We propose a new model of self-organized criticality. A particle is dropped
at random on a lattice and moves along directions specified by arrows at each
site. As it moves, it changes the direction of the arrows according to fixed
rules. On closed graphs these walks generate Euler circuits. On open graphs,
the particle eventually leaves the system, and a new particle is then added.
The operators corresponding to particle addition generate an abelian group,
same as the group for the Abelian Sandpile model on the graph. We determine the
critical steady state and some critical exponents exactly, using this
equivalence.Comment: 4 pages, RevTex, 4 figure

A. Sornette

Abhishek Dhar

B. Drossel

C. G. Langton

D. Dhar

Deepak Dhar

F. Harary

H. Flyvbjerg

K. Chen

P. Bak

P. Ruelle

P. W. Kasteleyn

Supriya Krishnamurthy

V. B. Priezzhev

English

arXiv

Crossref

Eulerian Walkers as a Model of Self-Organized Criticality

Restricted Access. An open-access version is available at arXiv.org (one of the alternative locations)We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then added. The operators corresponding to particle addition generate an Abelian group, same as the group for the Abelian sandpile model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalenc

Priezzhev, V.B.

Dhar, Deepak

Dhar, Abhishek

Krishnamurthy, Supriya

Raman Research Institute Digital Repository

Eulerian walkers as a model of self-organized criticality

We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then added. The operators corresponding to particle addition generate an Abelian group, same as the group for the Abelian sandpile model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalence

Priezzhe, V. B.

