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\def\jpa #1 #2 #3 {{\sl J. Phys.\ A} {\bf #1}, #2 (#3)}
\def\jcp #1 #2 #3 {{\sl J.\ Chem.\ Phys.} {\bf #1}, #2 (#3)}
\def\jpc #1 #2 #3 {{\sl J.\ Phys.\ Chem.} {\bf #1}, #2 (#3)}
\def\jsp #1 #2 #3 {{\sl J.\ Stat.\ Phys.} {\bf #1}, #2 (#3)}
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\def\pra #1 #2 #3 {{\sl Phys.\ Rev.\ A} {\bf #1}, #2 (#3)}
\def\prb #1 #2 #3 {{\sl Phys.\ Rev.\ B} {\bf #1}, #2 (#3)}
\def\pre #1 #2 #3 {{\sl Phys.\ Rev.\ E} {\bf #1}, #2 (#3)}
\def\prl #1 #2 #3 {{\sl Phys.\ Rev.\ Lett.} {\bf #1}, #2 (#3)}
\def\prsl #1 #2 #3 {{\sl Proc.\ Roy.\ Soc.\ London Ser. A} {\bf #1}, #2 (#3)}
\def\rmp #1 #2 #3 {{\sl Rev.\ Mod.\ Phys.} {\bf #1}, #2 (#3)}
\def\zpc #1 #2 #3 {{\sl Z. Phys.\ Chem.} {\bf #1}, #2 (#3)}
\def\zw #1 #2 #3 {{\sl Z. Wahrsch.\ verw.\ Gebiete} {\bf #1}, #2 (#3)}

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\def\gtwid{\mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\ltwid{\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}}}
\def\eg{{\it e.\ g.}}\def\ie{{\it i.\ e.}}\def\etc{{\it etc.}}
\def\vs{{\it vs.}}\def\vv{{\it vice versa}}\def\ea{{\it et al.}}
\def\apr{{\it a priori}}\def\apo{{\it a posteriori}}
\def\pd#1#2{{\partial #1\over\partial #2}}      %partial derivative
\def\p2d#1#2{{\partial^2 #1\over\partial #2^2}} %second partial derivative
\def\td#1#2{{d #1\over d #2}}      %total derivative
\def\t2d#1#2{{d^2 #1\over d #2^2}} %second total derivative
\def\ith{{$i^{\rm th}$}}
\def\jth{{$j^{\rm th}$}}
\def\kth{{$k^{\rm th}$}}
\def\nth{{$n^{\rm th}$}}
\def\dth{{$d^{\rm th}$}}
\def\eg{{\it e.\ g.}}\def\ie{{\it i.\ e.}}\def\etc{{\it etc.}}
\def\vs{{\it vs.}}\def\vv{{\it vice versa}}\def\ea{{\it et al.}}
\def\apr{{\it a priori}}\def\apo{{\it a posteriori}}


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\def\df{D_f}
\def\pxt{P(x,t)}
\def\pxi{P(x,0)}
\def\pyt{P(y,t)}
\def\mst{M(s,t)}
\def\pvt{P(V,t)}
\def\pxxt{P({\bf x},t)}
\def\pyyt{P({\bf y},t)}
\def\msst{M({\bf s},t)}
\def\mssf{M({\bf s}+{\bf 2})}
\def\fiz{\Phi(z)}
\def\Fz{\Phi_d(z)}
\def\fxt{\Phi\left(x \sqrt{t}\right)}
\def\fVt{\Phi_d\left(V \sqrt{t}\right)}
\def\vav{\langle V\rangle}
\def\lav{\langle l\rangle}
\def\xav{\langle x\rangle}
\def\inx{{\int \limits_x^1dy}}
\def\ini{{\int \limits_0^1}}
\def\inmo{{\int\limits_0^1\ldots\int\limits_0^1\prod_{j=1}^d dx_j x_j^{s_j-1}}}
\def\inxx{{\int\limits_{x_1}^1\ldots\int\limits_{x_d}^1}}
\def\a{\alpha}
\def\as{\alpha({\bf s})}
\def\b{\beta}
\def\bd{\beta_d}
\def\d{\delta}
\def\G{\Gamma}
\def\gd{\gamma_d}
\def\md{\mu_d}

