Crossover exponent for piecewise directed walk adsorption on Sierpinski fractals

TL;DR

This paper calculates the crossover exponent for piecewise directed walks adsorbing on Sierpinski fractals using renormalization group methods, revealing similarities to self-avoiding walks and discussing asymptotic behaviors.

Contribution

It provides exact calculations of the crossover exponent for a class of fractal lattices, extending understanding of adsorption phenomena on fractals.

Findings

Crossover exponent $$ closely matches that of self-avoiding walks.

Results show asymptotic behavior at the fractal to Euclidean crossover.

Exact renormalization group method effectively analyzes adsorption on fractals.

Abstract

We study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the Sierpinski gasket family. By applying the exact real space renormalization group method, we calculate the crossover exponent $\phi$, associated with the number of adsorbed steps, for the complete fractal family. We demonstrate that our results are very close to the results obtained for ordinary self-avoiding walk, and discuss the asymptotic behaviour of $\phi$ at the fractal to Euclidean lattice crossover.