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epl draft
Improved Upper Bounds on the Asymptotic Growth Velocity of
Eden Clusters
Aanjaneya Kumar and Deepak Dhar
Department of Physics, Indian Institute of Science Education and Research, Pune
Dr. Homi Bhabha Road, Pashan, Pune 411008, India
PACS 64.60.De – Statistical mechanics of model systems
PACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
PACS 05.50.+q – Lattice theory and statistics
Abstract – We consider the asymptotic shape of clusters in the Eden model on a d-dimensional
hypercubical lattice. We discuss two improvements for the well-known upper bound to the growth
velocity in different directions by that of the independent branching process (IBP). In the IBP, each
cell gives rise to a daughter cell at a neighboring site at a constant rate. In the first improvement,
we do not allow such births along the bond connecting the cell to its mother cell. In the second,
we iteratively evolve the system by a growth as IBP for a duration ∆t, followed by culling process
in which if any cell produced a descendant within this interval, who occupies the same site as the
cell itself, then the descendant is removed. We study the improvement on the upper bound on
the velocity for different dimensions d. The bounds are asymptotically exact in the large-dlimit.
But in d= 2, the improvement over the IBP approximation is only a few percent.
Introduction. – The first passage percolation prob-
lem (FPP) has a long history. Hammersley and Welsh
introduced First Passage Percolation as a lattice model of
fluid flowing through a random porous medium in 1965 [1].
Subsequently, it was shown by Richardson and improved
upon by Cox-Durrett and Kesten that for this model the
number of lattice sites that the fluid can wet grows linearly
with time and this cluster of wetted sites, asymptotically
converges to (under suitable normalization) a determinis-
tic subset of the lattice called the limit shape [2–4]. The
problem has attracted a lot of researchers, and a good
recent review of the subject may be found in [5]. How-
ever, there are many unanswered questions. In particular,
the exact solution for the asymptotic growth velocity is
not known for any regular translationally invariant lat-
tice, and not much is known about the exact asymptotic
shape of the wetted cluster, beyond its convexity.
In recent years, there has been a surge in interest in the
study of stochastic growth models [1, 5–7]. Apart from
their many applications, starting from growth of bacte-
rial colonies [8] to spreading of rumours in a society [9],
these models have given us a lot of insight into nonequilib-
rium phenomena by providing us with a platform to study
universal behavior [6, 10]. An important question about
these processes is understanding the extent of growth of
the cluster in different directions, and its fluctuations [5].
In this regard, in the context of the Eden model, bounds
on velocity of the growing cluster along the axis and the
diagonal directions have been obtained earlier [11–14]. On
ad-dimensional hypercubical lattice, the problem becomes
easier in the limit of large d. It was shown earlier [15] that
the asymptotic velocity along the axis uaxis(d) in large
dimensions is given by
lim
d→∞ uaxis(d)log d
d= 2 (1)
However, the convergence to this value is slow. We will
give an independent proof of this result below using the
fact that the Eden growth process is slower than the inde-
pendent branching process (IBP). For the velocity along
the diagonal direction (1,1,1, ..), one can get upper and
lower bounds on the velocity, both of which grow as √d,
with d.
The FPP problem is related to the problem of self-
avoiding walks, and bounds on the growth constant of
self-avoiding walks ( The growth constant is defined as
the limit of (CN)1
N, where CNis the number of self avoid-
ing walks of Nsteps). In the ’large d’ expansion tech-
nique, the growth constant µdof self-avoiding walks for
the d−dimensional hypercubical lattice, is expanded in a
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Aanjaneya Kumar and Deepak Dhar
asymptotic series of the form µd= 2d−1−1
2d+O(1/d2)
[16]. It is interesting to note that the value 2dis the growth
constant of fully random walks, and the leading correction
to the growth constant comes from disallowing immediate
retraversals. We will not need to invoke the connection
to self-avoiding walks in this paper, but will instead ex-
plore the idea that if we disallow immediate retraversals
in IBP, it would be expected to improve the upper bounds
on velocity significantly.
