| Summary: | 碩士 === 國立臺灣大學 === 應用力學研究所 === 87 === In this thesis, we briefly review the percolation theory and apply it to random resistor network problem.
We start with reviewing site percolation, bond percolation, percolation quantities and critical exponents. It has been conjectured that there exists a universal scaling law between critical exponents. We examine the scaling law by carrying out numerical simulation and use the finite-size scaling analysis to evaluate the various critical exponents.
To study the problem of random resistor network, we use real-space renormalization to work out the exact critical exponents of 3*3 bond square lattice and 2*2*2 bond simple cubic lattice. Although the exact renormalization solutions are not the exact solutions, we got reasonable approximations. In the case of square lattice, the fixed point of probability is equal to the exact percolation threshold and so we got very good estimates of other critical exponents. In the case of simple cubic lattice, renormalization method offers us a quicker approach of critical exponents. However, the results are not so good because the fixed point is not very close to percolation threshold. If we use the best numerical estimate of percolation threshold as the fixed point, we also can get very good approximations of critical exponents. In addition, we show the numerical solutions of percolation threshold by Monte Carlo Renormalization Method. It is found to be a good method to derive more precise estimate of percolation threshold.
The new contribution of this thesis is the exact renormalization results of 3*3 bond square lattice and 2*2*2 simple cubic lattice. We list all configurations in Appendix A and B. In two-dimensional lattice, we also work out the fixed-point p* and for several small lattices.
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