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%Paper: cond-mat/9408076
%From: Loic Turban <turban@lps.u-nancy.fr>
%Date: Wed, 24 Aug 1994 08:58:02 -0400

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\input preprint.sty \input epsf 
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\def\head{\eightsl Confined clusters in directed percolation}
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    \centerline{\bf Acknowledgments}
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    \noindent}
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\headline{cond-mat/9408076\lpsn{94}{2}}
\title{\centerline{FRACTAL DIMENSIONS OF CONFINED CLUSTERS} 
\centerline{IN TWO--DIMENSIONAL DIRECTED PERCOLATION}} 
\author{Christian Kaiser and Lo\"\i c Turban}
{\sl
\centerline{Laboratoire de Physique du Solide\footnote{\dag}
{\eightsl Laboratoire Associ\'e au CNRS No 155}}
\centerline{ Universit\'e Henri Poincar\'e (Nancy I), BP239}
\centerline{F-54506 Vand\oe uvre l\`es Nancy Cedex, France}} 
\medskip\centerline{\tenit (To appear in \JPA)} 
\vskip6truecm
{\par\begingroup\parindent=0pt\leftskip=1cm
\rightskip=1cm
\baselineskip=12pt
\eightrm\noindent {\eightbf Abstract.}  The fractal structure of directed percolation
clusters, grown at the percolation threshold inside parabolic--like
systems, is studied in two dimensions via Monte--Carlo simulations. With a
free surface at $\sst y=\!\pm Cx^k$ and a dynamical exponent $\sst z$, the
surface shape is a relevant perturbation when $\sst k<1/z$ and the
fractal dimensions of the anisotropic clusters vary continuously with
$\sst k$. Analytic expressions for these variations are obtained using a
blob picture approach.
\par\endgroup\par}
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Since the work of Cardy [1], one knows that a large--scale
alteration of the shape of a system may influence its local critical
behaviour (further references can be found in the review~[2]). 

Cardy studied systems limited by corners or wedges which are scale invariant
shapes for an isotropic system and thus, introduce a marginal perturbation
leading to varying local exponents. More generally, one may consider systems
limited by a parabolic--like surface at $y\!=\!\pm Cx^k$ in the
$(x,y)$--plane in two dimensions, a parabolic cylinder or a paraboloid in
higher dimensions. Under a uniform rescaling of the lengths by a factor $b$,
$C$ is changed into $C'\!=\! b^{k-1}C$. As a consequence, $C$ grows to
infinity for $k\!>\!1$ whereas it goes to zero when $k\!<\!1$. It remains
scale invariant when $k\!=\!1$, which corresponds to the marginal corner
geometry mentioned above. 

Since $1/C$ vanishes for the the semi--infinite system, it can be
considered as a scaling field for the reference fixed point corresponding
to the flat surface. This scaling variable is an irrelevant perturbation
for $k\!>\!1$ and a relevant one for $k\!<\!1$. 

In the relevant case, the shape renormalizes to a narrow system, so that the
local order is weakened. Using a conformal transformation from the
half--space to the interior of the parabola, one can show that the critical
correlation functions decay from the tip along the $x$--axis in a stretched
exponential way~[3]. The order parameter at the tip also displays an
essential singularity as a function of the reduced temperature~[3, 4, 5].
Similar results were obtained for the percolation problem [6]. In the
case of chain polymers (self--avoiding--walks) with one end fixed at the
tip, the exponent associated with the radius of gyration becomes 
anisotropic~[7].

When the unperturbed system is itself anisotropic, i.e. when the
correlation length exponents~$\nu_\parallel$ in the $x$--direction and
$\nu_\perp$ in the transverse direction are different with a dynamical
exponent $z$ given by the ratio $\nu_\parallel/\nu_\perp$, the
appropriate scaling factor becomes $b^z$ in the $x$--direction. Then the
scaling field $1/C$ transforms as 
$$
{1\over C'}=b^{1-zk}\ {1\over C}\eqno(1)
$$
and the critical behaviour is modified when $k\leq 1/z$ instead of $1$. The
directed walk (polymer) with $z\!=\!2$ was studied in [8, 9]. In the
marginal case, which then corresponds to the true parabola, the local
susceptibilities given by the numbers of walks with $N$ steps
starting at the tip and ending either anywhere or on the surface,
diverge with a $C$--dependent power of $N$ as expected.

In the present work, we study the structure of directed percolation
clusters confined inside a "parabola". We consider the site percolation
problem, directed along the diagonal of a square lattice, with the sites
inside or on the "parabola" belonging to the system. 