Publications of the IAS Fellows

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96
Eulerian Walkers as a model of Self-Organised Criticality
V. B. Priezzhev1, Deepak Dhar2, Abhishek Dhar2 and Supriya Krishnamurthy2
1 Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russia
2Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
(February 6, 2008)
We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions
specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs
these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then
added. The operators corresponding to particle addition generate an abelian group, same as the group for the Abelian Sandpile
model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalence.
PACS numbers: 05.70.Ln, 05.40.+j
In recent years, there has been much interest in the
study of systems showing self-organized criticality (SOC)
[1] and different models have been proposed for many sys-
tems such as sandpiles [1], earthquakes [2], forest-fires [3]
and biological evolution [4]. All these models involve a
slowly driven system, in which the externally-introduced
disturbance propagates in a random medium using de-
terministic or stochastic rules. In the process it modifies
the medium so that after many such disturbances, the
medium develops long-range spatial correlations [5].
The most analytically tractable of all these models has
been the Abelian sandpile model (ASM) [6,7]. In this
Letter we introduce a new model of SOC called the Eu-
lerian walkers model (EWM). This model is quite differ-
ent from the ASM in some ways, but shares with it the
abelian group property. This allows a determination of
the steady state and exact calculation of some critical
exponents. In fact we define a general abelian model of
which both the EWM and ASM are special cases.
In the EWM self-organization occurs due to activity
of a walker which moves deterministically in a medium
while also modifying it. We show that on closed graphs,
the walker finally settles into a limit cycle which is an Eu-
ler circuit visiting each directed bond exactly once in one
cycle. On open graphs, the particle eventually leaves the
system. We then add at a randomly chosen site another
particle, which then moves, and so on. We define particle
addition operators which act on the set of recurrent con-
figurations of the system. These operators generate an
abelian group, and satisfy same closure relations between
themselves as in the ASM on the same graph. We show
that recurrent configurations of the system are in one to
one correspondence with spanning trees on the lattice.
The model is defined for a general graph as follows:
consider a connected oriented graph G consisting of N
points i = 1, 2...N . A point j has τj outgoing bonds,
and an equal number of incoming bonds, connecting it
to other points. The outgoing bonds at j are labelled by
integers from 1 to τj . We associate with each point, an
arrow which can point along one of the outgoing bonds
(Fig. 1). Let nj (1 ≤ nj ≤ τj) denote the current direc-
tion of the arrow that is the label of the bond along which
the arrow points. The set {nj} specifies the arrow direc-
tions at all points and provides a complete description of
the arrow configuration of the medium .
We now put a walker at some point on the graph. At
each time step:
(i) the walker after arriving at a site j changes the arrow
direction from nj to nj + 1(modτj),
(ii) the walker moves one step from j along the new arrow
direction at j.
Thus the motion of the walker is deterministic, is af-
fected by the medium and in turn affects the medium .
We can interpret the rules (i),(ii) as an intention of the
walker to maximize intervals between successive visits of
the same bond each time the walker leaves a given site.
The current position of the walker along with the value
of the variable nj at every site j specifies completely the
state of the system [8].
In the absence of sinks the walker continues to walk
forever. Since the system (walker+medium) has a finite
number of possible states, it eventually settles into a limit
cycle. In general, one would expect the size of these cy-
cles to be of the order of Poincare recurrence times for the
system, and grow exponentially with N. The surprising
fact is that all the cycles are very short and, in fact, are
of the same length
∑N
j=1 τj . In each such cycle, all bonds
are visited exactly once (Fig. 1). Such walks are known
as Euler circuits [9] and their study has been an impor-
tant problem in lattice statistics (If the circuit visits all
sites exactly once, it is called a Hamilton circuit). There
is a one to one correspondence between Euler circuits and
spanning trees on the same graph. Clearly for any Eu-
ler walk ending at a site j, last exit bonds from all sites
other than j form a spanning tree rooted at j. Kasteleyn
also showed that each rooted spanning tree corresponds
to a unique Euler circuit [10]. The number of all possible
1
trees is known to be given in terms of the determinant
of the adjacency matrix by the well known matrix tree
theorem [10].
We now show that every limit cycle is an Euler circuit.
We start from some arbitrary initial state of the medium
with the walker at some point i. The walker leaves the
point i along some bond b1. We evolve it till after time T
it returns to b1 for the first time. Let the bond traversed
at the jth step of path be bj , so that the path is b1b2...bT
with bT+1 = b1. We can show that no other bond in this
path is visited twice. Proof: Assume the contrary and
suppose that during the T steps the bond c, originating
from the point j, is the first bond that is visited twice.
Each successive exit from j is along a different direction
so there will be τj + 1 exits. But the number of visits to
j equals number of exits. Hence there must exist some
bond going into j which is also passed more than once.
This contradicts the fact that c was taken to be the first
bond to be passed twice. Thus all bi, i = 1 to T are
distinct and hence T ≤
∑N
i=1 τi. If every bond in G is
visited we have an Euler circuit with T =
∑N
i=1 τi. If not,
consider the path b2b3...bT+1bT+2. If bT+2 = b2 we have