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\documentstyle[graphicx,aps,epsf,multicol]{revtex}
\begin{document}
\title{Multiscaling in Stochastic Fractals}
\author{P.~L.~Krapivsky$^1$ and E.~Ben-Naim$^2$}
\address{$^1$Center for Polymer Studies and Department of Physics, 
 Boston University, Boston, MA 02215, USA}
\address{$^2$The James Franck Institute, The University of Chicago, 
Chicago, IL 60637, USA}
\maketitle
\begin{abstract}
  We introduce a simple kinetic model describing the formation of a
  stochastic Cantor set in arbitrary spatial dimension $d$.  In one
  dimension, the model exhibits scaling asymptotic behavior.  For
  $d>1$, the volume distribution is characterized by a single scale
  $t^{-1/2}$, while other geometric properties such as the length are
  characterized by an infinite number of length scales and thus
  exhibit multiscaling.
\end{abstract}
%
%{\noindent  P. A. C. S. Numbers: 02.50.-r, 05.40.+j, 82.20.-w}
%\bigskip\noindent
%{\bf Keywords}: scaling, multiscaling, stochastic fractals, Cantor set

%\vfil\eject
\begin{multicols}{2}
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The notion of a fractal has been widely used to describe self-similar
structures [1]. The simplest way to construct a fractal is to repeat
a given operation over and over again. The classical example of such
a repetitive consruction is the Cantor's ``middle-third erasing'' set [1]. 
Recall the definition of this set: One divides an interval 
into three equal intervals and then removes the middle interval; 
on the next step, one repeats the same procedure with the two remaining 
intervals; \etc. The outcome of this process is a counterintuitive
uncountable set having a measure (``length'') zero. The Cantor set turns 
out to be a perfect fractal of dimension $\df=\ln(2)/\ln(3)\cong 0.63093$. 

The Cantor set is a {\it regular} fractal. In contrast, self-similar
structures arising in nature are usually random. Moreover,
fractals are usually formed by {\it continuous} kinetic processes
while the classical repetitive constructions are discrete in time.
In the present letter we introduce a stochastic process 
which may be considered as a natural kinetic counterpart to the
original Cantor construction. The resulting set turns out to be a
random fractal of dimension $\df=(\sqrt{17}-3)/2 \cong 0.56155$.  
We also investigate $d$-dimensional random Cantor sets and find
that several geometric characteristics such as the average length,
surface area, \etc, are characterized by different scales. 
In the following, the existence of multipole kinetic exponents 
characterizing the process, will be shortly called multiscaling.

In one dimension, our model can be defined as follows.  Starting with
the unit interval [0:1], cracks are deposited uniformly on the unit
interval with unit rate. When two cracks apear on the initial
interval, the middle is removed immediately and two new intervals are
formed.  The process continues independently for the surviving
intervals such that whenever a surviving interval contains two cracks,
the middle interval is removed.  In the classical Cantor
process, after $n$ stages we are left with $2^n$ intervals of length
$3^{-n}$. In the stochastic process the number of intervals and their
lengths at time $t$ are in principle arbitrary. The
distribution function $\pxt$ describing intervals of length $x$ at
time $t$ satisfies the following linear evolution equation,
\begin{equation}
\pd \pxt t = -{x^2\over 2}\pxt + 2\inx (y-x)\pyt,
\end{equation}
with the initial conditions $\pxi=\delta(x-1)$. The loss term on 
the right-hand side represents the decrease of intervals of 
length $x$, $x$-mers, due to the division process. Each division event 
consists of choosing two points at random and thus, the overall 
breakage rate is quadratic in the interval length, while the factor 
$1/2$ arises since the two points are indistinguishable. 
The gain term represents the increase of $x$-intervals due to 
breakups of longer intervals.

\begin{figure}
\narrowtext
%\centerline{\epsfxsize=8cm\epsfbox{fig1.ps}}
\centerline{\includegraphics[width=8cm]{fig1}}
\vspace{0.1in}
\caption{Fig. 1 Illustration of the process in two dimension.}
\end{figure}



