Asymmetric Abelian Avalanches and Sandpiles

33 Pages·2000·0.34 MB·English

Asymmetric Abelian Avalanches and Sandpiles
Andrei Gabrielov
Mathematical Sciences Institute
y
Cornell University, Ithaca, NY 14853
E-mail: [email protected]
Received May 31, 1994
ABSTRACT: We consider two classes of threshold failure models, Abel-
ian avalanches and sandpiles, with the redistribution matrices satisfying nat-
ural conditions guaranteeing absence of in(cid:12)nite avalanches. We investigate
combinatorialstructureofthesetofrecurrent con(cid:12)gurationsforthesemodels
and the corresponding statistical properties of the distribution of avalanches.
We introduce reduction operator for redistribution matrices and show that
thedynamicsof amodel withanon-reduced matrixiscompletely determined
by the dynamics of the corresponding model with a reduced matrix. Finally,
we show that the stationary distributions of avalanches in the two classes of
models: discrete, stochastic Abelian sandpiles and continuous, deterministic
Abelian avalanches, are identical.
Introduction. Di(cid:11)erent cellular automaton models of failure (sandpiles, avalanches,
forest (cid:12)res, etc.), starting with Bak, Tang and Wiesenfeld [BTW1, BTW2] sandpile model,
were introduced in connection with the concept of self-organized criticality. Traditionally,
all of these models are considered on uniform cubic lattices of di(cid:11)erent dimensions.
Recently Dhar [D1] suggested a generalization of the sandpile model with a general
(modulo some natural sign restrictions) integer matrix (cid:1) of redistribution of accumulated
particles during an avalanche. An important property of this Abelian sandpile (ASP)
model is the presence of an Abelian group governing its dynamics.
Dhar introduced the set of recurrent con(cid:12)gurations for an Abelian sandpile model,
the principal geometric object governing its dynamics in the stationary state. The burning
algorithm introduced in [MD] allows to recognize, for a symmetric sandpile model, when
y
Currently at Department of Mathematics, University of Toronto, Toronto M5S 1A1,
Canada
Asymmetric Abelian Avalanches and Sandpiles Page 2
a stable con(cid:12)guration is recurrent. A more sophisticated script algorithm suggested in
[S] plays the same role for asymmetric models. Both algorithms provide, in fact, certain
information on the combinatorial structure of the set of recurrent con(cid:12)gurations.
Abelian sandpiles were also studied in [WTM, D2, DM, GM, Cr, Go, CC, G1, G2, S,
P
DMan]. In a non-dissipative case ( (cid:1) = 0; for all i) an avalanche in the ASP model
j ij
coincides with a chip-(cid:12)ring game on a directed graph [BLS, BL] where −(cid:1) is the Laplace
matrix of the underlying digraph.
Another class of lattice models of failure, slider block models introduced by Burridge
and Knopo(cid:11) [BK] and studied in [CaL, Ca, N, MT, LKM], as well as models [FF, D-G,
OFC, CO, Z, PTZ, GNK] which are equivalent to quasistatic block models, have contin-
uous time and continuous quantity at the lattice sites which accumulates in time and is
redistributed during avalanches. This quantity is called the slope, height, stress or energy
by di(cid:11)erent authors. In slider block models it corresponds to force [OFC]. We use the term
height as in [D1].
Gabrielov [G1], introduced Abelianavalanche (AA)models, deterministiclatticemod-
els with continuous time and height values at the sites of the lattice, and with an arbitrary
redistribution matrix. For a symmetric matrix, these models are equivalent to arbitrarily
interconnected slider block systems. In the case of a uniform lattice, these models were
studied in [FF, D-G] and in [GNK] (as series case a).
The stationary behavior of the AA model is periodic or quasiperiodic, depending
on the loading rate vector. At the same time, the distribution of avalanches for a dis-
crete, stochastic ASP model is identical to the distribution of avalanches for an arbitrary
quasiperiodic trajectory (or to its average over all periodic trajectories) of a continuous,
deterministic AA model with the same redistribution matrix and loading rate [G1].
In this paper, we introduce general conditions on redistribution matrices that are
equivalent to the absence of in(cid:12)nite avalanches in the model. The models satisfying these
conditions include, in particular, the models with non-negative dissipation or codissipation
Asymmetric Abelian Avalanches and Sandpiles Page 3
considered before. We allow spatially inhomogeneous loading rate and show that the set of
recurrent con(cid:12)gurations does not depend on the loading rate vector, as long as a natural
condition guaranteeing absence of non-loaded components in the model is satis(cid:12)ed.
