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\begin{center}
\Large \bf
Computational Physics \\
\large 
\vspace{0.2cm}
4-February-2001
\end{center}

\noindent 
{\bf Ex.2 Site percolation on the square lattice}

Consider the model for site percolation for a finite square lattice 
of $N= L \times L$ sites. Construct the lattice for a few values of the 
probability $p$ and examine  the obtained occupancy  obtained for individual 
realizations and as average over many realizations.

Parts d) and e) are facultative.

\begin{itemize}
\item[a)]
Implement the Hoshen Kopelman
algorithm for cluster labelling. Check that it works by comparing with the 
results obtained by visual inspection for N=8 (just print the matrix denoting 
the sites with 0 for empty and 1 for filled sites). 

\item[b)] Consider several values of $p$ in the range $0.3$ to 
$0.8$ for different lattice sizes, $N=8,16,32$. Compute the probability $P_S$
that a spanning cluster occurs as well as the quantity
\begin{equation}
P_\infty(p)={\# \mbox{of sites in the spanning cluster} \over
\mbox{total number of occupied sites}}
\end{equation}
 Comment the behaviour. 

\item[c)]
Study the behaviour of $n_s(p)$ for $L$=16, $p$=0.2,~0.4,~0.5 ($p<p_c$), 
for $p=0.5927$ ($p=p_c$) and for $p$=0.8 ($p>p_c$) where
\begin{equation}
n_s(p)={\mbox{average \# of clusters of size s} \over 
\mbox{total number of lattice sites}}
\end{equation}
%Consider 5 blocks of 5 
%configurations for each value of $p$. 
%Does this choice give a comparable 
%precision at all values of $p$? 
To minimize the effect of the finite lattice, 
discard configurations with spanning clusters for $p<p_c$ and those without
spanning clusters for $p \ge p_c$. Does this procedure affect the results? 
Plot, at each $p$, $n_s$ as a function of 
$s$ and comment the dependence.

\item[d)]
With the same configurations as in c), calculate the mean cluster size
$S(p)$ (exclude the spanning cluster for $p>p_c$) where
\begin{equation}
S(p)={\Sigma_s n_s(p) s^2 \over \Sigma_s n_s(p) s}
\end{equation}

\item[e)]
Analyze the results of $P_S$ to extract the critical value $p_c$ and the quantity $P_\infty(p)$ to evaluate the critical exponent 
$\beta/\nu$ by finite size scaling.

\end{itemize}
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