{\bigskip\par\begingroup\parindent=0pt\leftskip=1cm
\rightskip=1cm\medskip
\baselineskip=12truept\epsfxsize=13.6truecm
\midinsert
\centerline{\epsfbox{f1-clus-para.eps}}
\smallskip
\Figcaption{1}{Structure of finite clusters generated at $\sst p_c$
for: (a) free growth, (b--c) parabolic systems with $\sst C=2,\ 4$ and
$\sst k=1/2$.} \par
\endinsert
\endgroup
\par\bigskip}


Using a standard Monte--Carlo method, $2.10^5$ clusters starting from the
tip at $x\!=\! y\!=\! 0$ were generated at the percolation threshold
$p_c\!=\! 0.705489(4)$ [10] as shown in figure 1. The square radii of
gyration for $s$--site clusters along the two directions
$$
X_s^2={1\over 2s^2}\sum_{i,j=1}^s(x_i-x_j)^2\qquad
Y_s^2={1\over 2s^2}\sum_{i,j=1}^s(y_i-y_j)^2\eqno(2)
$$
were put into bins with respect to the sizes $2^i\!\leq\! s\!<\!2^{i+1}$
($i\!=\!0,1,2,\dots$) and the bin averages $\langle X_s^2\rangle$, 
$\langle Y_s^2\rangle$, were calculated for "finite" clusters, i. e. for
clusters with $x_{\rm max}\!<\! L\!=\!1000$, the system size along the
$x$--direction.

Asymptotically, the following power laws are expected
$$
\overline{s}\sim\overline{X_s}^{\,d_\parallel(k)}
\sim\overline{Y_s}^{\,d_\perp(k)}\eqno(3)
$$
where $\overline{s}$ is the center of the bin, 
$\overline{X_s}\!=\!\langle X_s^2\rangle^{1/2}$, 
$\overline{Y_s}\!=\!\langle Y_s^2\rangle^{1/2}$ and ${d_\parallel}(k)$,
${d_\perp}(k)$, are the fractal dimensions in the two directions. In a
log--log plot, a linear behaviour is obtained for
$2^6\!<\overline{s}\!<\!2^{12}$--$\,2^{13}$, the deviations at larger sizes
being mainly due to the finite size of the system. There, the contribution
from longer clusters with the same values of $\overline{s}$ is not taken
into account. The fractal dimensions, shown in figure 2 versus $k$ for
different values of $C$, were obtained from least--square fits of the data
in the intermediate linear regime.

{\bigskip\par\begingroup\parindent=0pt\leftskip=1cm
\rightskip=1cm\medskip
\baselineskip=12truept\epsfxsize=10truecm
\midinsert
\centerline{\epsfbox{f2-clus-para.eps}}
\smallskip
\Figcaption{2}{Fractal dimensions $\sst d_\parallel(k)$,
$\sst d_\perp(k)$ and $\sst d_f(k)$ versus the system--shape exponent
$\sst k$. The lines give the analytic results obtained via a blob picture
approach. The fractal dimensions are varying when the perturbation is
relevant $\sst (k<1/z=0.633(2))$.}
\par
\endinsert
\endgroup
\par\bigskip}
 
In order to analyse our results, let us first consider unconfined
directed percolation [11]. In two dimensions, the best estimates for the
critical exponents, obtained through series expansions, are [10]:
$$
\nu_\perp=1.097(2)\quad\nu_\parallel=1.734(2)\quad\gamma=2.278(2)
\eqno(4)
$$
giving
$$
z=1.581(3)\qquad\beta=0.276(3)
\eqno(5)
$$
from scaling. Extending an argument of Stauffer for ordinary 
percolation~[12], one obtains the following relations for the fractal
dimensions of critical directed percolation clusters:
$$
{d_\parallel}={\beta+\gamma\over
\nu_\parallel}= 1.473(2)\qquad{d_\perp}={\beta+\gamma\over\nu_\perp}=2.329(3).
\eqno(6)
$$
The numerical value of ${d_\perp}\!>\! d$ may be at first sight surprising
for a fractal object. Actually, it can be traced to the anisotropy. Let
${l_\parallel}$ and ${l_\perp}$ be the length and width of a $s$--site
cluster with $s\sim{l_\parallel}^{d_\parallel}\sim{l_\perp}^{d_\perp}$. It
follows that $$
{l_\parallel}={l_\perp}^{\,{d_\perp}/{d_\parallel}}={l_\perp}^z\eqno(7)
$$
grows faster with $s$ than ${l_\perp}$. $s$ itself grows faster with
${l_\perp}$ than an isotropic massive object due to the parallel
growth. Introducing the characteristic length associated with the surface
of the cluster 
$$
l\sim({l_\parallel}{l_\perp})^{1/2}\sim l^{d_f}
\eqno(8)
$$
a single fractal dimension $d_f$ can be defined with [11]
$$
{1\over d_f}={1\over 2}\left({1\over{d_\parallel}}+{1\over{d_\perp}}\right)\eqno(9)
$$
and now, $d_f\!=\!1.805(2)\!<\!2$. 