As far as the asymptotic shape of the growing cluster is
concerned, most studies have been concerning the inequal-
ity of the growth velocity along the axes and diagonal. It
was first proved by Kesten [11] that in very high dimen-
sions (greater than about 106), the asymptotic shape of
the Eden cluster is not a Euclidean ball. Subsequently, di-
mensional improvements have been made [13, 14] and we
now know that the limit shape shows a departure from the
Euclidean ball in d > 22. While some numerical studies
of the asymptotic shape have been reported in d= 2 [17],
we could not find any general discussion of the equation of
the asymptotic shape, apart from a few simplified models
of first passage percolation [18].
In this paper, we first re-derive the already known up-
per bounds on the growth velocity of Eden clusters on
ad-dimensional hypercubical lattice by studying the In-
dependent Branching Process as an upper bound for the
Eden process. This is then easily extended to get bounds
in a general direction and its limit shape. Then, as an
improvement over our approximation, we study a birth
process in which any agent gives rise to progeny at neigh-
boring sites at a constant rate, except at the site of its
own parent. By analogy with self-avoiding walks, we ex-
pect that this takes care of the leading correction term in
the large dexpansion of the first passage velocity. In the
second, we iteratively evolve the system by a time-step
process of growth as IBP for a duration ∆t, followed by
culling process in which if any cell produced a descendant
within this interval, who occupies the same site as the cell
itself, then the descendant is removed. Thus, the finite-
time propagator of the branching process is modified so
that its value at any site does not exceed one.
The Eden Model. – The Eden Model was first
introduced by Murray Eden in 1961 [19] to investigate
the growth of biological cell colonies. Many variants
of this model have been studied since then −start-
ing from the model of skin cancer by Williams and
Bjerknes [20] to the SIR (Susceptible-Infected-Recovered)
and SIS (Susceptible-Infected-Susceptible) models of epi-
demics [21].
We begin by defining the Eden model as an epidemic
model. Consider an infection process on a d−dimensional
hypercubical lattice where each site can either be in-
fected or healthy. We denote the coordinates of each
site by (x1, x2, ..., xd). At time t= 0, only the origin
O=(0,0, ..., 0) is infected and all other sites are healthy.
The evolution is a continuous time Markov process in
which an infected site infects each of its healthy neigh-
bours at rate 1. We consider the process in which an
infected site never recovers. This model is equivalent to
first passage percolation with exponentially distributed in-
dependent passage times. In the context of this epidemic
process, the question of finding upper bounds to the shape
of the growing cluster is natural, and is equivalent to es-
timating the size of region in which the residing people
could be exposed to a particular infection given that the
infection started from the origin. We will provide upper
bounds to the asymptotic shape of the Eden cluster using
the Independent Branching Process (IBP) and its variants.
The independent branching process. – The in-
dependent branching process on the Zdlattice is defined
as follows: We consider an infection process in which the
number of cells present at a site can be arbitrarily large.
Let n(~
R, t) denote the number of cells present at the site ~
R
at time t. At the time t= 0, there is only one cell present
in the system, and it is placed at the origin ~
O. Then, we
have n(~
R, t = 0) = δ~
R, ~
O.
The time evolution is a continuous-time Markov process.
At any time t, a cell gives birth to a descendant cell, that
sits at a nearest neighbor site. Then number of cells at
the neighbor increases by one. Once born, a cell never
dies. We assume that the rate at which a cell gives birth
to a daughter cell is 1 along each bond, independent of the
number of cells present at the site, or at neighbours.
In this model, the number of cells present at time t
increases exponentially with t. Each cell gives birth to a
daughter cell at a rate 2dper unit time (because each has
2dneighbors). Hence the average number of cells present
at time tis exp(2dt), for all t > 0. Also, with time, the
region occupied by at least one cell, also called the region
invaded by the cells, grows with time. The outer boundary
of the region invaded by the cells is called the invasion
front. We define u(~
Ω) as the velocity of the invasion front
in the direction ~
Ω as
u(~
Ω) = lim
t→∞(1/t)~
Rt(~
Ω) (2)
where ~
Rt(~
Ω) is position of the invasion front in the direc-
tion ~
Ω) at time t. It is easily seen that u(~
Ω) has a non-zero
limit, and the fluctuations in (1/t)~
Rt(~
Ω) tend to zero as
time increases.