The formation of the random Cantor set is equivalent to 
random sequential parking on a line with a uniform 
distribution of lengths of parking intervals. 
While the latter problem has been investigated in Ref.~[2], 
we present an alternative solution method [3] that can be easily 
generalized to higher dimensions. This method focuses on the 
leading asymptotic behavior of the moments of the length distribution
$M(s,t)$, defined by 
\begin{equation}
M(s,t)=\ini \pxt x^{s-1}dx.
\end{equation}

The rate equation (1) yield the following kinetic equation for the moments, 
\begin{equation}
\pd \mst t = \left[-{1\over 2}+{2\over s(s+1)}\right]M(s+2,t). 
\end{equation}
Asymptotically, the moments exhibit the power-law behavior
\begin{equation}
M(s,t) \simeq A(s)t^{-\a(s)}.
\end{equation}
By inserting the anticipated power-law behavior into Eq.~(3)
and solving the resulting difference equations one gets
\begin{equation}
\a(s)={s-\b\over 2}, \quad  A(s)={\G(s)\G\left(\b+{1\over 2}\right)
\over \G\left({s+\b+1\over 2}\right)\G(\b)}2^{(\b-s)/2}.
\end{equation}
In the above equation we have introduced a shorthand notation 
$\beta=(\sqrt{17}-1)/2$.

Eq. (4) implies that in the long-time limit $\pxt$ approaches
the scaling form,
\begin{equation}
\pxt \simeq t^{\b/2}\fxt,
\end{equation}
with the scaling function $\fiz$ being the inverse Mellin transform
of $A(s)$. In the limit of small $z, \fiz$ approaches a constant while
in the large-$z$ limit, $\fiz \sim z^{-\b}\exp(-z^2/2)$.
 

Furthermore, the total number of intervals $N(t)$, $N(t)\equiv
M(1,t)$, grows as $t^{(\b-1)/2}$, while the typical interval size
$\xav$, $\xav=M(2,t)/N(t)$, decays as $t^{-1/2}$.  This simple scaling
relation follows directly from the rate equations, since the loss rate
is quadratic in the interval length. However, it is interesting that
the simple rate equation (1) leads to non-trivial asymptotic
exponents. 


Knowledge of the asymptotic behavior of the average length and the
average number enables calculation of the fractal dimension.  Since $N
\sim \xav^{-(\b-1)}$, the fractal dimension of the stochastic Cantor set
is given by 
\begin{equation} 
\df=\b-1=(\sqrt{17}-3)/2 \cong 0.56155.   
\end{equation} 
The
above dimension is smaller than the fractal dimension of the classic
Cantor set, $\df=\ln2/\ln3 \cong 0.63093$.


We turn now to the general $d$-dimensional version of the model.
In two dimensions, the model describes the formation of the stochastic 
Cantor gasket. The governing rule of the model is sketched 
in Fig.~1 for the two-dimensional situation. Denote by $\pxxt, 
{\bf x}=(x_1,\ldots,x_d)$, the distribution function for 
(hyper)rectangles of size $x_1 \times \ldots \times x_d$. 
The rate equation governing $\pxxt$, is given by a  
straightforward generalization of Eq.~(1), 
\begin{eqnarray}
\pd \pxxt t&= &{\pxxt \over 2^d}\prod_{j=1}^dx_j^2+ \\
&&\left(3^d-1\right)\inxx \pyyt \prod_{j=1}^d (y_j-x_j)dy_j.\nonumber
\end{eqnarray}
Similarly, the moments of the distribution function $\pxxt$,
\begin{equation}
\msst = \int\limits_0^1\ldots\int\limits_0^1 \pxxt
\prod_{j=1}^d  x_j^{s_j-1}dx_j,
\end{equation}
satisfy the kinetic equation
\begin{equation}
\pd \msst t = \left[-{1 \over 2^d}
+\left(3^d-1\right)\prod_{j=1}^d {1\over s_j(s_j+1)}
\right]M({\bf s}+{\bf 2},t),
\end{equation}
where ${\bf s}=(s_1,\ldots,s_d)$ and ${\bf 2}=(2,\ldots,2)$.


