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Pattern formation in growing sandpiles
Deepak Dhar,∗Tridib Sadhu,†and Samarth Chandra‡
Department of Theoretical Physics, Tata Institute of Fundamental Research, 1 Homi Bhaba Road, Mumbai 400005 India
(Dated: August 12, 2008)
Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful
complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional
square lattice with directed edges (each site has two arrows directed inward and two outward),
starting with a periodic background with half the sites occupied. The size of the pattern formed
scales with the number of grains added N as
pattern, in terms of the position and shape of different features of the pattern.
√N. We give exact characterization of the asymptotic
Many complicated and intricate patterns found in na-
ture can be modelled by deterministic dynamics [1]. In
Turing patterns [2] the final outcome is random due to
the randomness in initial conditions. In the game of life
[3], one can get a very wide variety of patterns from sim-
ple deterministic cellular automaton evolution rules, de-
pending on the initial condition.
While the real sand, poured at one point on a flat sub-
strate produces a rather simple pyramidal shape, much
more complex patterns are produced in the theoreti-
cal models of sandpiles, like the Abelian sandpile model
(ASM) [4]. Earlier studies have usually concentrated on
determining the asymptotic shape of the growing clus-
ter [5, 6]. Other special configurations in the model, like
the identity [7], or the stable state produced from special
unstable states also show complex internal self-similar
structures [8]. The limiting shape has been determined
in the related rotor-router model, and the model of di-
visible sandpiles with multiple sites of addition [9].
In this paper, we study the asymptotic pattern pro-
duced by adding N grains of sand at a single site on a
two dimensional Abelian sandpile model starting from a
periodic background, and allowing the system to relax.
It is easy to see that the diameter of the pattern grows
as
proportionate growth, with different parts of the pattern
all growing as
studied models of growth such as diffusion limited aggre-
gation, Eden model etc. [10], where the growth occurs
mainly at the surface.
The standard square lattice produces a rather compli-
cated pattern (Fig.1a), and it has not been possible to
characterize it so far. We consider two variations, assign-
ing orientations to the edges of the lattice, as shown in
Fig.2a and 2b. The initial state was chosen to be a pe-
riodic checkerboard arrangement of sites with heights 0
and 1. The asymptotic pattern produced in the two cases
turns out to be the same, and is shown in Fig 1b. Taking
some qualitative features of the observed pattern ( e.g.
only two types of patches are present, and they are all
3- or 4- sided polygons) as input, we show how one can
get a complete and quantitative characterization of the
pattern. We show that the pattern has exact 8-fold ro-
tational symmetry, and determine the exact coordinates
√N. Interestingly, for large N, the pattern shows a
√N. This is thus different from earlier-
FIG. 1: Stable configurations for the Abelian sandpile model
obtained by adding particles at one site.
square lattice, initial configurations with all heights 2, and
2×105particles added, color code: red=0, blue=1, green=2,
yellow=3. (b) The F-lattice of Fig.2a with initial checker-
board configuration, with 2×105particles added, color code:
green=0, yellow=1. The apparent green regions in the picture
represent the patches with checkerboard configuration.
(a) Undirected
FIG. 2: The directed square lattices studied in this paper (a)
the F-lattice (b) the Manhattan lattice.
of all the boundaries in the asymptotic pattern. We dis-
cuss some other cases, where exactly the same pattern is
obtained.
In the two lattices we studied (Fig.2), each bond of
the lattice is directed with two in-arrows, and two out-
arrows at each vertex. The ASM on these is defined by
the toppling rule: A site (x,y) is unstable if the number
of grains at the site zx,y≥ 2, and then transfers one grain
each in the direction of its outward arrows. We start with
an initial configuration in which zx,y= 1, for sites with
(x + y) = even, and 0 otherwise.
arXiv:0808.1732v1 [cond-mat.stat-mech] 12 Aug 2008
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We used a lattice large enough so that no avalanches
started from the origin reach the boundary. Using the
Abelian property, we add all N particles in the beginning,
and relax the configuration to get the final pattern. The
result of adding N = 2×105particles on the F-lattice is
shown in Fig.1b. The pattern formed on the Manhattan
lattice is indistinguishable at large scales. The pattern
is identical to Fig.1b, except that the thin lines of 1’s
forming two triangles outside the octagon are rotated by
45◦in the Manhattan case. Since the lattices are quite
different, this is quite intriguing.