We continue the study of the combinatorial structure of the set of recurrent con(cid:12)gu-
rations started in [D1, DM, MD, S] and introduce a new description of this set, essentially
improving the script algorithm suggested in [S] for asymmetric redistribution matrices,
Additional possibility for the study of the models with asymmetric matrices arises
from the matrix reduction operations. These operations act on the redistribution matrices
in the same way as the topple operations act on unstable con(cid:12)gurations, and satisfy the
same property of the independence of the resulting reduced matrix on the possible change
of the order of reductions. Each reduction operation simpli(cid:12)es the redistribution matrix,
and replaces the original model by a simpler reduced model such that the combinatorics
of the set of recurrent con(cid:12)gurations for the reduced model completely determines the
combinatorics for the original non-reduced model.
In the (cid:12)rst section, we de(cid:12)ne con(cid:12)gurations, redistribution matrices and avalanche
operators. We show, followingaconstruction implicitlypresent in [S], that alegal sequence
of topples satis(cid:12)es certain minimality condition among all (possibly illegal) sequences of
topples with the same (cid:12)nal stable con(cid:12)guration (lemma 1.1). This minimality condition
provides, in particular, a new proof of the principal Abelian property (theorem 1.2). We
introduce the class of avalanche-(cid:12)nite redistribution matrices satisfying eight equivalent
conditions and check these conditions for the matrices with non-negative dissipations and
codissipations.
In the second section, we de(cid:12)ne, following[G1], the AA model as a sequence of loading
periods and avalanches and describe the dynamics of the model on its attracting set of
recurrent con(cid:12)gurations. The arguments here are similar to the arguments of Dhar [D1]
for the ASP models.
In the third section, we study the combinatorial structure of the set of recurrent
Asymmetric Abelian Avalanches and Sandpiles Page 4
con(cid:12)gurations of the AA model. The principal result here, the theorem 3.8, describes this
set asthe complement inthe set of all stablecon(cid:12)gurations tothe union of negativeoctants
with the vertices in a (cid:12)nite set N. An explicit constructive description of this set N is
given in the theorem 3.11.
In the fourth section, we show the possibilities to extract the information on the dy-
namicsofanAAmodelfromthedynamicsofanothermodelwithasimpli(cid:12)edredistribution
matrix. We introduce the total reduction operator for redistribution matrices, similar to
the avalanche operator for con(cid:12)gurations. We show that the stationary dynamics of an
AA model with a redistribution matrix (cid:1) is completely determined by the dynamics of
the corresponding model with a reduced redistribution matrix, the total reduction of (cid:1).
In the (cid:12)fth section, we introduce marginally stable con(cid:12)gurations and derive formulas
for the mean number of avalanches. The arguments here are again similar to those of Dhar
[D1], modi(cid:12)ed for the more general situation considered here.
In the sixth section, we establish the identity between the distributions of avalanches
for AA and ASP models with the same redistribution matrices.
Some of the results of this paper were announced in [G2].
1. Redistribution matrices and avalanches. Let V be a (cid:12)nite set of N elements
(sites), and let (cid:1) be a N (cid:2)N real matrix with indices in V. We call (cid:1) a redistribution
matrix when
(cid:1) > 0; for all i; (cid:1) (cid:20) 0; for all i 6= j: (1)
ii ij
A real vector h = fh ; i 2 Vg is called a con(cid:12)guration. The value h is called the height
i i
at the site i. For every site i, a threshold H is de(cid:12)ned, and a site i with h < H is called
i i i
stable. A con(cid:12)guration is stable when all the sites are stable.
For i 2 V, a topple operator T is de(cid:12)ned as
i
T (h) = h−(cid:14) (2)
i i
Asymmetric Abelian Avalanches and Sandpiles Page 5
where (cid:14) = ((cid:1) ;:::;(cid:1) ) is the i-th row vector of (cid:1). Obviously, every two topple
i i1 iN
operators commute. The topple T (h) is legal if h (cid:21) H , i.e. if the site i is unstable.
i i i
No topples are legal for a stable con(cid:12)guration. A sequence of consecutive legal topples is
called an avalanche if it is either in(cid:12)nite or terminates at a stable con(cid:12)guration. In the
latter case, the integer vector n = fn ; i 2 Vg where n is the number of topples at a site
i i
P
i during the avalanche is called its script, and the total number n of topples in the
i i
avalanche is called its size.
The following lemma shows that the avalanches are \extremal" among all the se-
quences of (possibly, illegal) consecutive topples with the same endpoints. It allows, in
particular, to give an alternative proof of the principal property of avalanches | the script
and the (cid:12)nal stable con(cid:12)guration depend only on the starting con(cid:12)guration, not on the
possible choice in the sequence of topples (theorem 1.2 below).