A surprising feature of Eq.~(10) is that it implies the existence 
of an infinite number of conservation laws: on the hypersurface
$\prod_{j=1}^d s_j(s_j+1)=2^d\left(3^d-1\right)$, the moments 
$\msst$ are independent of time. Thus the competition between 
creation and destruction of the (hyper)rectangles gives birth 
to an infinite number of integrals of motion. Similar hidden 
conserved integrals have been found in recent studies of 
multidimensional fragmentation [3,4]. In contrast to the 
fragmentation problem where at least one integral - the total
volume - is an obvious conserved quantity, in the present
model we could not physically explain the appearance of any
conserved integral in any dimension.


These integrals play an important role in the dynamics of the system, 
$\eg$, they are  responsible for the absence of scaling solutions 
to Eq.~(8). Indeed, trying a scaling solution of the form 
$\pxxt=t^wQ(t^z{\bf x})$, one derives infinitely many scaling 
relations, $w=z\sum s_j$, which should be valid for {\it all} 
points ${\bf s}$ on the hypersurface. This infinite set of 
scaling relations cannot be satisfied by just two scaling exponents, 
$w$ and $z$.


In analogy with one-dimensional case, one can expect a power-law 
behavior of the moments in the general $d$-dimensional situation: 
$\msst \sim t^{-\alpha({\bf s})}$ as $t \to \infty$.
Substituting this asymptotic form into Eq.~(10)  we obtain the 
difference equation for the exponent $\alpha({\bf s})$,
\begin{equation}
\alpha({\bf s}) + 1=\alpha({\bf s}+{\bf 2}).
\end{equation}
In addition, on the hypersurface 
$\prod_{1 \le j \le d} s_j(s_j+1)=2^d\left(3^d-1\right)$, 
one has $\alpha({\bf s})=0$. 
The solution to Eq.~(11) with this boundary condition is given
by the formal expression
\begin{equation}
\alpha({\bf s}) \equiv \alpha({\bf s^*}+ k {\bf 2}) = k,
\end{equation}
where the point ${\bf s}^*$ lies on the hypersurface. 
Geometrically, the exponent $\alpha({\bf s})$ gives a 
(normalized) distance from the point ${\bf s}$ to the 
hypersurface in the ${\bf 1}=(1,\ldots,1)$ direction.


For ordinary scaling distributions the exponent $\alpha({\bf s})$
should be linear in the variable $\sum s_j$. This property is
equivalent to the existence of a single length scale in the system.
However, in our stochastic process the exponent $\alpha({\bf s})$ is a
function of {\it all} of its variables.  This manifests the
non-trivial scaling properties of the process.  On the other hand, 
since all the moments still show a power-law behavior we conclude that
the model exhibits a multiscaling asymptotic behavior.


As a manifestation of the existence of multiple length scales
in the system let us consider the ratio of the average volume
$\vav, \vav=M({\bf 2},t)/N(t)$, to the \dth power of the 
average length $\lav, \lav=M(2,1,\ldots,1,t)/N(t)$. For scaling 
distributions characterized by a single scale, the ratio $\vav/\lav^d$ 
{\it cannot depend} on time. Compute now this ratio for the 
present system. First using Eq.~(12) we derive the asymptotic
behavior of the total number of (hyper)rectangles 
$N(t), N(t) \equiv M({\bf 1},t)$:
\begin{equation}
N(t) \sim t^{(\bd-1)/2}, \quad 
\bd={\sqrt{1+24(1-3^{-d})^{1/d}}-1 \over 2}.
\end{equation}
In the above equation $\beta_d$ is the larger root of the 
equation $[\beta_d(\beta_d+1)]^d=2^d(3^d-1)$. 
Analogously, one  finds that $M({\bf 2},t) \sim t^{(\bd-2)/2}$
and that $M(2,1,\ldots,1,t)\sim t^{(\gd-1)/2}$ with $\gd$ being
the largest positive root of algebraic equation
$(\gd+1)(\gd+2)[\gd(\gd+1)]^{d-1}= 2^d\left(3^d-1\right)$.
Finally we obtain
\begin{equation}
{\vav \over \lav^d} \sim t^{-\md}, \quad
\md={1+d(\gd-\bd) \over 2},
\end{equation} 
indicating that the ratio {\it depends} on time. Computing the 
exponent $\md$ yields $\md$=0, 0.05534, 0.06566, and 0.07054 
for $d$=1,2,3, and 4, respectively. 


