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RMS displacement

After large number of steps $t$the mean square displacement $\langle R^2 \rangle$of the RW on Percolation behaves as

\begin{displaymath} \langle R^2 \rangle =t^\frac{2}{dew} \end{displaymath}

(1)


Here the exponent $dew$is related to fractal dimension $D$of the cluster and fracton $\tilde{d}$dimension of the walk by

 

\begin{displaymath} d_w=2D/\tilde{d} \end{displaymath}

(2)


Alexander and Orbach conjectured the value of $\tilde{d}$as $4/3$[3]. Various simulations have supported this. Our results for the best estimate can be read by the following figure[]. It was generated for $200\times 200$lattice with $1500$walks and averaged over $5000$runs.

\includegraphics[width=3in,height=2.5in]{rmsdistabovepc2.eps}

Figure 8: log(Mean square displacement)/2 v/s log(number of steps)

 

Our best linear fit for $log\langle R^2 \rangle$gave $d_w=2.307$.

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SHIVAKUMAR JOLAD 2005-11-17