Cluster Monte Carlo distributions in fractal dimensions between two and three: Scaling properties and dynamical aspects for the Ising model

Abstract

We study the Wolff cluster size distributions obtained from Monte Carlo simulations of the Ising phase transition on Sierpinski fractals with Hausdorff dimensions ${D}_{f}$ between 2 and 3. These distributions are shown to be invariant when going from an iteration step of the fractal to the next under a scaling of the cluster sizes involving the exponent $(\ensuremath{\beta}/\ensuremath{\nu})+(\ensuremath{\gamma}/\ensuremath{\nu}).$ Moreover, the decay of the autocorrelation functions at the critical points enables us to calculate the Wolff dynamical critical exponents z for three different values of ${D}_{f}.$ The Wolff algorithm is more efficient in reducing the critical slowing down when ${D}_{f}$ is lowered.