Lemma 1.1. Let h be an arbitrary con(cid:12)guration and let m be an integer vector with
P
non-negative components m such that g = h− m (cid:14) is a stable con(cid:12)guration. For any
i i i
(cid:12)nite sequence of consecutive legal topples started at h, with n topples at a site i, we have
i
m (cid:21) n .
i i
P
Proof. The arguments appear implicitly in [S]. We use induction on the size n = n
i
of the sequence of legal topples. For n = 0, the statement is trivial. Let it be true, i.e.
m (cid:21) n , for a sequence with n topples at a site i. If a site j is unstable for a con(cid:12)guration
i i i
P
f = h − n (cid:14) then g < f . Due to (1), this implies m > n , hence the statement
i i j j j j
remains true when we add a topple at the site j to the sequence.
Theorem 1.2. (Sf. [D1], [BLS], [BL].) Every two avalanches starting at the same con-
(cid:12)guration h are either both in(cid:12)nite or both (cid:12)nite. In the latter case, the scripts of both
avalanches coincide. In particular, both avalanches terminate at the same stable con(cid:12)gu-
ration and have the same size.
Proof. The statement follows easily from the lemma 1.1.
Asymmetric Abelian Avalanches and Sandpiles Page 6
Remark 1.3. If we consider a con(cid:12)guration as an initial state of a game, and every legal
topple as a legal move, an avalanche becomes a (solitary) game. The theorem 1.2 means
that this game is strongly convergent in the de(cid:12)nition of [E].
Lemma 1.4. Forevery siteithat toppledat least once duringanavalanche, h (cid:21) H −(cid:1)
i i ii
till the end of the avalanche.
The statement follows from (1) and (2).
Let RV = fh (cid:21) 0; for all ig and RV = −RV denote positive and negative closed
+ i − +
octants in RV, and let R_ V = fh > 0; for all ig be an open positive octant. Let (cid:1)0 be the
+ i
transpose of the matrix (cid:1).
Theorem 1.5. For a redistribution matrix (cid:1), the following properties are equivalent.
i. Every avalanche for (cid:1) is (cid:12)nite.
ii. (cid:1)(RV nf0g)\RV = ;:
+ −
iii. (cid:1)(RV) (cid:19) RV, i.e. (cid:1)−1 exists and all its elements are non-negative.
+ +
iv. (cid:1)(RV)\R_ V 6= ;.
+ +
0 0
i : Every avalanche for (cid:1) is (cid:12)nite.
ii0: (cid:1)0(RV nf0g)\RV = ;:
+ −
iii0: (cid:1)0(RV) (cid:19) RV, i.e. (cid:1)0−1 exists and all its elements are non-negative.
+ +
iv0: (cid:1)0(RV)\R_ V 6= ;:
+ +
Proof. (ii0) ) (i): Let us show that for (cid:1) satisfying (ii0); every avalanche is (cid:12)nite. If
there exists an in(cid:12)nite avalanche started at a con(cid:12)guration h, let r(k) = fk =k; i 2 Vg
i
where k is the number of topples at a site i after a total number of topples k. According
i
to (2), the con(cid:12)guration after k topples is h(k) = h − k(cid:1)0r(k). Let r 2 RV n f0g be
an accumulation point for r(k) (it exists because all these vectors have unit length) and
p = −(cid:1)0r an accumulation point for (h(k)−h)=k. According to lemma 1.4, components
of h(k)−h are bounded from below. Hence all components of p are non-negative, and (cid:1)
0
does not satisfy (ii ):
Asymmetric Abelian Avalanches and Sandpiles Page 7
(i) ) (iii0): Let h be a con(cid:12)guration in RV, and let a (cid:12)nite avalanche starting at kh
+
terminates at a stable con(cid:12)guration h(k). Let r(k) = fk =k; i 2 Vg where k is the
i i
number of topples at a site i during this avalanche. We have (cid:1)0r(k) = h−h(k)=k. Let r
be an accumulation point for r(k), as k ! 1. Then r 2 RV and (cid:1)0r = h, because h(k)
+
remains bounded as k ! 1.
(iv) ) (ii0): Suppose that (cid:1) does not satisfy (ii0): This means that there exists a linear
P
form l 6= 0 with non-negative coe(cid:14)cients such that l((cid:14) ) (cid:20) 0; for all i. Hence l( c (cid:14) ) (cid:20) 0
i i i
for any combination of the vectors (cid:14) with non-negative coe(cid:14)cients c . At the same time,
i i
l((cid:14)) > 0; for every (cid:14) 2 R_ V. This means that (cid:1)(RV)\R_ V = ; and (cid:1) does not satisfy (iv).
