Scale-dependent lacunarity and fractal Ising models

Abstract

As an alternate way of measuring the lacunarity of Sierpinski carpets, the authors have computed the site to site fluctuation of the quantity M(n), namely the total number of lattice sites within the chemical distance of n. In long scales of n, the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method. In short scales of n, however, the lacunarity shows a different trend among fractals; it varies monotonically with the Ising correlation length exponent, a trend which was sought before but not found with the previous method. This applies only for the carpets of the first kind.