Note that in the limit of infinite dimension the exponent saturates at
$\mu_{\infty}={1 \over 2}-{3 \over 5}\ln 2 \cong 0.0841117$.  Since
the exponent $\md$ measures the deviation between the asymptotic
behavior of the length and the volume we conclude that this
discrepancy becomes more pronounced as the spatial dimension 
increases.  It is also easy to find that different directions behave
independently in the limit of infinite dimension. Mathematically, it
follows from the relation 
\begin{equation}
\langle\prod_{j=1}^{\infty}x_j^{n_j}\rangle=
\prod_{j=1}^{\infty}\langle x_j^{n_j}\rangle,  
\end{equation}
 which is valid
if $n_j=0$ for all indices $j$ except a finite number.  Thus the
average of a finite product decouples onto a product of
single-variable averages. However, further decoupling is impossible:
\eg, $\langle x^2\rangle \ne \langle x\rangle^2$.  The general
asymptotic formula for the \nth moment in the limit $d \gg 1$ reads:
\begin{equation} 
\langle l^n \rangle \sim t^{-\nu_n}, \quad \nu_n={3\over 5d}
\ln\left[\left(1+{n\over 2}\right)\left(1+{n\over 3}\right)\right].
 \end{equation} This equation again indicates the presence of an infinite
number of length scales.


In principle, it is possible to write down an explicit solution
to Eq.~(10) in terms of the generalized hypergeometric function
$_{2d}F_{2d}$ and then obtain a formal expression for the
distribution function $\pxxt$ by performing the inverse $d$-fold
Mellin transform. However, such a complete solution is very
cumbersome even in one dimension [2]. 
Thus, we restrict ourselves to the {\it volume} 
distribution function $\pvt$,
\begin{equation}
\pvt = \int\limits_0^1\ldots\int\limits_0^1 \pxxt
\d\left(\prod_{j=1}^d x_j - V\right) \prod_{j=1}^d dx_j,
\end{equation}
for which we can derive more complete results. The moments of the
volume distribution function $\pvt$ are just the diagonal
moments $M(s,\ldots,s;t)$. Asymptotically, they exhibit 
the power-law behavior
\begin{equation}
M(s,\ldots,s;t) \simeq A_d(s)t^{(\bd-s)/2},
\end{equation}
with prefactor $A_d(s)$ satisfying the difference equation
\begin{equation}
A_d(s+2)=A_d(s)2^{d-1}(s-\bd)\left[1-{2^d(3^d-1)\over s^d(s+1)^d}\right]^{-1},
\end{equation}
and the boundary condition $A_d(s=\bd)=1$.


Eq.~(18) indicates that in the long-time limit the volume
distribution function approaches the scaling form
\begin{equation}
\pvt \simeq t^{\bd/2}\fVt,
\end{equation}
with $\Fz$ being the inverse Mellin transform of $A_d(s)$.
The most interesting asymptotic behavior of $\Fz$ may be found
from the corresponding asymptotics of $A_d(s)$. By solving 
Eq.~(19) asymptotically we obtain 
\begin{equation}
A_d(s) \sim \cases {s^{-\bd/2}2^{ds/2}
                   \G\left({s\over 2}\right),   &for $s \to \infty$,\cr 
                   s^{-d}, &for $s \to 0$.  \cr}  
\end{equation} 
By performing the inverse Mellin transform we find the following 
limiting behaviors of the scaling function $\Fz$: 
\begin{equation} \Fz \sim \cases {z^{-\bd}\exp(-z^2/2^d),
&for $z \to \infty$,\cr
                 \ln^{d-1}(1/z),           &for $z \to 0$.     \cr}
\end{equation}
Hence for all $d>1$ the volume distribution 
function diverges logarithmically in the small-volume limit. 