We start by setting up some general theoretical frame-
work, which is independent of the details of the partic-
ular lattices studied. Formally, we can characterize the
asymptotic pattern in terms of the rescaled coordinates,
ξ = x/√N, η = y/√N and the density function ρ(ξ,η)
which gives the local density of grains in the pattern in
a small rectangle of size ∆ξ, ∆η about the point (ξ,η),
with N−1/2? ∆ξ,∆η ? 1.
Equivalently, we can describe the asymptotic pattern
in term of the rescaled toppling function φ(ξ,η).
TN(x,y) be the numbers of toppling at site (x,y) when
N particles are added at the origin, and the configuration
is relaxed. We define
1
2NTN(?√Nξ?,?√Nη?)
where floor function ?x? is the largest integer less than
x. From the conservation of sand grains, it is easily seen
that φ(ξ,η) is related to the density function ρ(ξ,η) by
Let
φ(ξ,η) = lim
N→∞
(1)
(δ2
δξ2+
δ2
δη2)φ(ξ,η) = ∆ρ(ξ,η) − δ(ξ)δ(η)
where excess density ∆ρ(ξ,η) is the difference between
ρ(ξ,η) and the initial density ρ0(ξ,η).
It was already noted [11] that for N large ρ(ξ,η) tends
to a nontrivial limit, and the asymptotic pattern is made
of distinct regions, called ‘patches’. Typically inside a
patch the heights are periodic in space, and there are
few defect-lines, which move with N, but do not change
the macroscopic density ρ(ξ,η). Then, the coarse grained
function ρ(ξ,η) takes constant rational value in each
patch. Also in each patch of constant ∆ρ(ξ,η), φ(ξ,η) is
a quadratic function, and was first noted in [11]. We in-
dicate the proof here. For all patches the function φ(ξ,η)
is Taylor expandable around any point inside the patch.
Consider any term of order ≥ 3 in the expansion, for ex-
ample the term ∼ (∆ξ)3. This can only arise due to a
term ∼ (∆x)3/√N in T(x,y). Then the integer func-
tion T(x,y) will change discontinuously at intervals of
∆x ∼ O(N1/6) leading to infinitely many defect-lines
in the asymptotic pattern. However there are no such
feature in Fig.1a or Fig.1b. Therefore inside a patch of
constant ∆ρ(ξ,η), φ(ξ,η) can at most be quadratic in
ξ and η, and in each periodic patch, the toppling func-
tion T(x,y) is sum of two terms: a part that is a sim-
ple quadratic function of x and y, and a periodic part.
(2)
The periodic part averages to zero, and does not con-
tribute to the coarse-grained function φ(ξ,η). In some
patterns, there are regions of finite fractional area which
show aperiodic height patterns. In these regions φ(ξ,η)
is not quadratic and are harder to characterize.
Now consider two neighboring periodic patches P and
P?with mean densities ρ and ρ?respectively. Let the
quadratic toppling function be Q(ξ,η) and Q?(ξ,η) in
these patches. Then the boundary between the patches
is given by the equation Q(ξ,η) = Q?(ξ,η).
