We relate the fractal dimension of the backbone, and the spectral dimension
of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal
dimension of backbone equal to 4/3 and spectral dimension of trees equal to
5/4. In three dimensions, it provides us a new way to estimate z numerically.
We get z=1.617 +/- 0.004.Comment: 6 pages, Latex, and 4 postscript figures, uuencoded. Minor
  typographical errors, and problems with postscript files fixe

Dhar, D.

Manna, S. S.

English

arXiv

We relate the fractal dimension of the backbone, and the spectral dimension of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In three dimensions, it provides us with another way to estimate z numerically. We get z=1.617&#177;0.004

Indian Academy of Sciences

Fractal dimension of backbone of eden trees

S. S. Manna

D. Dhar

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Fractal dimension of backbone of Eden trees

Publications of the IAS Fellows

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Fractal Dimension of Backbone of Eden Trees
S. S. Manna1 and D. Dhar2
1Department of Physics, Indian Institute of Technology
Powai, Mumbai 400076, India
e-mail : manna@niharika.phy.iitb.ernet.in
2Theoretical Physics Group, Tata Institute of Fundamental Research
Homi Bhabha Road, Mumbai 400005, India
e-mail : ddhar@theory.tifr.res.in
Abstract
We relate the fractal dimension of the backbone, and the spectral dimension of
Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of
backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In three dimensions,
it provides us a new way to estimate z numerically. We get z = 1.617± 0.004.
Dense branching patterns are found in many different physical situations in nature
e.g., coral reefs, river networks, collapsed phase of branched polymers, very slowly evapo-
rated films of sugar dissolved in water [1-3]. In all these systems, the Hausdorff dimension
of the structure is equal to that of the embedding space but the detailed structure is
different depending on the different physical processes involved.
The Eden model has been studied a lot in the last decade, mainly for the surface
properties [4]. Eden trees [5,6] are simple theoretical model of dense branching structures.
In [5], it was argued that classical diffusion on Eden trees is anomalous because of trapping
in dead end branches, and the root mean square deviation of a random walker on the
tree increases with time as tx, where the exponent x does not satisfy the usual relation
x = d˜/2d¯, where d˜ is the spectral dimension, and d¯ is the (Hausdorff) fractal dimension
of the lattice. Dhar and Ramaswamy expressed the exponents x and d˜ in terms of an
exponent θ related to the fractal dimension of the backbone of the trees [5]. Using a
different method of analysis, and somewhat larger simulations, Nakanishi and Herrmann
[6] also calculated these exponents. However, these exponents have not been determined
analytically so far.
In this Rapid Communication, we show that the backbone exponent of the Eden
trees can be related to the dynamical exponent of KPZ model, and show that in two
dimensions,θ = 1/3, d˜ = 5/4 and x = 3/8. In three dimensions, our method gives us a
new way to determine the dynamical exponent numerically. The numerical determined
value is z = 1.617± 0.004.
We shall consider Bond-Eden Trees (BET) in this report. The model is defined in
1
terms of the spreading of an infectious disease along the bonds of a d-dimensional lattice.
Each site may be in one of two states : healthy or infected. At time t=0, a fixed set of ‘seed’
sites are infected. This seed set is a single site in the so-called point seed geometry, and
a (d− 1) dimensional hyper-plane in layer seed geometry. At each time-step, we select at
random one site from the set of healthy sites having at least one infected neighbour. This
site then becomes infected, and once infected, a site never recovers. We connect this site
to existing infected cluster by the bond connecting it to its first-infected neighbour. Each
new infected site adds exactly 1 bond to the cluster. The resulting cluster so generated
is called a BET.
However, the bulk of the Bond Eden trees is a spanning tree, and has a complex
internal structure. This may be quantified in terms of the fractal dimension of the chemical
paths along the tree, distribution of branch sizes, spectral dimension of the tree, structure
of its backbone etc. We shall see below that all these measures can be determined in
terms of a single critical exponent, which we choose to be the dynamical critical exponent
z.
There is a unique path which connects any two sites of a tree. We define the
backbone of the tree corresponding to radius R as the set of all sites which lie on a path
connecting one of the seed sites to any of the sites of the tree at a distance R from it.
