 0 < 4d (2M +1) d C (M +1) d 1 0 as M, where It follows that v (M) n = { ṽ (M) n if n Q M, otherwise. lim ξ g (v v (M) ) = 0 M in the topology of coordinate-wise convergence. Since v n v (M) {v 4d 1 : 0 v n < 2d for every n Q M } for every v 4d 1 and M 1, we conclude that ξ g ( 2d ) is dense in X f (d).asξ g ( 2d ) is also closed, this implies that ξ g ( 2d ) = X f (d), as claimed.

20 Abelian Sandpiles and the Harmonic Model 739 Remark 3.6. Although we have not yet introduced sandpiles and their stabilization (this will happen in Sect. 4), the second part of the proof of Lemma 3.5 is effectively a sandpile argument, and ṽ (M) is a stabilization of v in Q M. Proof of Proposition 3.3. If g lies in Ĩ d, Lemma 3.5 shows that ξ g ( 2d ) = ξ g (l (Z d, Z)) = X f (d). On the other hand, if g = h f (d) for some h R d, then g w (d) R d, and hence ξ g (v) n Z for every n Z d and v l (Z d, Z), implying that ξ g (v) = Kernels of covering maps. Having found compact shift-invariant subsets V l (Z d, Z) such that the restrictions of ξ g to V are surjective for every g Ĩ d (cf. Lemma 3.5), we turn to the problem of determining the kernels of the group homomorphisms ξ g : l (Z d, Z) X f (d), g I d (cf. (3.15)). We shall see below that ker(ξ g ) depends on g and that ker ξ gh ker(ξ g ) for g I d and 0 = h R d.inviewofthisit is desirable to characterize the set K d = g I d ker(ξ g ) (3.31) of all v l (Z d, Z) which are sent to 0 by every ξ g, g I d. In the following discussion we set, for every ideal J R d, and put X J ={x T Zd : g(α)(x) = 0 for every g J} = g J X f (d) = ker g(α), (3.32) I d /( f (d) ) = X f (d)/ X Id. (3.33) In order to explain (3.33) we note that the dual group of X f (d) is a subgroup of X f (d) = R d /( f (d) ), hence X f (d) is a quotient of X f (d) by a closed, shift-invariant subgroup, which is the annihilator of I d /( f (d) ) and hence equal to X Id.TheZ d -action α f (d) on X f (d) induces a Z d -action α f (d) on X f (d). Note that α n is dual to multiplication by u n f (d) on I d /( f (d) ). With this notation we have the following result. Theorem 3.7. There exists a surjective group homomorphism η : l (Z d, Z) X f (d) with the following properties: (1) The homomorphism η is equivariant in the sense that η σ n = α n η for every f (d) n Z d ; (2) ker(η) = K d ; (3) The topological entropy of α f (d) coincides with that of α f (d) (cf. (3.4)). For the proof of Theorem 3.7 we choose and fix a set of generators G d ={g (1),..., g (m) } of I d (for d = 2 we may take, for example, G 2 ={g (1), g (2), g (3) } with g (1) = (1 u 1 ) 2 (1 u 2 ), g (2) = (1 u 1 ) (1 u 2 ) 2 and g (3) = (1 u 1 ) 2 + (1 u 2 ) 2 );for d 3 we can use the set of generators G d ={f (d) } {(u i 1) (u j 1) (u k 1) : i, j, k = 1,...,d}). With such a choice of G d we define a map ξ Id : l (Z d, Z) X m f (d) (3.34)

