Adsorption of piecewise directed random walks on sierpinski fractals

Abstract

Studies of the random walk problem on fractal lattices with adsorbing boundaries are important for practical (as a model for polymer adsorption on rigid impenetrable surfaces, when the polymer-solvent system is in a nonhomogenous-fractal container), as well as in the wider context of the surface critical phenomena physics. In this work we study the problem of adsorption of piecewise directed random walks (PDW) on fractal lattices that belong to the Sierpinski-gasket (SG) family. By applying the renormalization group (RG) method in real space we calculate exactly critical exponent φ associated with the number of the adsorbed steps, for the first 1000 members of the SG family. The exact results are numerically analyzed, the possible relation between the exponent φ and the fractal properties is discussed, as well as the asymptotic behavior of the exponent φ in the vicinity of the fractal-homogeneous lattice crossover, when PDW on SG fractal becomes the common directed walk on the triangular lattice. In addition, we compare our results with the exact and Monte Carlo RG results for the self-avoiding walk model on SG fractals, and, finally we test the Bouchaud-Vannimenus bounds for φ.