another circuit of length T . We keep shifting the path
thus till we reach a t such that bT+t 6= bt. Such a t < T
exists so long as there are points j on the path which have
not been visited τj number of times. Let T
′ be the first
time when bT ′+t = bt. Clearly T < T
′. Now define the
new circuit formed by the T ′ steps starting with the tth
step. Iterating this we get circuits of increasing lengths
T < T ′ < T ′′... where each is ≤
∑N
i=1 τi and so finally
we will get an Euler circuit when T =
∑N
i=1 τi. All the
configurations which the system goes through before it
enters the cycle are transients.
To illustrate the process of self-organization, consider
the motion of a walker on an infinite line starting with a
random initial configuration of the medium. This walk
has a simple structure (Fig. 2). The walker turns the
arrow at the origin and starts the motion along the new
direction of this arrow reversing the arrows at all sites it
passes through. It moves on till it encounters a site with
an arrow pointing in the direction of motion. The walker
now reverses its direction and retraces its path entirely,
passing over all the sites traversed since the last reversal
of its direction. Then it continues to move ahead till it
again encounters an arrow pointing in the direction of
motion and so on. Thus the arrows in the region already
visited get organized into an almost Eulerian circuit so
that, if at time t the number of sites visited is S(t), then
in the previous 2S(t) time steps, most of these sites have
been visited exactly 2 times. In addition, the boundary
of the cluster advances by a finite amount ∆, as some
new sites are visited. For compact clusters S(t) ∼ R(t),
the average distance of the of the walker from the origin,
at time t. Thus we get
dR(t)
dt
∼
∆
R
. (1)
which implies that R ∼ t
1
2 for large t. The average num-
ber of sites visited till time t, S(t), goes as t
1
2 .
In higher dimensions, the motion of the EW is not so
simple.In Fig. 3 we show the results of a simulation of
the model on a square lattice with random initial config-
uration of arrows. Before each step the arrow is turned
clockwise by 90o. The sites visited at least once by the
walker form a cluster with few holes, whose radius R(t)
increases with time t. In the region visited by the walker,
all arrows are not aligned parallel, but are organized into
an almost Euler circuit so that in the time between T
and T − 4S(T ), only a very small fraction of sites is not
visited exactly 4 times (here τj = 4 for all sites j). Ar-
guing as in the one dimensional case we get R ∼ t
1
3 for
large t. We have carried out Monte-carlo simulations and
verified this to very good accuracy.
However, for d > 2, a random walker does not return
to previously visited sites often enough, and we expect
the motion of a walker in an initially random medium to
be diffusive (R2 ∼ t). Our numerical simulations show
that this is indeed the case for d = 3.
Now consider an open graph for which all the external
perimeter sites are identified with a single sink site, i0,
at which the walker gets absorbed. We place a walker
at some point i, with probability pi (
∑
pi = 1), and let
it evolve according to the rules specified before, until it
leaves the system [The walker will not get into a cycle as
every cycle would contain all points of the graph includ-
ing i0]. Now the system is specified only by the values
of ni, i = 1, N . We define operators ai acting on the
space of recurrent configurations of the EWM as follows:
for any recurrent configuration C, aiC = C
′, where C′
is the resulting configuration of the medium obtained by
adding a particle at site i on the configuration C, and
evolving it until it leaves the system.
It is easy to see that the operators at different sites
commute. Treat the motion of each walker as a sequence
of elementary steps. Then if two particles (walkers) are
added to the lattice at sites j and j
′
, the elementary
moves on two sites j 6= j
′
commute. If j = j
′
, they also
commute due to identity of particles. Therefore
[ai, aj ] = 0 . (2)
Within the space of recurrent configurations the opera-
tors ai will have unique inverses. If we define the N ×N
matrix, ∆, such that ∆ii gives the number of outgoing
bonds from i and −∆ij gives the number of bonds from
i to j then
∏
j
a
∆ij
i = I , for all i (3)
which simply reflects the fact that τi particles added at
i produce the same effect as 1 particle added at each
2
nearest neighbor of i. Thus, operators ai satisfy the same
algebra as the particle addition operators in the ASM.
In fact one can define commuting operators ai(ǫ), which
have a phase factor proportional to the number of steps
taken by the walker in going from C to C
′
as in the case
of the ASM [11].
So, as in the ASM, the number of recurrent configura-
tions R = Det(∆) and they occur with equal probability.
The first result also follows from the one to one correspon-
dence between steady state configurations of the EWM
and spanning trees which is obtained by drawing the last
exit directions at all points. Since the ai’s commute, they
can be diagonalized simultaneously. Then as for the ASM
Eq.(3) determines all the eigenvalues of a’s. Thus one can
diagonalize the evolution operatorW =
∑
piai, where pi
is the probability that a new particle is added at site i.
For a lattice of size L on a d-dimensional lattice, this im-
plies that the largest relaxation time of the system varies
as Ld.
Let Gij be the expected number of full rotations of the
arrow at the site j due to addition of a walker at the site
i. During the walk, the expected number of steps leaving
j is Gij∆jj whereas −
∑
k 6=j Gik∆kj is the average flux
into j. Equating both fluxes one gets
∑
k
Gik∆kj = δij or
Gij = [∆
−1]ij . (4)
Average number of steps n taken by the walker till it
leaves the system is given by < n >= z < s >ASM ,
where s is the number of topplings in ASM avalanches,
for regular graphs with coordination number z. Hence
< n >∼ L2, where L is the length of the system.
It is quite straightforward to calculate the arrow-arrow
correlation function in the steady state using the equiv-