21 740 K. Schmidt, E. Verbitskiy by setting ξ Id (v) = (ξ g (1)(v),...,ξ g (m)(v)) (3.35) for every v l (Z d, Z). Lemma 3.8. There exists a continuous shift-equivariant group isomorphism θ d : ξ Id (l (Z d, Z)) X f (d). (3.36) Proof. We define a continuous group homomorphism θ : X f (d) θ (x) = (g (1) (α)(x),..., g (m) (α)(x)) for every x X f (d). According to (3.15) and (3.16), X m f (d) by setting ξ g h(σ )(v) = ρ(g w (d) h v) = g(α) ξ h (v) for every g, h Ĩ d and v R d, and hence, by continuity, for every g, h Ĩ d and v l (Z d, Z). Since ξ h (l (Z d, Z)) = X f (d) by Lemma 3.5 we conclude that On the other hand, ξ Id (l (Z d, Z)) ξ Id h(σ )(l (Z d, Z)) = θ (X f (d)). ξ Id (v) = (g (1) (α) v (α)(x ),...,g (m) (α) v (α)(x )) θ (X f (d)) for every v R d and hence, again by continuity, for every v l (Z d, Z). Wehave proved that ξ Id (l (Z d, Z))) = θ (X f (d)). The homomorphism θ has kernel X Id and induces a group isomorphism θ : X f (d) θ (X f (d)). The proof is completed by setting θ d = (θ ) 1. Proof of Theorem 3.7. We set η = θ d ξ Id (cf. (3.34) (3.36)). By definition, K d = ker(ξ Id ) = ker(η). The equivariance of η is obvious. Furthermore, h top ( α) h top (α f (d)), since X f (d) is an equivariant quotient of X f (d). On the other hand, X f (d) = ξ Id (l (Z d, Z)), and the first coordinate projection π 1 : ξ Id (l (Z d, Z) X f (d) issurjectivebylemma3.5. This implies that h top (β 1 ) = h top ( α) h top (α f (d)), so that these entropies have to coincide. In order to characterize the kernel K d of η further we need a lemma and a definition. Lemma 3.9. For every y l (Z d ) with ρ(y) X I 3 there exists a unique c(y) [0, 1) d with f (d) y+ c(y) l (Z d, Z), where c(y) denotes the element of R with c(y) n = c(y) for every n Z d.

22 Abelian Sandpiles and the Harmonic Model 741 Proof. Let x X I 3 d and y l (Z d ) with ρ(y) = x. According to the definition of X I 3 d this means that g(α)(x) = ρ(g y) = 0 for every g I 3 d. Since g j = (u j 1) f (d) I 3 d for j = 1,...,d, g j(α)(x) = ρ(g j y) = 0for j = 1,...,d, which implies that f (d) (α)(x) is a fixed point of the Z d -action α on X I 3. d Hence there exists a unique constant c(y) [0, 1) with f (d) y + c(y) l (Z d, Z). Definition We call points v l (Z d ) and x T Zd periodic if their orbits (under σ and α, respectively) are finite. If Ɣ Z d is a subgroup of finite index we denote by l(z d ) (Ɣ), l (Z d, Z) (Ɣ) and K (Ɣ) d the sets of all Ɣ-invariant elements in the respective spaces. Theorem (1) For every y l (Z d ) with ρ(y) X I 3 d, v = f (d) y + c(y) + m K d l (Z d, Z) (3.37) for every m Z (cf. (2.10), (3.11), (3.32) and Lemma 3.9). (2) Let Ɣ Z d be a subgroup of finite index. An element v l (Z d, Z) (Ɣ) lies in K d if and only if it is of the form (3.37) with y l (Z d ) (Ɣ), ρ(y) X I 3 d and m Z. We start the proof of Theorem 3.11 with two lemmas. Lemma For every g I d and every constant element m l (Z d, Z), ξ g ( m) = 0. In other words, Z K d. Proof. We know that g I d if and only if it satisfies (2.13) (2.16). We fix g = k Z d g ku k I d, put v = g w (d) l 1 (Z d ), and set c = k Z d g kk 2 j Z (note that this value is independent of j {1,...,d} by (2.16)). For every n Z d, v n = (g w (d) ) n = k g k w (d) n+k = e 2πi n,t g ke 2πi k,t k Z d T d 2d 2 d j=1 cos(2πt j ) dt. Hence v = (v n ) is the sequence of Fourier coefficients of the function H g (t) = k g ke 2πi k,t 2d 2 d j=1 cos(2πt j ). Since these Fourier coefficients are absolutely summable by assumption, we get that n Z d v n = H g (0). (3.38)