+ + +
(iii0) ) (iv0): The implication is obvious.
Combining the four implications, we have (iv) ) (ii0) ) (i) ) (iii0) ) (iv0): The same
arguments applied to (cid:1)0 instead of (cid:1) imply (iv0) ) (ii) ) (i0) ) (iii) ) (iv): This com-
pletes the proof.
De(cid:12)nition 1.6. A redistribution matrix satisfying the conditions of the theorem 1.5 is
called avalanche-(cid:12)nite.
Remark 1.7. Let (cid:1) be an avalanche-(cid:12)nite matrix, and let t 2 R_ V; (cid:1)t 2 R_ V. Such a
+ +
vector t always exists due to the property (iii) or (iv). Let jhj = (h;t) be the t-weighted
t
length of a con(cid:12)guration h. Then jT (h)j < jhj , for all i 2 V, i.e. every topple operator
i t t
dissipates the t-weighted length. This can be also used to prove the implications (iii) ) (i)
and (iv) ) (i).
P
De(cid:12)nition 1.8. The value s = (cid:1) is called the dissipation at the site i, and the
i j ij
P
0
value s = (cid:1) is called the codissipation at the site j. A site i is called dissipative
j i ij
(non-dissipative) if s > 0 (s = 0). A site j is called codissipative (non-codissipative) if
i i
0 0
s > 0 (s = 0).
j j
An underlying digraph Γ = Γ((cid:1)) of a redistribution matrix (cid:1) is de(cid:12)ned by the vertex
set V(Γ) = V and an edge from a site i to a site j drawn i(cid:11) (cid:1) < 0.
ij
Asymmetric Abelian Avalanches and Sandpiles Page 8
0 0 0
Let s be a diagonal matrix with s = s , and let
ii i
(cid:1) = (cid:1)−s0 (3)
0
be the non-codissipative part of (cid:1). The matrix (cid:1) coincides with the Kirchho(cid:11) matrix of
0
−!
Γ, with conductance of an edge ij de(cid:12)ned as −(cid:1) [T, p.138].
ij
A subset W of V is called a sink in Γ if there are no edges from sites in W to sites
outside W, and a source if there are no edges from sites outside W to sites in W.
A matrix (cid:1) is called weakly dissipative if all the dissipation values s are non-negative
i
and the digraph Γ((cid:1)) has no non-dissipative sinks, i.e. from every site there exists a
directed path in Γ((cid:1)) to a dissipative site.
Proposition 1.9. A matrix with non-negative dissipation values is avalanche-(cid:12)nite if
and only if it is weakly dissipative.
P
Proof. If the graph Γ((cid:1)) has a non-dissipative sink W (cid:18) V then h does not
i2W i
decrease during an avalanche, hence the avalanche started at a con(cid:12)guration with large
enough values of h ; i 2 W, cannot be (cid:12)nite.
i
Suppose now that (cid:1) is weakly dissipative. It follows from the de(cid:12)nition 1.8 that
P
(cid:27) = h does not increase at any topple and decreases when a dissipative site topples.
i i
Suppose that there exists an in(cid:12)nite avalanche, and let W (cid:26) V be the subset of sites
that topple in(cid:12)nite number of times in this avalanche. Then all the sites in W are non-
dissipative, otherwise (cid:27) would decrease inde(cid:12)nitely, in contradiction to the lemma 1.4. At
the same time, W is a sink of Γ, otherwise h would increase inde(cid:12)nitely at any site j 62 W
j
such that (cid:1) < 0, for some i 2 W. This contradicts the de(cid:12)nition 1.8.
ij
De(cid:12)nition 1.10. A matrix (cid:1) is called weakly codissipative if all the codissipation values
0
s are non-negative and the digraph Γ((cid:1)) has no non-codissipative sources, i.e. to every
j
site there exists a directed path in Γ((cid:1)) from a codissipative site.
Proposition 1.11. A matrix with non-negative codissipation values is avalanche-(cid:12)nite
if and only if it is weakly codissipative.
Asymmetric Abelian Avalanches and Sandpiles Page 9
Proof. The statement follows from the theorem 1.5 and proposition 1.9, because the
transpose of a weakly codissipative matrix is weakly dissipative.
