We present new results of Monte Carlo simulations for self-avoiding walks on randomly diluted square and simple-cubic lattices performed very close to the percolation thresholds. Our results indicate the asymptotic behavior of the walk dimension to be rather similar to the undiluted lattice even at critical dilution.
The equilibrium-restricted solid-on-solid growth model on fractal substrates is studied by introducing a fractional Langevin equation. The growth exponent beta and the roughness exponent alpha defined, respectively, by the surface width via W approximately t(beta) and the saturated width via W(sat) approximately L(alpha), L being the system size, were obtained by a power-counting analysis, and the scaling relation 2alpha+d(f)=z(RW) was found to hold. The numerical simulation data on Sierpinski gasket, checkerboard fractal, and critical percolation cluster were found to agree well with the analytical predictions of the fractional Langevin equation.
The optimization results by conformational space annealing are presented for an off-lattice protein model consisting of hydrophobic and hydrophilic residues in Fibonacci sequences. The ground-state energies found are lower than those reported in the literature. In addition, the ground-state conformations in three dimensions exhibit the important aspect of forming a single hydrophobic core in real proteins. The energy landscape for the population of local minima is also investigated.
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