23 742 K. Schmidt, E. Verbitskiy On the other hand, given the Taylor series expansion of H g at t = 0, wehave ( ) k g kk 2 j + h.o.t 2π 2 d j=1 t 2 j H g (t) = 4π 2 d j=1 t 2 j + h.o.t and hence H g (0) = c/2. = 2π 2 c dj=1 t 2 j + h.o.t 4π 2 d j=1 t 2 j + h.o.t, We are going to show that H g (0) Z. Indeed, since k g kk j = 0 for all j by (2.14), we have that H g (0) = 1 g k k 2 j 2 = 1 g k k j (k j 1) = k j (k j 1) g k Z. (3.39) 2 2 k k k Finally, for any g I d and m Z, wehave ξ g ( m) = m v n = mh g (0) Z (3.40) n Z by (3.38), and hence ξ d g ( m) = 0 X f (d). Lemma For every g I 3 d,h g(0) = 0(cf. (3.38)). Proof. Every element of I 3 d is of the form h g with h R d and g = (u i 1) (u j 1) (u k 1) for some i, j, k {1,...,d}.Wesetv = g w (d) and obtain from (3.39) that H g (0) = n Z d v n = 0. If w = (hg) w (d) = h v, then H hg (0) = n Z d w n = k Z d h k n Z d v n k = 0. Proof of Theorem Let x X I 3, y l (Z d ) with ρ(y) = x, m Z, and v = d f (d) y + c(y) + m l (Z d, Z) (cf. Lemma 3.9). Then g(α)(x) = ρ(g y) = 0 for every g I 3 d.wesetw = g w (d) and obtain from (3.16), (3.18) and Lemma 3.12, that ξ g (v) = ξ g ( f (d) y + c(y) + m) = ξ g ( f (d) y + c(y)) = ρ(g w (d) f (d) y + g w (d) c(y)) = ρ(g y + w c(y)) = ρ(g y) = 0, since n Z d w n = 0 by Lemma This proves that every v l (Z d, Z) of the form (3.37) lies in K d. For (2) we assume that Ɣ Z d is a subgroup of finite index. In view of (1) we only have to verify that every v l (Z d, Z) (Ɣ) K d has the form (3.37). Assume therefore that v l (Z d, Z) (Ɣ) K d. We choose a set C Ɣ Z d which intersects each coset of Ɣ in Z d in a single point and set l (Ɣ) 0 ={w l (Z d ) (Ɣ) : n C Ɣ w n = 0}. Asl (Ɣ) 0 is finite-dimensional and ker( f (d) (σ )) = R there exists, for every y l (Ɣ) 0, a unique y l (Ɣ) 0 with f (d) y = y. Put ã = ( ) n C Ɣ v n / Z d /Ɣ, regarded as an element of R. Ifv = v ã, then v l (Ɣ) 0 and f (d) y = v for some y l (Ɣ) 0.

24 Abelian Sandpiles and the Harmonic Model 743 Since v K d, ξ g (v) = 0 for every g I d.forg I 3 d, Lemma 3.13 shows that ξ g (v) = g w (d) v = g w (d) v + g w (d) ã = g y + g w (d) ã = g y l (Z d, Z). Hence ρ(g y) = g(α)(ρ(y)) = 0 for all g I 3 d, so that ρ(y) X I 3 d. We obtain that v = f (d) y + ã for some y l (Z d ) with ρ(y) X I 3 and some ã R, which completes the proof d of (2). Theorem 3.11 implies that there exist nonconstant elements v K d f (d) (σ )(l (Z d, Z)). However, if two elements v, v l (Z d, Z) differ in only finitely many coordinates, then they get identified under ξ Id (i.e., their difference lies in K d ) if and only if they differ by an element in ( f (d) ) l (Z d, Z). This is a consequence of the following assertion: Proposition For every g Ĩ d, ker(ξ g ) R d = ( f (d) ) = f (d) R d. Proof. Suppose that h R d ker(ξ g ). Then v := ξ g (h) = g w (d) h l (Z d, Z). (3.41) Since g I d, g w (d) l 1 (Z d ) and hence v R d = l 1 (Z d ) l (Z d, Z). Ifwe multiply both sides of (3.41) by f (d) we get that f (d) v = g h. As R d has unique factorization this implies that h f (d) R d. Remarks (1) One can show that the periodic points are dense in K d, so that every v K d is a coordinate-wise limit of elements of the form (3.37) in Theorem (2) Theorem 3.11 (1) gives a lower bound for the kernel K d of the maps ξ g, g I d. There is also a straightforward upper bound for that kernel: an element v l (Z d, Z) lies in K d if and only if ξ g (v) = g w (d) v =: w g l (Z d, Z) for every g I d. By multiplying this equation with f (d) we obtain that K d {v l (Z d, Z) : g v f (d) l (Z d, Z) for every g I d }=: K d. (3.42) It is not very difficult to see that the inclusion in (3.42) is strict. In fact, K d /K d turns out to be isomorphic to T d. (3) In [18], the kernel K d of ξ Id was studied using methods of commutative algebra.