Fig. 1 shows the backbone of a two dimensional Eden tree grown from a single seed for
two values R = 80 and 130 lattice spacings. As the cluster grows, some of the growing
branches became dead ends, and are removed from the backbone. While the part of the
backbone near the surface changes quite fast, structure of the backbone deep inside the
cluster does not change much with time, and gets frozen in.
From Fig. 1, it is easy to see that the backbone of Eden clusters has a branching
structure, where each branch is directed radially outwards. A branch of the backbone of
length R has transverse fluctuations of order R1/z , where clearly z ≥ 1 and is called the
dynamical exponent. This exponent is inverse of the exponent specifying the transverse
fluctuations of a directed polymer in a random medium [7].
We now argue that the backbone has a fractal dimension dB given by
dB = 1 + (d− 1)(1− 1/z) (1)
Consider the backbone of the cluster when its radius is R, and at a later stage when its
radius is R+ h, h << R. Only a small fraction of the perimeter sites at radius R remain
part of the backbone at the later stage. Each such site gives rise to a cluster of active
sites of transverse size h1/z. This implies that the density of backbone sites at radius R,
when cluster has grown to size (R+ h) vanishes as h−(d−1)/z for large h << R. Once h is
of order R, it gets frozen to the value R−(d−1)/z , and does not change further. Thus the
number of sites in the frozen backbone upto radius R varies as RdB , where dB is given by
Eq.(1).
The spectral dimension of the Eden trees is clearly a bulk property and can also be
expressed in terms of z. Consider a spring network on the tree, such that at each bond
of the tree, there is a spring of spring constant κ, and a mass m is attached to each site
of the tree. Let F (ω2) be the fractional number of eigenvalues with frequency less than
ω. Then if F (ω2) ∼ ωd˜ for small ω, d˜ is the spectral dimension of the tree.
2
To determine d˜, we use a decimation argument analogous to that in [8]. Define the
‘burning time’ at a site on the tree T as the length of the longest directed path from the
site to a leaf site of the tree. We construct a decimated tree T ′, consisting of only those
sites whose burning time is an exact multiple of an integer scale factor b. The resulting
tree T ′ is not related to T by a simple scale change (for some figures of decimated trees see
[9]). However, small parts of T are related to corresponding parts of T ′ by local self-affine
transformation, where radial distances are scaled by a factor b, and transverse distances
by a factor b1/z . The spring constants renormalize by κ′ = κ/b, and masses according to
the formula M ′ = b1+(d−1)/zM . Thus we get
F (κ′/M ′) = b−1−(d−1)/zF (κ/M). (2)
which implies that
d˜ = 2.
z + d− 1
2z + d− 1
. (3)
For d = 2, the dynamical exponent takes the well known KPZ value 3/2. Corre-
spondingly, we get dB = 4/3 and d˜ = 5/4. This value is in good agreement with the
earlier numerical estimates of [5] and [6]. Dhar and Ramaswamy found that for d = 2,
d˜ = 1.22 ± 0.04, x = 0.42 ± 0.04, and for d = 3, d˜ = 1.30 ± 0.12, x = 0.44 ± 0.04 [5].
Nakanishi and Herrmann estimated d = 2, d˜ = 1.22± 0.02 and x = 0.39± 0.02 where as
for d = 3, d˜ = 1.32± 0.02 and x = 0.30± 0.02 [6]. As d → ∞, d˜ tends to 2, which again
agrees with the exact result [10].
For the root mean square displacement of order L the average number of branch
site per backbone site visited is of order L(d−1)/z . Hence the diffusion constant is of order
L−(d−1)/z and time T scales as L2+(d−1)/z , so we get 1/x = 2 + (d − 1)/z. We note that
x 6= d˜/2d¯, where d¯ is the fractal dimension of the substrate. However, the equality sign
holds if d¯ is replaced by dm = 1 + (d− 1)/z, which is the effective mass dimension of the
graph defined by the relation that the number of distinct sites within a distance r along
the tree from a randomly chosen site (not near origin) varies as rdm for large r << size
of cluster R.
We have checked these predictions against numerical simulations. Figure 2 shows
a plot of the average number of sites in the backbone < Mb > of a 2-dimensional Bond
Eden tree within a (Euclidean) distance R from the origin multiplied by a factor R−4/3
versus R−1/3. We averaged over 12 million independently generated configurations for
R = 16 decreased to 12000 configurations for R = 1024 on the square lattice. We see a
reasonably good fitting of a straight line in fair agreement with the theoretical prediction
of dB =4/3 which also indicates that the correction to scaling is likely to be R
−1/3. We
estimate the error in the quoted values of both the fractal dimension and the correction
to scaling exponent to be about 0.03.
In Fig. 3 we show the results of simulations of Eden trees on a triangular lattice
grown from a line seed on a 6000× 10000 lattice, averaged over 2500 configurations. We
