Page 1

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

Page 2

2

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,

Page 3

3

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)

Page 4

4

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)

Page 5

5

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

[1] M. C. Cross, P. C. Hohenberg, Rev of Mod Phys. 65, 851

(1993).

[2] John E. Pearson, Science, New Series, Vol. 261, No. 5118

(1993), 189.

[3] L. S. Schulman and P. E. Seidon, J. Stat. Phys. 19

293(1978).

[4] D. Dhar, Physica A 369, 29 (2006).

[5] D. Dhar, arXiv:cond-mat/9909009.

[6] Anne Fey-den Boer, Frank Redig, J. Stat. Phys. 130, 579

(2008).

[7] Y. Le Borgne and D. Rossin, Discr. Math., 256, 775

(2002);M. Creutz. Comput. Phys. 5 198 (1991).

[8] S. H. Liu, T. Kaplan and L. J. Gray, Phys. Rev. A 42,

3207 (1990).

[9] Lionel Levine and Yuval Peres, Indiana Univ. Math. J.

57 (2008), 431-450. [arXiv:math/0503251].

[10] Fractal concepts in surface growth, L. Barabasi and H. E.

Stanley, Cambridge Univ. Press, Cambridge, 1995.

[11] S. Ostojic, Physica A 318 187 (2003).

www.theory.tifr.res.in/

www.theory.tifr.res.in/