25 744 K. Schmidt, E. Verbitskiy 4. The Abelian Sandpile Model Let d 2, γ 2d, and let E Z d be a nonempty set. For every n E we denote by N E (n) the number of neighbours of n in E, i.e., N E (n) = E {n ± e (i) : 1 = 1,...,d}, (4.1) where e (i) is the i th unit vector in Z d.weset (cf. Lemma 3.5) and put P (γ ) E γ ={0,...,γ 1} Zd (4.2) ={v {0,...,γ 1} E : v n N E (n) for at least one n E}, R (γ ) E = =F E 0< F < P F. (4.3) In the literature the set R (γ ) E is called the set of recurrent configurations on E. A configuration v {0,...,γ 1} E is recurrent if and only if it passes the burning test, which is described as follows: given v {0,...,γ 1} E, delete (or burn) all sites n E such that v n N E (n), thereby obtaining a configuration v {0,...,γ 1} E(1) with E (1) E. We repeat the process and obtain a sequence E E (1) E (k). If at some stage E (k) = E (k+1) = we say that v fails the burning test, and v is a forbidden (or nonrecurrent) configuration. The closed, shift-invariant subset R (γ ) = R (γ ) Z d γ l (Z d, Z) (4.4) is called the d-dimensional sandpile model with parameter γ.forγ = 2d, R = R (2d) is called the critical sandpile model, and for γ>2d, the model R (γ ) is said to be dissipative. In order to motivate this terminology we assume that E Z d is a nonempty set. An element v Z+ E is called stable if y n <γfor every n E. Ifv Z+ E is unstable at some n E, i.e., if v n γ, then v topples at this site: the result is a configuration T n (v) with v n γ if k = n, T n (v) k = v k +1 if n k max = 1, otherwise. v k If v m,v m γ for some m, n E, m = n, then T n (T n (v)) = T m (T n (v)), i.e., toppling operators commute. A stable configuration ṽ {0,...,γ 1} E is the result of toppling of v, if there exist n (1),...,n (k) E such that ( k ) ṽ = T n (i) (v). i=1

26 Abelian Sandpiles and the Harmonic Model 745 If the set E is finite (in symbols: E Z d ), then every v Z+ E will lead to a stable configuration ṽ by repeated topplings. However, if E is infinite, then repeated toppling of a configuration v Z+ E will, in general, lead to a stable configuration ṽ {0,...,γ 1}E only if γ>2d, i.e., in the dissipative case. 3 We denote by σ = σ (γ ) the shift-action of Zd R on R (γ ) l (Z d, Z) W d (cf. (3.7)). For the following discussion we introduce the Laurent polynomial f (d,γ ) = γ d (u i + ui 1 ) R d = Z[u ± 1,...,u± d ]. (4.5) i=1 For γ = 2d, f (d,γ ) = f (d) (cf. (2.4)). Proposition 4.1. Let d 2 and γ 2d. The following conditions are equivalent for every v γ : (1) v R (γ ) ; (2) For every nonzero h R d with h n {0, 1} for every n Z d, ( f (d,γ ) h) n + v n γ for at least one n supp(h) ={m Z d : h m = 0}. (3) For every h R d with h n > 0 for some n Z d, ( f (d,γ ) h) n + v n γ for at least one n {m Z d : h m > 0}. Furthermore, if v, v R (γ ) and 0 = v v R d, then v v / f (d,γ ) R d. Proof. Fix an element v γ.ifh R d with h n {0, 1} for every n Z d and E = supp(h), then ( f (d,γ ) h) n + v n {0,...,γ 1} for every n E if and only if v n N E (n) 1 for every n E, in which case π E (v) / P E and v/ R (γ ) (cf. (4.3)). This proves the equivalence of (1) and (2). Now suppose that h l (Z d, Z) with M h = max m Z d h m > 0, and that f (d,γ ) h + v γ.weset and observe that for every n S max (h), so that S max (h) ={n Z d : h n = M h } (4.6) v n + ( f (d,γ ) h) n v n + M h (γ N Smax (h)) <γ v n N Smax (h) 1 for every n S max (h). (4.7) If h R d, then S max (h) is finite and (4.7) yields a contradiction to the definition of R (γ ). This proves the implication (1) (3), and the reverse implication (3) (2) is obvious. The last assertion of this proposition is a consequence of (3). The proof of Proposition 4.1 has the following corollary. 3 Even in the dissipative case stable configurations will, in general, only arise as coordinate-wise limits of infinite sequences of topplings of v.