In the Eden process(EP), the number of cells at any site
is at most 1. It is easily that if we have two configurations
Cand C0, where Cevolves according to the rules of EP, and
C0evolves as an IBP, and at any given time t, the number
of cells in C0at any site ~
Ris greater than or equal to the
number at the corresponding site in C. Then, this property
will be preserved at subsequent times. This implies that
the invasion front velocity in IBP provides an upper bound
to the invasion front velocity in EP in all directions ~
Ω.
It is straight forward to determine the growth velocity
in IBP. Let the average number of cells in the IBP at time
t, at the site ~
Rbe denoted by ¯n(~
R, t). We use the fact
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Improved Upper Bounds on the Asymptotic Growth Velocity of Eden Clusters
that in IBP, these variables satisfy a linear equation
d
dt ¯n(~
R, t) = X
nn
¯n(~
R0, t) (3)
where the sum runs over the 2dnearest neighbors of ~
R.
This is a linear equation, and is easily solved, by Fourier
transformation. We define the variables ˜n(~
k, t) as
˜n(~
k, t) = X
~
R
¯n(~
R, t)exp(−i~
k. ~
R) (4)
Then, these variables satisfy the equation
d
dt ˜n(~
k, t) = λ(~
k)˜n(~
k, t) (5)
with λ(~
k)=2Pd
i=1 cos(ki).
This equation is easily solved, and by inverse Fourier
transformation, we get
¯n(~
R, t) = Zd~
k
(2π)dexp(λ(~
k)t+i~
k. ~
R).(6)
It is easily seen that for fixed ~r, ¯n(~
R, t) increases as
exp(2dt). We are interested in the case where as tin-
creases, ~
Ralso becomes bigger with time as ~
R=~vt. Then,
the integral becomes
¯n(~
R, t) = Zd~
k
(2π)dexp(thλ(~
k) + i~
k.~vi).(7)
In the limit of large t, this is evaluated easily, using the
steepest descent method. The stationary point occurs at
a imaginary value of ~
k=i~κ, given by
κj= sinh−1(vj/2).(8)
We define the large deviation function F(~v) by the condi-
tion that for large t,
¯n(~v t, t)∼exp [tF(~v)] (9)
with
F(~v) =
d
X
i=1 h2p1 + vi2/4−visinh−1(vi/2)i(10)
We note that Fis a decreasing function of its argument.
At ~
V= 0, it has a value 2d. And for large |v|, it varies as
−Pi|vi|log |vi|).
At the cluster boundary, ¯nis of O(1). So, the boundary
of the cluster, scaled by t, is given by equating the growth
rate of ¯nto zero. Thus, we get that the scaled boundary
of the cluster in the IBP is given by
d
X
i=1 h2p1 + vi2/4−visinh−1(vi/2)i= 0.(11)
(a) (b)
Fig. 1: (a) A contour plot for the independent branching
process in d= 3 for z= 0,1,2,3,4,5,5.5,5.65 starting
with the outermost curve. (b) A demonstration of the
departure from the spherical ball. The plot in blue is a
numerical plot of Eq(10) and orange is an arc of a circle
with the radius 5.67.
Note that from Eq.(7), ¯n(~
R, t) is also equal to the num-
ber of walkers expected at ~
Rat time t, if exp(2dt) walkers
are released at the origin at time t= 0, and perform inde-
pendent random walks.
As a check, we see that along the diagonal direction
(1,1,1,1..), we set vi=v∗, for all i. Then, for all d, we
get V∗is the solution of the equation
2q1 + v∗2/4 = v∗sinh−1(v∗/2) (12)
which gives v∗≈3.01776. This gives the well-known up-
per bound to the speed along the diagonal in ddimensions
(measured in Euclidean norm) as
vdiag,EP ≤vdiag ,IB P = 3.01776√d. (13)
One can also verify that this agrees with the al-
ready known results about the asymptotic velocity along
one of the axes in the IBP. In this case, we set ~v =
(Vaxis,IB P ,0,0,0..). The corresponding equation becomes
2s1 + V2
axis,IB P
4+ 2d−2 = Vaxis,IB P sinh−1(Vaxis,IB P
2)
(14)
This is easily solved, and gives Vaxis,IBP =
4.4668,5.67295,6.75371 and 7.75405, for d= 2,3,4
and 5 respectively. For large d,Vaxis varies as 2d/log(d).