derivatives of φ are also continuous across the bound-
ary, the boundary between two periodic patches must be
a straight line, and
As the
Q?(ξ,η) = Q(ξ,η) +1
2(ρ?− ρ)l2
⊥
(3)
where l⊥is the perpendicular distance of (ξ,η) from the
boundary. We can start with a periodic patch P, and
go to another patch P?by more than one path. Since
the final quadratic function at P?should be the same
whichever path we take, this imposes consistency con-
ditions which restricts the allowed values of slopes of
boundaries. Consider a point z0where n periodic patches
meet, with n > 2 (Fig.3a). If the jth boundary at this
point makes an angle θjwith the x-axis, and the density
of the patch in the wedge θj≤ θ ≤ θj+1is ρj+1(Fig.3a)
then using Eq.3 repeatedly for all n patches around z0
we get that the following equation must be true for all θ:
n
?
j=1
(ρj+1− ρj)sin2(θ − θj) = 0,
(4)
with ρn+1= ρn. This is equivalent to the condition:
n
?
j=1
(ρj+1− ρj)e2iθj= 0(5)
For n = 3, with ρ1 ?= ρ2 ?= ρ3, this equation has only
trivial solutions with θjequal to 0 or π for all j. Hence,
only n ≥ 4 are allowed.
We now discuss how the exact function ρ(ξ,η) can be
determined for our problem.
there are no aperiodic patches, only two types of periodic
patches, where ρ(ξ,η) only take values 1 or 1/2. Also,
the slopes of the boundaries between patches only take
values 0, ±1, ∞. The patches are typically dart shaped
quadrilaterals, and some triangles (which may be consid-
ered as degenerate quadrilaterals with one side of length
zero). These simplifications, not present in Fig.1a, make
possible a full characterization of the pattern in Fig.1b.
Given that there are only these two types of patches,
we only need to look for possible patterns where ∆ρ takes
piecewise constant values 1/2 or 0. From Eq.(2), we see
that we can think of φ(ξ,η) as the potential produced
by a point charge at the origin, and a charge cloud with
We note that in Fig.1b,
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θ
θ
θ
θ
z
ρ
1
2
3
3
j
θn
ρ2
ρj+1
0
ρ1
j+1
θ
FIG. 3: (a) n different periodic patches of density ρ1,...,ρn
meeting at point z0. (b) The pattern in Fig.1b is obtainable
by putting together square tiles of different sizes. Each of the
tiles is divided into two halves of different density.
areal density −∆ρ(ξ,η), with total charge zero. The ba-
sic principle which selects the actual stable pattern out
of many is a version of the principle of minimum dissi-
pation: It is a stable state reached by minimum number
of toppling. (This follows immediately from the toppling
rules, where no toppling occurs unless forced).
The requirement that φ(ξ,η) be exactly zero, in the
region outside the pattern, implies that all the multi-
pole moments of the charge distribution ∆ρ(ξ,η) are
exactly zero. We show below that the conditions that
∆ρ takes only two values, the potential function is ex-
actly quadratic within a periodic patch, and the slopes
of the boundaries are only 0,±1,∞, fix the allowed pat-
tern uniquely.
We start by determining the exact asymptotic size of
the pattern. We note from Fig.1b that the boundary of
the pattern is an octagon ( we shall prove later that this
is a regular octagon ). In fact there are four lines of 1’s
outside the octagon. But these has zero areal density in
the limit N → ∞, and do not contribute to ρ(ξ,η). We
will ignore these in the following discussion.
Let B be the minimum boundary square containing
all (ξ, η) that have a non-zero charge density ρ(ξ,η). We
observe that B can be considered as a union of disjoint
smaller squares, each of which is divided by diagonal into
two parts where ∆ρ(ξ,η) takes values 1/2 and 0 [Fig.3b].
This is seen to be true for the outer layer patches. To-
wards the center, the squares are not so well resolved.
Assuming that this construction remains true all the way
to the center, in the limit of large N, the mean density
of the negative charge in the bounding square = 1/4.