27 746 K. Schmidt, E. Verbitskiy Corollary 4.2. If v R (γ ), and if h l (Z d, Z) satisfies that max m Z d h m > 0 and v + f (d,γ ) h R (γ ), then every connected 4 component of S max (h) is infinite (cf. (4.6)). Proof. If S max (h) has a finite connected component C then (4.7) shows that ( f (d,γ ) h) n + v n ( f (d,γ ) h) n + v n = γ N C (n) for every n C, where h n = { h n if n C, 0 otherwise. As in (4.7) we obtain a contradiction to (4.3). Remark 4.3. Proposition 4.1 implies that ( f (d,γ ) (σ )(h) + R (γ ) ) R (γ ) = for every nonzero h R d. However, if h {0, 1} Zd satisfies that the set S(h) ={n Z d : h n = 1} is infinite and connected, then one checks easily that there exists a v R (γ ) f (d) (σ )(h) + v R (γ ). In spite of this the following result holds. Proposition 4.4. The set with V ={v R (γ ) : v + w/ R (γ ) for every nonzero w f (d,γ ) (σ )(l (Z d, Z))} (4.8) is a countable intesection of dense open sets (i.e., a dense G δ -set) in R (γ ). Proof. Let v R (γ ) and h l (Z d, Z) be such that max n Z d h n 0, f (d,γ ) h = 0 and v + f (d,γ ) h R (γ ).WesetM h = max m Z d h m, define S max (h) Z d as in (4.6), and put S max (h) ={n S max (h) : m n max = 1forsomem Z d S max (h)}. As ( f (d,γ ) h) n intersection with > 0 for every n S max (h), theset S max (h) must have empty F(v) ={n Z d : v n = γ 1}. Now suppose that v R (γ ) has the following properties: (a) The set F(v) is connected. (b) Every connected component of Z d F(v) is finite. (c) min n Z d v n = 0. According to Corollary 4.2, every connected component C of S max (h) is infinite. If C = Z d, then the hypotheses (a) (b) above guarantee that the boundary C = C S max (h) of C is a union of finite sets, each of which is contained in one of the connected components of Z d F(v). Let C and D be connected components of S max (h) and Z d F(v), respectively, with D C =. Since C is infinite and connected and F(v) is connected, we must have that h m = M h = 0 for every m F(v). 4 AsetS Z d is connected if we can find, for any two coordinates m and n in S, a path p(0) = m, p(1),...,p(k) = n in S with p( j) p( j 1) max = 1forevery j = 1,...,k.

28 Abelian Sandpiles and the Harmonic Model 747 Define h by { h n if n D h n = 0 otherwise. Then ( f (d,γ ) h) n = ( f (d,γ ) h) n for every n D, and 0 ( f (d,γ ) h) n ( f (d,γ ) h) n for every n F(v). Forn Z d (F(v) D), ( f (d,γ ) h) n = 0. By combining these statements we see that v + f (d,γ ) h R (γ ). Since 0 = h R d we obtain a contradiction to Proposition 4.1. This shows that v + f (d,γ ) h / R (γ ) for every v R (γ ) satisfying conditions (a) (b) above and every nonzero h l (Z d, Z) with max n Z d h n 0. If γ = 2d and h l (Z d, Z) satisfies that f (d) h = 0, then we may add a constant to h, if necessary, to ensure that max n Z d h n 0. Since such an addition will not affect f (d) h, we obtain that v+ f (d) h / R (γ ) = R for every v R satisfying conditions (a) (c) above and every nonconstant h l (Z d, Z). If γ>2dand h l (Z d, Z) satisfies that max n Z d h n < 0, then ( f (d,γ ) h) n < 0 for every n Z d, and v + f (d,γ ) h / R (γ ) for every v R (γ ) satisfying condition (c) above. Let V R (γ ) be the set of all points satisfying conditions (a) (c) above. This set is clearly dense and V V ={v R (γ ) : v + w/ R (γ ) for every nonzero w f (d) (σ )(l (Z d, Z))}. (4.9) The set V is therefore dense, and it is obviously shift-invariant. In order to verify that V is a G δ we write its complement as an F σ of the form R (γ ) V = M 1 N 1 0 =c Z Q M π ( {(v, h) R (γ ) B N (l (Z d, Z)) : v + f (d) h R (γ ) and π Q M ( f (d,γ ) h) = c} where B N (l (Z d, Z)) ={h l (Z d, Z) : h N}, Q M appears in (3.26) and π : R (γ ) l (Z d, Z) R (γ ) is the first coordinate projection. ), 5. The Critical Sandpile Model Throughout this section we assume that d 2 and γ = 2d. We write R = R (2d) for the critical abelian sandpile model, define the harmonic model X f (d) T Zd by (2.4) and (3.3), and use the notation of Sect Surjectivity of the maps ξ g : R X f (d). For every g Ĩ d (cf. (3.17)) we define the map ξ g : l (Z d, Z) X f (d) by (2.9) and (3.15). We shall prove the following results.