Sang Bub Lee

Exact enumeration study of self-avoiding walks on two-dimensional percolation clusters

Quenched disorder averages for the number and size of the self-avoiding walks on two-dimensional percolation clusters very near pc are calculated by exact enumeration of all walks which start at the seed site from which clusters have been generated by Monte Carlo simulation. The authors results are in good agreement with previous work, which generated the walks by Monte Carlo simulation (rather than by enumeration) on the incipient infinite cluster.

True self-avoiding walks on fractal lattices above the upper marginal dimension

We study by Monte Carlo simulations the critical behaviour and the cross-over scaling of the true self-avoiding walks (TSAW) on fractal lattices above the upper marginal dimension. We estimate the Flory exponent nu which characterizes the RMS end-to-end distances of SAW on a percolation cluster at percolation thresholds both in three and four dimensions and on a DLA cluster in three dimensions. Results were in good agreement with the predictions of the known Flory formulae. We also discuss the fractal-to-Euclidean and the RW-to-TSAW cross-over scaling. Monte Carlo data appear to collapse in the scaling regions for both cases; however, we found that the scaling function for the latter is different from that on the regular lattices.

Finite size analysis of eigenvalue spectrum for random walks on a critical percolation cluster in four dimensions

We study by Markov chain analysis the random walks on a critical percolation cluster embedded in a four-dimensional hypercubic lattice. We calculate the number of dominant eigenvalues of the transition probability matrix and estimate the spectral and fractal dimensions ds and dw of random walks from the eigenvalues and their distribution. The estimates of ds and dw obtained from the data for a given size S of the percolation cluster exhibit some S dependence. Extrapolating the results to S limit, we obtain ds = 1.330±0.010 close to the previous result by other methods and a new result dw = 4.50±0.15. These values are also confirmed by direct Monte Carlo simulations of random walks on a percolation cluster.

Markov matrix analysis of random walks in disordered continuous media

We study by a Markov matrix analysis of the equivalent random walks the dynamic properties of continuous media consisting of both correlated and uncorrelated equal-size spheres. We employ a blind ant random-walk model using the rule that a walker jumps among centers of the directly connected spherical particles on an infinite network. The dominant eigenvalues and eigenvectors of the transition probability matrix of the random walks are calculated, yielding estimates of the spectral dimension ds and the fractal dimension dw of random walks on the continuous network. We find that, for the present model, the estimates are very close to the corresponding lattice percolation values, though only after the finite-size effects have been carefully taken into account. We also show that the finite-size scaling of the largest nontrivial eigenvalues holds for our model with the same exponents as for the lattice percolation.

True self-avoiding walks on a percolation cluster

The authors investigate by Monte Carlo simulations the critical behaviour of true self-avoiding walks (TSAWs) on a percolation cluster performed very close to the percolation threshold. Specifically they generate TSAWs on a site-percolated incipient infinite cluster, for various values of the self-avoidance parameter g)0. They found that such walks exhibit critical behaviour different from that of ordinary-self-avoiding walks and also from that of random walks of no constraint. The flory exponent obtained was v=0432+or-0.005 for all g)0, which agrees well with the Flory-type formula suggested by Rammal (1984).

Cell renormalisation study of biased self-avoiding walks

The authors present a cell renormalisation approach for the biased self-avoiding walks and show that a stiff-to-isotropic crossover exponent is exactly one in all dimensions and for all cell sizes. They also show, by use of renormalisation flow diagrams, a substantial difference between two and three dimensions in the crossover from stiff limit to isotropic limit as the length of walk N to infinity for fixed gauche weight p. In three dimensions, a crossover seems to occur first to the random walk limit and then to the self-avoiding walk limit, while, in two dimensions, it seems to occur directly to the self-avoiding walk limit in agreement with recent observations based on Monte Carlo simulations.