Given that the total amount of negative charge is −1,
the area of the bounding square should be 4. Hence we
conclude that the equation of the boundary of the mini-
mum bounding square are
|ξ| = 1,
|η| = 1(6)
Let Nbbe the minimum number of particles that have to
be added so that at least one site at y = b topples. We
find that for b = 10, 50, 100, and 300,√Nb = 10.770,
49.436, 98.894 and 297.798. Clearly the boundary dis-
tance b tends to
√N for large N.
m
n
FIG. 4: Two representations of the adjacency graph of the
pattern.Here the vertices are the patches, and the edges
connect the adjacent patches. (a) Representation as a planar
graph (b) as a graph of wedge of angle 4π formed by glueing
together the eight quadrant graphs at the origin.
We now describe the topological structure of the pat-
tern.This is characterized by its adjacency graph
[Fig.4a], where each vertex denotes a patch, and a bond
between the vertices is drawn if the vertices share a com-
mon boundary. It is convenient to think of the triangular
patches in the pattern as degenerate quadrilaterals, with
one side of length zero. Then we see that the adjacency
graph is planar with each vertex of degree four, except a
single vertex of coordination number eight corresponding
to the exterior of the pattern. The graph has the struc-
ture of a square lattice wedge, with wedge angle 4π. The
square lattice structure of the adjacency graph is seen
most directly by applying a z?= 1/z2transformation to
the picture (used earlier in [11]), where z = ξ + iη, and
view it in the complex z?-plane. Thus, one can equiva-
lently represent the graph as a square grid on a Riemann
surface of two sheets (fig.4b).
We now use the qualitative information obtained from
the adjacency matrix of the observed pattern, to obtain
quantitative prediction of the exact coordinates of all the
patches. Consider an arbitrary patch P, having an excess
density 1/2. The potential function in the patch is a
quadratic function of (ξ,η) and we parametrize it as
φP(ξ,η) =
1
8(mP+ 1)ξ2+1
+dPξ + ePη + fP
4nPξη +1
8(1 − mP)η2
p
(7)
The potential function in a patch P having zero excess
density will be parametrized as
φP(ξ,η) =1
8mP(ξ2−η2)+1
4nPξη+dPξ+ePη+fP(8)
Now consider two neighboring patches P and P?with
excess densities 1/2 and 0 respectively. Then using the
matching condition Eq.(3), it is easy to show that if the
boundary between them is a horizontal line η = ηP, we
must have
mP? = mP+ 1,
eP? = eP+ ηP/2,
nP? = nP,
fP? = fP− η2
dP? = dP
P/4 (9)
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Similar calculation for other boundaries show that across
a vertical boundary, going from a patch of higher density
to the one of lower density, we have ∆mP= −1 and
∆nP= 0. Across a boundary with slope ±1, ∆mP= 0,
and ∆nP= ±1.
In the outermost patch, clearly φ(ξ,η) = 0, and for
this patch both m and n are zero. It follows that all mP
and nPtake integer values. In the following, we denote
a patch by integers (m,n), and write the corresponding
coefficients dP, eP, and fPas dm,n, em,nand fm,n. With
this convention, the matching conditions in Eq.(9) can
be rewritten as
dm+1,n= dm,n, em+1,n− em,n= ηm,n/2,(m + n) odd
Using similar matching conditions for the boundary of
patch (m, n) with slope ±1, we get the conditions
dm,n+1− dm,n = em,n− em,n+1,(m + n) odd
dm,n−1− dm,n = em,n−1− em,n,(m + n) odd (11)
We can eliminate the variables dm,nand em,nwith (m+
n) even using Eq.(10) and Eq.(11). Then the equations
become
(10)
em+2,n− em,n = ηm,n/2
dm−2,n− dm,n = ξm,n/2
dm−1,n−1− dm,n = em+1,n−1− em,n
dm−1,n+1− dm,n = −[em+1,n+1− em,n]
It is convenient to introduce the complex variables z =
ξ + iη, M = m + in and D = d + ie. In these variables
we can write Eq.(7) as
(12)
(13)
(14)
(15)
φ(z) =1
8z¯ z +1
8Re[z2¯
M +¯Dz] + f
(16)
Under a rotation of axes by an angle θ, z → z?= zeiθ,
the requirement that φ is invariant is satisfied if we have
M?= Me2iθ;
D?= Deiθ
(17)
On the (m,n) lattice, with (m + n) odd, the natural
basis vectors are (1,1) and (1,−1). Let us call these α
and β. We define the finite difference operators ∆±αand
∆±βby
∆±αf(z) = f(z ± α) − f(z)
∆±βf(z) = f(z ± β) − f(z)
Then the equations (14-15) can be written as
(18)
∆−αd = ∆βe
∆−βd = −∆αe
(19)
These equations are the discrete analog of the famil-
iar Cauchy-Riemann conditions connecting the partial
derivatives of real and imaginary parts of an analytic
function where the role of the analytic function is played
by D = d + ie.