{\bigskip\par\begingroup\parindent=0pt\leftskip=1cm
\rightskip=1cm\medskip
\baselineskip=12truept\epsfxsize=10truecm
\midinsert
\centerline{\epsfbox{f3-clus-para.eps}}
\smallskip
\Figcaption{3}{In the blob picture approach, confined clusters are built
through piling up successive anisotropic blobs, with a transverse size
$\sst l_{n\perp}$ fixed by the geometry of the system. The correlations
inside each blob are the same as for unconfined clusters.}
\par
\endinsert
\endgroup
\par\bigskip}

When the perturbation introduced by the surface is relevant, the structure
of confined clusters can be studied studied using a blob picture approach
[13, 14, 7]. Inside the "parabola", the cluster configuration results from
the piling up of anisotropic blobs as shown in figure 3. Within each blob,
the correlations are the same as for unconfined clusters. The $n$th~blob
at~$x_n$ has a width~${l_{n\perp}}\!=\!Cx_n^k$ which is fixed by the
geometry of the system. Using equation~(7), its length is 
$$
{l_{n\parallel}}=C^zx_n^{zk}\sim C^un^v\eqno(10)
$$
where the last expression corresponds to a continuum approximation in the
blob index~$n$. {}From $x_n\!-\!~x_{n-1}\!=\!{l_{n\parallel}}$, one deduces the
following differential equation in the continuum limit,
$$
{\d x_n\over\d n}\sim C^un^v\sim C^{(u-z)/zk}n^{-1+v/zk}\eqno(11)
$$
where the last term follows from the $n$--dependance of $x_n$ in~(10).
Identifying powers of~$C$ and~$n$, one obtains $u\!=\!~z/(1-zk)$ and
$v\!=\! ku$. These exponents are positive for a relevant perturbation
only. Otherwise, when $k\!>\!1/z$, the cluster grows freely, i. e. there
is a single blob. Let $s_n\!\sim\!({l_{n\parallel}})^{d_\parallel}$
be the number of sites inside the $n$th blob. For the whole cluster, we
have: $$ s=\sum_{n=1}^Ns_n\sim C^{z{d_\parallel}/(1-zk)}\!\int_1^N\!\!\d n\ 
n^{zk{d_\parallel}/(1-zk)}\sim C^{z{d_\parallel}/(1-zk)}\  N^{[1+zk({d_\parallel}-1)]/(1-zk)}.
\eqno(12)
$$
In the same way, the cluster length is given by
$$
{l_\parallel}=\sum_{n=1}^N{l_{n\parallel}}\sim C^{z/(1-zk)}\!\int_1^N\!\!\d n\  n^{zk/(1-zk)}\sim
C^{z/(1-zk)}\  N^{1/(1-zk)}.\eqno(13)
$$
Eliminating $N$ between (12) and (13), one finally obtains:
$$
s\sim C^{z({d_\parallel}-1)}\  {l_\parallel}^{d_\parallel(k)}\qquad
d_\parallel(k)=1+zk({d_\parallel}-1)\qquad0\leq k\leq 1/z
\eqno(14) 
$$ 
which, together with
${l_\perp}\!=\! C{l_\parallel}^k$, also gives: 
$$
s\sim C^{-1/k}\  {l_\perp}^{d_\perp(k)}\qquad
d_\perp(k)={d_\parallel(k)\over k}\qquad0<k\leq1/z .
\eqno(15) 
$$

The variations of the fractal dimensions are shown in figure~2. For a
relevant perturbation, $d_\parallel(k)$ and $d_\perp(k)$ vary continuously
with $k$ until they reach their unperturbed values for $k\!=\!1/z$. As
a result of finite--size effects, the  Monte--Carlo values deviate
from the analytical ones when $C$ increases. For large $C$ values the
system is more open and the number of blobs inside it remains small for a
finite system. As a consequence, the Monte--Carlo estimates for the fractal
dimensions are then closer to the unperturbed ones. 

Results for the tip percolation probability will be discussed elsewhere.

\ack
C. K. thanks Henri Poincar\'e University for hospitality. This work was
supported by CNIMAT under project No 155C93b. 
\references
\numrefjl{[1]}{Cardy J L 1983}{\JPA}{16}{3617}
\numrefjl{[2]}{Igl\'oi F, Peschel I and Turban L 1993}{Adv. Phys.}{42}{683}
\numrefjl{[3]}{Peschel I, Turban L and Igl\'oi F 1991}{\JPA}{24}{L1229}
\numrefjl{[4]}{Davies B and Peschel I 1992}{Ann. Phys. (Leipzig)}{2}{79}
\numrefjl{[5]}{Blawid S and Peschel I 1994}{\ZP\ \rm B}{95}{73}
\numrefjl{[6]}{Berche B, Debierre J M and Eckle P}{1994}{}{Nancy I
preprint}  
\numrefjl{[7]}{Turban L and Berche B 1993}{\JP\  I}{3}{925}
\numrefjl{[8]}{Turban L 1992}{\JPA}{25}{L127} 
\numrefjl{[9]}{Igl\'oi F 1992}{\PR\ \rm B}{45}{7024} 
\numrefjl{[10]}{Essam J W, Guttmann A J and De'Bell K 1988}{\JPA}{21}{3815}
\numrefjl{[11]}{Kinzel W 1983}{Ann. Israel Phys. Soc.}{5}{425}
\numrefbk{[12]}{Stauffer D 1985}{Introduction to Percolation
Theory}{(London: Taylor \& Francis) p 64}
\numrefjl{[13]}{Pincus P 1976}{Macromolecules}{9}{386}
\numrefbk{[14]}{de Gennes P G 1979}{Scaling Concepts in Polymer
Physics}{(Ithaca: Cornell University Press) p 50}
\vfill\eject\bye
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end