These results about velocity along the axes, or along the
main diagonal have been known for a long time. However,
we did not find an explicit discussion of the asymptotic
shape of the cluster for the IBP in a general direction in
the literature. In Fig. 1, we show the asymptotic shape in
3−dcalculated numerically using Eq.(11. In the EP, Alm
and Deijfen found that the cluster shape is not exactly
circular, with the Euclidean speed along the diagonal and
p-3
Aanjaneya Kumar and Deepak Dhar
Fig. 2: A plot of velocity vas a function of direction θ
for the IBP in d= 2. Note that the velocity is maximum
along the axis (θ=nπ/2 for integer n) and minimum
along the diagonal direction (θ= (2n+ 1)π/4).
along the axis being 2.4420 and 2.4742 respectively. Thus
the speed along the diagonal is smaller by about 1.3%. For
the IBP, we found these to be 4.26775 and 4.4668, with
the diagonal speed being smaller that that along the axis
by a bigger amount(about 4.5%).
The First Variation of IBP (IBP1). – We will now
define a variation that is a bit more complicated than the
IBP defined above. This is also a continuous-time Marko-
vian evolution independent branching process. Here, we
consider the process on a hyper-cubical lattice in ddi-
mensions. The number of cells at any site can be a non-
negative integer. As before, We start with a single cell
at the origin at t= 0, with the rest of the lattice empty.
However, We note that all cells, other than the original
‘eve’-cell have a mother cell. The evolution rule is still
Markovian. Each cell gives rise to a daughter cell along
each bond at rate 1, independent of the state of the other
sites, except along the bond that connects it to its mother
cell. Thus, such a cell will have (2d−1) bonds along which
it can The ‘eve’-cell gives rise to a child along each of the
2dbonds connected to it, at rate 1. We call this process
the IBP1.
For the IBP1process also, we can write a closed set of
coupled linear evolution equations for the average number
of cells at site ~
Rat time t. But we need to define 2dvari-
ables at each site. Let ¯n(~
R, t, eα) denote the average num-
ber of cells residing at the site ~
Rat time t, whose mother
cell is along the bond eα. Here αtakes 2dpossible values
±1,±2,...,±d, and e1is the unit vector along coordinate
x1, and e−1=−e1. Then, the variables ¯n(~
R, t, eα) evolve
according to the equations
d
dt ¯n(~
R, t, eα) = X
α06=−α
n(~
R+eα, t, eα0) (15)
Again, we define the Fourier transform variables of
¯n(~
R, t, eα) as ˜n(~
k, t, eα) as
˜n(~
k, t, eα) = X
~
R
exp(−i~
k. ~
R)¯n(~
R, t, eα) (16)
Then, the equations for different ~
kdecouple, and the in-
finite set coupled equations reduces to that of 2dcoupled
variables ˜n(~
k, t, eα), for the 2dvalues of αfor fixed ~
k.
These are easily seen to be
d
dt ˜n(~
k, t, eα) = exp(ikα)hS(~
k)−˜n(~
k, t, −eα)i(17)
d
dt ˜n(~
k, t, −eα) = exp(−ikα)hS(~
k)−˜n(~
k, t, eα)i(18)
where α= 1,2..d and S(~
k) = Pd
β=1[fβ+f−β] with fβ
and f−βrepresent ˜n(~
k, t, eβ) and ˜n(~
k, t, −eβ) respectively.
This may be written as
d
dt ˜n(~
k, t, α) = X
α0
Mα,α0(~
k) ˜n(~
k, t, α0) (19)
The eigenvalues for this 2d×2dmatrix for a fixed ~
k-block
are easily determined. Explicitly, the matrix elements are
Mα,β = exp(ikα),ifβ6=−α; (20)
= 0,ifα=−β. (21)
Here the indices αand βtake values (±1,±2,±3.. ±d)
For the eigenvalue λ, let the eigenvector of the matrix be
fα, we have, for α= 1 to d
λfα=eikα[S(~
k)−f−α] (22)
λf−α=e−ikα[S(~
k)−fα] (23)
where
S(~
k) =
d
X
i=1
= [fi+f−i].(24)
We try to solve the coupled equations for fαand f−αin
terms of S(~
k). Two cases arise. If λ26= 1, then, we can
solve these equations to give
fα=eikαλ−1
λ2−1S(~
k); f−α=e−ikαλ−1
λ2−1S(~
k) (25)
Then, the consistency condition 24 then becomes
λ2−1+2d= 2λ"d
X
i=1
cos ki#(26)
This is a quadratic equation in λ, and has two roots. We
denote the larger one by λmax. this gives
λmax = Λ + pΛ2−2d+ 1.(27)
where we have used the abbreviation Λ = Pd
i=1 cos(ki).