have plotted h2/3Nh versus h, where Nh is the fractional number of distinct trees which
survive up to height h. From the scaling hypothesis Nh ∼ h
−α, where α = (d − 1)/z
In this case α = 2/3, and hence the graph should be a horizontal line. For large h we
3
see that this expectation is very well satisfied, and the numerically determined value of z
from this plot gives z = 3/2± 0.002.
The main advantage of our simulation over earlier simulations of various versions
of Eden model, e.g., models A, B and C of Jullien and Botet [12] is that here we are in
effect studying bulk quantities (The quantity Nh can be related to the probability that
randomly chosen site in the bulk of the tree has at least one descendant left after h more
generations). A related quantity, the distribution of branch sizes which are disconnected
on removing a randomly chosen bond from the tree has been studied in [9].
Encouraged by the good convergence of simulation results in our model to the asymp-
totic values in 2-dimensions, we extended our studies to 2+1 dimensions. In Fig. 4, we
plot the results of simulation of Eden growth in 2+1 dimension in the layer geometry.
We used as 180× 180× 500 simple cubic lattice, and averaged over 10500 configurations.
Fig. 4 shows h5/4Nh versus h on a log-log plot. We estimate the slope of the curve in this
plot to be 0.013± .003. This implies that α = 5/4− 0.013± 0.003, which corresponds to
z = 1.617± .004. The value α = 5/4 corresponds to the Kim-Kosterlitz conjecture [13],
which is clearly ruled out by our data. Our values are in good agreement with the current
best numerical estimate of α = 1.240 ± 0.001 by Forrest and Tang [14] and Ala-Nissila
and Venalainen’s result of α = 1.240± 0.002 [15].
The extension to higher dimensions seems possible, but would require higher com-
putational power than available to us at present.
We thank Drs. T. Halpin-Healy and L. H. Tang for providing some useful references.
4
Figure Captions
Figure 1 : Backbone of a Bond Eden Tree generated on the square lattice from a point
seed at two stages of growth of radii 80 and 130 lattice constants.
Figure 2 : Variation of average mass of the backbone < Mb > of Bond Eden Trees of
radius R. The fractal dimension of the backbone is ≈ 4/3, in complete agreement with
our theoretical prediction and the correction to scaling decreases as R−1/3.
Figure 3 : Plot of h2/3Nh versus h for Bond Eden Trees grown on a base line of length
6000. Here Nh is the fractional number of distinct trees which survive upto height h .
The average slope of this plot for h between 1000 and 10000 is at most ±0.003.
Figure 4 :Plot of h5/4Nh versus h for Bond Eden Trees grown on a square base of size
180× 180. The average slope between h = 100 and 500 is 0.013. A straight line of slope
0.013 is plotted as a guide to the eye.
5
References
[1.] S. S. Manna and B. Subramanian, Phys. Rev. Lett. 76 (1996) 3460; A. Rinaldo, I.
Rodriguez-Iturbe, R. Rigon, E. Ijjasz-Vasquez and R. L. Bras, Phys. Rev. Lett. 70, 822
(1993); A .Giacometti, A. Maritan, and J. R. Banavar, Phys. Rev. Lett. 75, 577 (1995).
[2.] B. Derrida and H. J. Herrmann, J. Phys. (Paris) 44 (1983) 1365.
[3.] A. S. Paranjape, Phys. Lett. A 176 (1993) 349.
[4.] For a review see P. Meakin, Phys. Rep. 235 (1993) 189; T. Halpin-Healy and Y. C.
Zhang, Phys. Rep. 254 (1995) 215.
[5.] D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 54, (1985) 1346.
[6.] H. Nakanishi and H. J. Herrmann, J. Phys. A 26 (1993) 4513.
[7.] S. Roux, A. Hansen and E. L. Hinrichsen, J. Phys. A 24, (1991) L295; M. Cieplak,
A. Maritan and J. R. Banavar, Phys. Rev. Lett. 76, 3754 (1996).
[8.] S. S. Manna, D. Dhar and S. N. Majumdar, Phys. Rev. A 46 (1992) R4471.
[9.] P. Meakin, Physica Scripta 45 (1992) 733.
[10.] D. Dhar, J. Phys. A 18 (1985) L713.
[11.] J. Krug and P. Meakin, Phys. Rev. A 40 (1989) 2064.
[12.] R. Jullien and R. Botet J. Phys. A 18 (1985) 2279.
[13.] J. M. Kim and J. M. Kosterlitz, Phys. Rev. Lett. 62 (1989) 2289.
[14.] B. M. Forrest and L. H. Tang, Phys. Rev. Lett 64 (1990) 1405.
[15.] T. Ala-Nissila and O. Venalainen, J. Stat. Phys. 76 (1994) 1083.
6

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Fig. 2   Manna and Dhar  Fractal Dimension of Backbone of eden Trees
100 101 102 103 104
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Fig 3 : Manna and Dhar  Fractal Dimension of Backbone of Eden Trees
100 101 102 103
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Fig 4 : Manna and Dhar Fractal Dimension of Backbone of Eden Trees


We relate the fractal dimension of the backbone, and the spectral dimension of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In three dimensions, it provides us with another way to estimate z numerically. We get z=1.617+/-0.004

Fractal Dimension of Backbone of Eden Trees

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