Sierpinski Gasket Family Of Fractals Research Articles

Comment on "Critical behavior of the chain-generating function of self-avoiding walks on the sierpinski gasket family: the euclidean limit"

We refute the claims made by Riera and Chalub [Phys. Rev. E 58, 4001 (1998)] by demonstrating that they have not provided enough data (requisite in their series expansion method) to draw reliable conclusions about criticality of self-avoiding walks on the Sierpinski gasket family of fractals.

Critical behavior of the chain-generating function of self-avoiding walks on the Sierpinski gasket family: The Euclidean limit

We study self-avoiding walks (SAW's) on the generalized Sierpinski gasket family of fractals. Each fractal can be labeled by an integer b $(2l~bl~\ensuremath{\infty}),$ so that the fractal and spectral dimensions tend to the Euclidean value 2 when $\stackrel{\ensuremath{\rightarrow}}{b}\ensuremath{\infty}$. By using an exact enumeration technique to obtain the series expansion for the chain-generating function of SAW's on these lattices, we calculate the associated critical exponent ${\ensuremath{\gamma}}_{b}$ for $2l~bl~100$. The large-b behavior of ${\ensuremath{\gamma}}_{b}$ is the first numerical result consistent with the asymptotic convergence toward the Euclidean value ${\ensuremath{\gamma}}_{E}.$ We also give an analytic argument supporting the assumption that ${limop}_{\stackrel{\ensuremath{\rightarrow}}{b}\ensuremath{\infty}}{\ensuremath{\gamma}}_{b}\ensuremath{\rightarrow}{\ensuremath{\gamma}}_{E}.$

Critical exponent gamma for self-avoiding walks on the Sierpinski gasket family of fractals

The authors apply the Monte Carlo renormalization group (MCRG) analysis of self-avoiding walks (SAWs) on fractals to calculate the critical exponent gamma , associated with the total number of distinct SAWs. In the case of the Sierpinski gasket family of fractals (whose members are labelled by an integer b, 2<or=b< infinity ) they have calculated gamma for 2<or=b<or=80. Their MCRG results deviate at most 0.2% from the available exact results (2<or=b<or=8). The entire set of their results demonstrates that gamma , being always larger than the Euclidean value 43/32, monotonically increases with b.

Self-avoiding walks on fractals studied by the Monte Carlo renormalization group

The authors have applied the Monte Carlo renormalization group (MCRG) method to study the problem of self-avoiding walks on the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b, 2<or=b<or= infinity , and when b to infinity both the fractal df and spectral ds dimension approach their Euclidean value 2. They have calculated the critical exponent nu , associated with the mean square end-to-end distance, up to b=80. Their MCRG results deviate at most 0.03% from the available exact results (for 2<or=b<or=9). The obtained data show clearly that nu monotonically decreases with b and crosses the Euclidean value nu =3/4 at b approximately=27, that is, before entering the fractal to Euclidean crossover region that occurs in the limit b to infinity .

Piecewise directed random walk on the Sierpinski gasket family of fractals

We have introduced a new type of nontrivial directed random walk on the Sierpinski gasket family of fractals. We demonstrate that the behaviour of the critical exponent ν, associated with the mean squared end-to-end distance, at the fractal to Euclidean lattice crossover is governed by the fractal dimension alone.

Asymptotic form of the spectral dimension at the fractal to lattice crossover

The authors study the spectral dimension of the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b(2<or=b< infinity ), so that when b to infinity both the fractal and spectral dimension approach the Euclidean value 2. The asymptotic law for the spectral dimension was recently suggested to be dS approximately=2-B/(1nb)beta , with B and beta being some constants. Here the authors demonstrate that this form should be replaced by the asymptotic law ds approximately=2-1n(1nb)+ constant/1nb. Their analysis is based on the exact calculation of the electric resistances Rb for all members of the family up to b=650.