On the other hand, if λ=±1, then to get a non-zero
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Improved Upper Bounds on the Asymptotic Growth Velocity of Eden Clusters
solution, we must have S(~
k) = 0, and fα=−λeikαf(−α),
for α= 1 to d.
Writing ~
R=~ut allows us to write
¯n(~ut, t) = Zd~
k
(2π)dexp hti~
k.~u +Λ+pΛ2−2d+ 1i
(28)
The above integral can be easily evaluated in the long time
limit using the method of steepest descent giving
iuj−sin kj"√Λ2−2d+ 1 + Λ
√Λ2−2d+ 1 #= 0 (29)
The stationary point occurs at a imaginary value of ~
k=i~κ
which upon substitution gives
κj= sinh−1uj
β(30)
where
β=√Λ2−2d+ 1 + Λ
√Λ2−2d+ 1 (31)
This gives us
Λ =
d
X
j=1 r1+(uj
β)2(32)
We can self-consistently determine Λ and βfor given ~u
from Eqs.(31,32). Now we can write
¯n(~ut, t)∼exp
t(Λ + pΛ2−2d+ 1 −
d
X
j=1
ujsinh−1uj
β)
(33)
Let Λ∗(~u), β∗(~u) be solutions of Eqs.(31,32). Then the
equation of the surafce is obtained to be
d
X
j=1 ujsinh−1uj
β∗(~u)= Λ∗(~u) + q(Λ∗(~u))2−2d+ 1.
(34)
By a straightforward numerical calculation, we find that
in IBP1, the velocity along axis is 4.3466, 5.3533, 6.4485
and 7.4602 while along the diagonal direction is 3.9770,
4.9772, 5.8150 and 6.5485 for d= 2, 3, 4 and 5 respectively.
The Second Variation of IBP (IBP2). – We de-
fine the second variation of the IBP as follows: we start
with a single eve cell at the origin at t= 0, with the
rest of the lattice empty. We break the time evolution
of the process in time intervals of ∆. In the time interval
z∆≤t≤(z+1)∆ where zis nonnegative integer, the sys-
tem evolves according to the IBP rules. At times z∆, we
introduce a culling process: If any cell present at time z∆
at any site ~
R, generates a descendant ( this is second-order
descendant, or higher order descendant) within the subse-
quent interval ∆, and this descendant occupies the same
site ~
R, then the descendant is removed. This modified
evolution + culling process is captured by the modified
propagator G0(R, ∆). This new propagator G0is obtained
from the IBP propagator G0by a process of clipping: we
write
G0(R, ∆) = [G0(R, ∆)]clipped (35)
where the clipping process on a function is here defined as
replacing its value by 1 if it is greater than 1, and leaving
it unchanged, if the value is ≤1. In our calculations, we
work with low enough values of ∆ so that the propagator
G0at all sites other than the origin is less than 1 at time
∆. Then, the propagator for this variation is given by
G0(−→
R , ∆) = G0(−→
R , ∆) + [1 −G0(0,∆)] δ−→
R ,0,(36)
Where we have In Fourier space,
f
G0(−→
k , ∆) = e2∆ Pcos ki+ 1 −Id
0(2∆) (37)
To evolve this system up to time t=n∆, it is im-
mediately noted that the propagator needs to be applied
iteratively and the result takes the form
f
G0(−→
k , n∆) = f
G0(−→
k , ∆)n(38)
Upon taking an inverse Fourier transform and substi-
tuting for −→
R=−→
u t, we get
G(−→
u t, t) = Z−→
dk
(2π)det1
∆log(e2∆ Pcos kj+1−Id
0(2∆))+i−→
k .−→
u
(39)
The above integral can be easily evaluated in the long
time limit using the method of steepest descent giving
iuj−2 sin kje2∆ Pcos ki
e2∆ Pcos kj+ 1 −Id
0(2∆) = 0 (40)
The stationary point occurs at a imaginary value of ~
k=
i~κ which upon substitution gives
uj=2
Γsinh κj(41)
where
Γ = e2∆ Pcosh κj−Id
0(2∆) + 1
e2∆ Pcosh κj(42)
We can eliminate κfrom the above equation. For nota-
tional convenience, we define
A= 2∆
d
X
j=1
cosh κj= 2∆
d
X
j=1 s1 + Γ
2uj2
.(43)
Then Eq.(42) can be rewritten as
ln (1 −Γ) = ln Id
0(2∆) −1−A(44)
p-5
Aanjaneya Kumar and Deepak Dhar
Let A∗(~u) and Γ∗(~u) be the solution to Eqs.(43,44) for
a given ~u. Then the equation of the shape of IBP2cluster