Compare now the asymptotic behavior (13) for the number of 
(hyper)rectangles with $\vav \sim t^{-1/2}$ for their average volume. 
These asymptotics imply $N \sim \vav^{-(\bd-1)}$ providing 
the following value of the fractal dimension of the
$d$-dimensional stochastic Cantor set:
\begin{equation}
\df=d(\bd-1)=
d{\sqrt{1+24(1-3^{-d})^{1/d}}-3\over 2}.
\end{equation}
Hence $\df \cong 1.860804, 2.954859, 3.985106$ for $d$=2, 3, 4,
respectively. For comparison: the regular Cantor set
has dimension $\df=\ln(3^d-1)/\ln 3$, \ie,
$\df \cong 1.89279, 2.96565, 3.98869$ for $d$=2, 3, 4,
respectively. It is easy to check that the fractal
dimension of the stochastic Cantor set is always smaller 
than the corresponding value for regular set.


Formally, the moments provide a complete analytical description of the
division process. However, the snapshot of the system at the later
stages remains intriguing. Figure 2 represents a single realization of
the process at time $t=1000$.  A simple Monte Carlo algorithm was chosen 
where pairs of points are deposited on the unit square with unit rate. 
If both points belong to the same rectangle, then that rectangle is
divided into 9 rectangles as illustrated in Figure 1.  Finally, the
central rectangle is removed from the system. In Figure 2 the existing
rectangles are shaded.  This unexpectedly rich pattern arising in such
a simple kinetic process can be viewed as a consequence of the fact
that the process is not fully self-similar.  Instead, the pattern is
formed of sets of different scales which are spatially interwoven.
Figure 2 also shows that a number of removed 
rectangles have large aspect ratio.
This qualitative observation is in a good agreement with a power-law
behavior of the moments of aspect ratio, $\langle(x_1/x_2)^n\rangle
\sim t^{\lambda_n}$ with $\lambda_n=-\alpha(1+n,1-n)-(\b_2-1)/2$. One
can check that $\lambda_n \ge 0$ and hence the aspect ratio grows with
time.


In conclusion, we have investigated a kinetic process describing the
formation of stochastic counterparts to the Cantor set ($d=1$),
Cantor gasket ($d=2$), Cantor cheese ($d=3$), \etc. 
In the long-time limit, the volume of these stochastic Cantor sets 
is characterized by the single scale $t^{-1/2}$. 
The volume distribution function exhibits a weak logarithmic 
singularity near the origin. Similar logarithmic singularity 
has been observed recently in multidimensional fragmentation [3]. 
We have found that other geometrical characteristics such as the
average length, the surface area, $\etc$, decay nonuniversally 
in time because of the existence of an infinite amount 
of different scales, $\ie$, due to multiscaling. 
We have shown that the intrinsic reason for multiscaling 
is the existence of infinitely many hidden conservation laws. 
This feature again resembles multidimensional fragmentation 
prosesses [3,4]. Finally, we have found that the fractal dimension 
characterizing stochastic Cantor set in $d$ dimensions is always 
smaller than its classic counterpart.



P.L.K. was supported by ARO grant \#DAAH04-93-G-0021 
and NSF grant \#DMR-9219845. E.B. was supported by the MRSEC, 
NSF grant\#DMR-9400379, and by NSF grant \#92-08527. 
This support is greatly appreciated. 







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{thebibliography}{99}

\bibitem{} B.~B.~Mandelbrot, {\sl The Fractal Geometry of Nature}
      (W.~H.~Freeman: San-Francisco, 1982).
\bibitem{} P.~L.~Krapivsky, {\sl J. Stat. Phys.} {\bf 69} (1992) 125.
\bibitem{} P.~L.~Krapivsky and E.~Ben-Naim, Scaling and multiscaling in 
      models of fragmentation, {\sl Phys. Rev. E} (1994) to be published.
\bibitem{} D.~Boyer, G.~Tarjus, and P.~Viot, Fluctuation-induced shattering
      transition in a multivariable fragmentation model,  
      {\sl Phys. Rev. Lett.} (1994) to be published.

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\centerline{\bf Figure Captions}

{Figure 2.}  Realization of the process on a unit square at time $t=1000$.  

\end{multicols}
\end{document}