alence of the problem to spanning trees. For two given
sites ~R1 and ~R2 the probabilities that arrows at the sites
are in the directions ~e1 and ~e2 respectively is the ratio
of spanning trees with these bonds occupied to the num-
ber of all spanning trees. This is easily calculated. For
large R12 the leading term in the connected part of this
probability is given by
C( ~R12; ~e1, ~e2) ∼ (~e1.~∇φ( ~R12))(~e2.~∇φ( ~R12)) (5)
where φ( ~R12) = G ~R1 ~R2 − G ~R1 ~R1 . In d dimensions φ(R)
varies as R2−d, hence the correlation function C(R)
varies as R2−2d for large separations R [12]. Thus the
steady state of the model has long range correlations and
hence exhibits self-organized criticality.
As pointed out earlier, when the walker has left the
system, the medium is in a recurrent state, and the ar-
rows form a spanning tree. This is not true for inter-
mediate times where the motion of the EW may lead to
a cyclic configuration of arrows. Thus a typical evolu-
tion of medium has periods of cyclicity interspersed be-
tween ‘normal’ acyclic states. In the EWM, the durations
of these intervals of cyclicity have a power-law distribu-
tion. In two dimensions, numerical simulations [13] show
that the probability of intervals of cyclicity of duration τ
varies approximately as 1/τ1.75.
Though we can establish a one to one correspondence
between the recurrent configurations of the EWM and
ASM, the relaxation process in the two models is quite
different. In the latter (in more than one dimensions) in
almost all cases particle addition leads to a stable configu-
ration after a finite number of topplings, and the fraction
of avalanches which reach the boundary is very small. In
contrast, in the present model, each walker must travel to
the boundary before it leaves the system, and thus the
fraction of events involving a finite number of steps of
the walker is zero in the limit of large system sizes. This
leads to the interesting conclusion that the statistics of
avalanches is not completely determined by the operator
algebra of ASM.
However, in one dimension, the probability distribu-
tion of number of steps can be computed exactly, and we
find that apart from trivial numerical factors, it is ex-
actly the same form as the limiting distribution found by
Ruelle and Sen [14] for avalanches in the 1d ASM.
To bring out the relationship of the present model to
the ASM more clearly, we observe that due to the abelian
nature of the evolution rules, we can add and evolve two
or more walkers in the system in arbitrary order without
effecting the final state. Let us choose the following rules:
each walker arriving at a site waits there until the number
of particles waiting at that site is ≥ r. Then these r parti-
cles take 1 step each in the directions nj+1,nj+2...nj+r,
and the arrow is reset to nj + r(modτj). Clearly r = 1
corresponds to the EWM, and r = τj corresponds to the
ASM. In the latter case the arrow configuration does not
evolve at all, and may be omitted from discussion.In Fig.
4, we have shown the results of Monte Carlo simulation
of this general model on a square lattice of size 200×200
for r = 1 to 4. We see that we get the same general be-
haviour of distribution of avalanches for all r > 1, but the
case r = 1 is special. It belongs to a different universality
class. For small s, the distribution P (s) is dominated by
boundary avalanches, therefore, it does not have a sim-
ple thermodynamic limit. However, the model has long
range correlations, and hence is critical.
In brief, we have introduced a new analytically
tractable model of SOC. It is hoped that further studies
of the model will contribute to a better understanding of
self-organizing systems in general.
3
[1] P. Bak, C. Tang and K. Weisenfeld, Phys. Rev. Lett. 59,
381 (1987).
[2] A. Sornette and D. Sornette, Europhys. Lett. 9, 197 (1989);
K. Chen, P. Bak and S. P. Obukhov, Phys. Rev. A 43, 625
(1990).
[3] B. Drossel and F. Schwabl, Phys. Rev. Lett. 69, 1629,
(1992).
[4] P. Bak and K. Sneppen Phys. Rev. Lett. 71, 4083 (1993).
[5] H. Flyvbjerg, Phys. Rev. Lett. 76, 940 (1996).
[6] D. Dhar, Phys. Rev. Lett. 64, 1613 (1990).
[7] V. B. Priezzhev, D. V. Ktitarev, and E. V. Ivashkevich,
Phys. Rev. Lett. 76, 2093 (1996).
[8] A model of self-organization using somewhat similar rules
was earlier defined by Langton. The emphasis was, how-
ever, on evolution of complex structures by deterministic
rules, starting from special simple initial conditions. Lang-
ton, Physica D 22, 120 (1986)
[9] “Graph Theory” by F. Harary, Addison-Wesley, chapters
7 and 16.
[10] P. W. Kasteleyn in “Graph Theory and Theoretical
Physics” by F. Harary, Academic Press, London and NY,
76 (1967).
[11] D. Dhar, Physica A 224, 162 (1995).
[12] The decay of correlations can be faster in special directions
if ~e1 or ~e2 are perpendicular to ~R12 so that the coefficient
of R2−2d vanishes.
[13] R. R. Shcherbakov, V. V. Papoyan, A. M. Povolotsky: Pri-
vate Communication.
[14] P. Ruelle and S. Sen, J. Phys. A: Math. Gen. 25, L1257
(1992).
Figure Captions
FIG. 1. (a) A directed graph. The outgoing bonds at each
site are labelled by integers 1, 2.... An initial state with a
configuration of arrows as in (b) and a walker starting at the
site a moves along the path abcb... which eventually settles to
the Euler circuit abcdcbaca .
FIG. 2. A random initial state of a lattice in one dimension
and the motion of a walker on this lattice. The medium is
organized into a state in which all arrows point in the same
direction.
FIG. 3. Simulation of the Euler walk on a square lattice
with random initial conditions. The whole cluster consists of
sites covered by the walker after 105 steps. The white region
shows the cluster of approximately 12500 sites visited exactly
four times in the last 50, 000 steps. The grey sites at the
boundary of the cluster are visited less than four times.
FIG. 4. Plot of probability P (s) of avalanche of size s vs.
s for different values of r.
4
ab
c
d a
b
c
d
1
3
2
1
1
1
2
2
(a) (b)
figure 1
figure 2
gure 3
1
100 102 104 106
s
10−10
10−8
10−6
10−4
10−2
100
P
(
s
)
figure 4
r=1
r=2
r=3
r=4


http://adsabs.harvard.edu/abs/1996PhRvL..77.5079P