Proposition 1.12. For every avalanche-(cid:12)nite matrix (cid:1); det((cid:1)) > 0:
Proof. Let a redistribution matrix (cid:1) satisfy the condition (ii) of the theorem 1.5. For
t 2 [0;1], all the matrices (cid:1) = t(cid:1)+(1−t)E from the segment connecting with the unit
t
matrix E satisfy (ii). Hence all these matrices are avalanche-(cid:12)nite. Due to the condition
(iii), all the matrices in this segment are non-singular, hence their determinants have the
same (positive) sign.
In the following, we consider only avalanche-(cid:12)nite redistribution matrices.
De(cid:12)nition 1.13. For a con(cid:12)guration h, the avalanche operator Ah is de(cid:12)ned as the
stable con(cid:12)guration that terminates an avalanche initiated at h. Due to the theorem 1.2,
this stable con(cid:12)guration is unique. If h is stable, Ah = h.
Example 1.14. The sandpile model introduced in [BTW], n(cid:2)n square lattice with the
nearest neighbor interaction and particles dropping from the boundary, is de(cid:12)ned by a
symmetric redistribution matrix (cid:1) of the size n2 (cid:2) n2. The rows and columns of (cid:1) are
speci(cid:12)ed by a vector index i = (i ;i ) with 1 (cid:20) i (cid:20) N, for (cid:23) = 1;2; (cid:1) = 4; (cid:1) =
1 2 (cid:23) i;i i;j
−1, for i = j ; i = j (cid:6)1, and for i = j (cid:6)1; i = j ; (cid:1) = 0 otherwise. This matrix
1 1 2 2 1 1 2 2 i;j
is weakly (co-) dissipative, hence avalanche-(cid:12)nite.
Example 1.15. The 1-dimensional model with the failure depending on the local slope,
introduced in [BTW] (for m = 1) and studied in [KNWZ, LLT, LT, S, CFKKP], is de(cid:12)ned
as follows. At every site i; 1 (cid:20) i (cid:20) N, we place k particles, and set k = 0. The site i
i N+1
topples when k −k (cid:21) m. The topple operator removes m particles from the site i and
i i+1
addsoneparticletoeachsitej = i+1;:::;i+massoonasj (cid:20) N. Afterthetransformation
h = k −k , for 1 (cid:20) i (cid:20) N, this model can be de(cid:12)ned by a redistribution matrix (cid:1) with
i i i+1
(cid:1)i;i = m+1, for i < N; (cid:1)N;N = m (cid:1)i;i−1 = −m, for i > 1; (cid:1)i;(cid:23) = −1, for i < N
Asymmetric Abelian Avalanches and Sandpiles Page 10
and (cid:23) = max(i+m;N) (sf. [S]). This matrix is not symmetric when m > 1. It is weakly
(co-) dissipative, hence avalanche-(cid:12)nite.
Example 1.16. A chip-(cid:12)ring game introduced in [BLS, BL] is de(cid:12)ned by a (directed)
graph Γ with a certain number of chips placed at each of its vertices, and a sequence of
legal moves ((cid:12)res), when one particle is allowed to be moved from a vertex i to the end of
each edge directed from i, in case the total number of chips at the vertex i is not less than
the total number of the edges directed from i. The corresponding redistribution matrix is,
after a sign change, the Laplace matrix of Γ. It is always degenerate (all the dissipations
are equal 0) hence not avalanche-(cid:12)nite.
Example 1.17. For N (cid:20) 3, a redistribution matrix (cid:1) is avalanche-(cid:12)nite i(cid:11) det((cid:1)) > 0.
However, for N (cid:21) 4 there exist redistribution matrices with positive determinant which
are not avalanche-(cid:12)nite. Consider, for example, an 4(cid:2)4-matrix
0 1
1 −3 −1 0
B−3 1 −1 0C
(cid:1) = @ A:
−1 −1 1 −2
0 0 −2 1
We have det((cid:1)) = 16 > 0. At the same time, the matrix (cid:1) is not avalanche-(cid:12)nite, because
(cid:1)(1;1;0;0)= (−2;−2;−2;0), in contradiction to the condition (ii) of the theorem 1.5.
2. Abelian avalanche model. In this section, we de(cid:12)ne the Abelian avalanche model
as a sequence of slow loading periods and fast redistribution events (avalanches). Many
of the statements in this section are similar to the corresponding statements of Dhar [D1]
for the ASP models. We present these statements with short proofs to make the paper
self-contained. Also, the class of the redistribution matrices and loading vectors considered
here is more general than in [D1].
Let v = fv ; i 2 Vg be a non-zero vector with non-negative components. For an
i
(avalanche-(cid:12)nite) redistribution matrix (cid:1), an Abelian avalanche (AA) model [G1] with a
loading rate vector v is de(cid:12)ned as follows.