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\Large \bf
Computational Physics \\
\large 
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29-January-2001\\
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\noindent 
{\bf Ex.~1b: True Self Avoiding Random Walk in one dimension} 

Consider first a RW in one dimension 
where the walker can go left or right
with equal probability and calculate $\nu$ as in exercise 1a. 
Consider now a self-avoiding walk (SAW) in one dimension as defined in 
Ref.~\cite{Bernasconi} where a 
walker can go to the 
left or right site with a probability that depends on the number of times these 
sites have already been visited. In this case you have to define a lattice of 
$M\sim2N$ sites, right and left from the origin.  
The probability that at time $t+1$ the walker 
will jump to site $i+1$ is given by:
\begin{equation}
P_{i+1}={e^{-gn_{i+1}} \over {e^{-gn_{i+1}}+e^{-gn_{i-1}}}}
\end{equation}
where $n_{i \pm 1}$ is the number of times the site $i \pm 1$ has been visited,
$P_{i-1}=1-P_{i+1}$, and the parameter $g>0$ is a measure of the desire of the 
path to avoid itself. Perform a Monte Carlo simulation of a SAW. Store the 
values of $n$. At each step calculate the probability $P_{i+1}$, draw a random 
number $r$ and perform the move to site $i+1$ if $r\le P_{i+1}$ or to the left 
otherwise. Calculate the mean square displacement $<\Delta x^2(N)>$ 
(where $x$ is the distance from the origin) as a function of $N$ 
and estimate $\nu$. Consider for instance $g=0.1$ and $N=10^3$ and average
over $\sim 10^3$ trials. Compare, comment ....
\begin{thebibliography}*
\bibitem{Bernasconi} J. Bernasconi, L. Pietronero Phys. Rev. {\bf B 29}, 
5196 (1994)
\end{thebibliography}

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