From Eq.(14) and Eq.(15), it is easy to deduce that D
satisfies the discrete Laplace’s equation
[∆α∆−α+ ∆β∆−β]D = 0 (20)
If m and n are large, the corresponding patch is near
the origin (|ξ| + |η| is small), and where the leading be-
havior of φ(ξ,η) is given by˜φ(ξ,η) ∼ −1
Consider a point z0, such that at z0
4πlog(ξ2+ η2).
∂2˜φ/∂ξ2≈ m/4;
∂2˜φ/∂ξ∂η ≈ n/4,
(21)
Then, z0would be expected to lie in the patch labeled
by (m,n). This gives z0≈ ±(π¯
∂˜φ/∂z equal to¯D/2 gives us
M/2)−1/2. Then, setting
Dm,n? ±
1
√2π
√m + in
(22)
The equation (20), subjected to the behavior at large
|m| + |n| given by Eq.(22) on the 4π-wedge graph (for
each value of (m,n), Dm,nhas two values) has an unique
solution. Clearly the solution has eight fold rotational
symmetry about the origin in the (m,n) space.
implies that
This
D−n,m= i1/2Dm,n; for all (m,n).
Given Dm,n, its real and imaginary parts determine dm,n
and em,n, and using Eq.(12, 13) we determine the exact
positions of all the patch corners. The exact eight-fold ro-
tational symmetry of the adjacency graph of the pattern,
and the fact that D satisfies Eq.(20) on the adjacency
graph together imply the eight-fold rotational symmetry
of all the distances in the pattern.
We have not been able to find a closed-form formula
for Dm,n. But the system of coupled linear equations
(20) can be determined numerically to very good pre-
cision by solving it on a finite grid −L ≤ m, n ≤ L
with the condition in Eq.(22) imposed exactly at the
boundary. We determined dm,n and em,n numerically
for L = 100,200,400, and extrapolated our results for
L → ∞. We find d1,0 = 0.5000 and d2,1 = 0.6464, in
perfect agreement with the exact theoretical values 1/2
and 1 − 1/2√2 respectively.
Our arguments above can be extended to other two
dimensional lattices, so long as there are only two allowed
values of ∆ρ. While this is not clear why, this seems
to happen for the Manhattan lattice (Fig.2b), for initial
density 1/2. Also, this happens on the F-lattice, with
a periodic background pattern with initial density 5/8
[zi,j = 1 if i + j even, or (i,j) congruent to (0, 1) or
(2, 3) mod 4]. In some other cases, like the F-lattice,
with initially all sites empty, the pattern is very similar,
but there are some non periodic patches in the outermost
(23)
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ring. Since the behavior of φ(ξ,η) in such patches is not
known, the equations for Dm,ndo not close in this case.
We thank L. Levine for very useful discussions. The
special features of growth pattern studied here were
noted first in numerical studies by Mr.
Singha. DD would like to thank J. P. Eckmann for get-
ting him interested in this problem, and B. Nienhuis for
discussions.
Subhendu B.
∗ddhar@theory.tifr.res.in;
∼ddhar
†tsadhu@gmail.com; www.theory.tifr.res.in/∼tridib
‡schandra@tifr.res.in;
∼schandra
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