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

Abstract

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 ${limop}_{\stackrel{\ensuremath{\rightarrow}}{b}\ensuremath{\infty}}{\ensuremath{\gamma}}_{b}\ensuremath{\rightarrow}{\ensuremath{\gamma}}_{E}.$