is determined to be
d
X
j=1
ujsinh−1Γ∗(~u)
2uj−2s1 + ujΓ∗(~u)
22
=log Γ∗(~u)
∆(45)
It is clear that if ∆ is close to zero, then the capping is
ineffective as for small ∆, evolution is effectively already
capped. For very large ∆, this method would be ineffective
as only the origin would be capped and the evolution of
the IBP2would be very much like the IBP. It is also easily
verified if we take the ∆ →0 and the ∆ → ∞ limits, we
recover Eq.(11) from Eq.(45). We can choose the optimum
value of ∆ that gives the best bound.
In IBP2, the velocity along axis is evaluated to be
4.1134, 5.3826, 6.4927 and 7.5107 while along the diag-
onal direction, it is 3.9653, 5.0019, 5.8492 and 6.5838 for
d= 2, 3, 4 and 5 respectively.
Summary and discussion. – In this paper, we de-
veloped improved upper bounds to the asymptotic shape
of Eden clusters. We first found the exact equation that
gives the exact asymptotic shape of the IBP cluster. We
found that even in the IBP cluster, a departure from the
circular shape of cluster is seen as pointed out by Alm and
Deijfen for the Eden cluster. Then we improved upon the
bounds by considering two independent modifications to
the IBP - one in which each cell independently gives rise to
daughter cells at neighbouring sites except along the bond
that connects it to its mother cell and the other, in which
we iteratively evolve the system and in each iteration, im-
pose the condition at a non-empty site, no more cells can
be added due to the descendants of the cells present at
that site. Contrary to our initial expectation, we found
that both these modifications improve of the upper bound
only by a few percent, even for small dimensions like 2 to
5.
Of more interest is the structure of the equation that
describes the dependence of the velocity on direction. We
note that the equation that describes the direction depen-
dent velocity in the IBP is eq.(11), which is a condition of
the form d
X
i=1
g(ui) = constant.(46)
Here g(u) is a convex function of its argument, and the
equation has explicitly the permutation symmetry over
directions. For example, if we g(u) was of the form
g(u) = constant|u|a, with 1 < a < 2, we get a set of
shapes that interpolate between d-dimensional sphere and
d-dimensional diamond-shape.
From the explicit form of the function, we see that
g()∼2, for small . This implies that if we look at
the a direction only slightly away from axes, at a small
angle θfrom the axis, we get the velocity in this direction
depends only on θ, and is given by Vaxis,IBP −aθ2, where
ais some constant. Of course, this is shown only for the
IBP, which only gives an upper bound the the Eden cluster
size.
It it an interesting question to see if the true Eden clus-
ter shape is also described by an equation of the form of
Eq.(46), with a different convex function g. The structure
of shape in the IBP2bound is more complicated. That
can only written as a couple of equations of the form
X
i
g(ui,Λ) = constant,X
i
h(ui,Λ) = constant.
where g(u, Λ) and h(u, Λ) are convex functions of the ar-
gument u, and Λ is a parameter, which has to be elim-
inated from the two equations, to generate an equation
only amongst the variables {ui}. Of course, higher order
approximations may give rise to coupled set of equations,
symmetric in {ui}, but with more parameters. This seems
like an interesting question to explore further.
This article is dedicated to Joel Lebowitz for his gentle
mentoring of many generations of scientists as the Chief
Editor of Journal of Statistical Physics for 40 years.
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