Scaling behavior in the number theoretic division model of self-organized criticality.

Abstract

We revisit the number theoretic division model of self-organized criticality [B. Luque et al.Phys. Rev. Lett. 101, 158702 (2008)10.1103/PhysRevLett.101.158702]. The model consists of a pool of M-1 ordered integers {2,3,⋯,M}, and the aim is to dynamically form a primitive set of integers, where no number can be divided or divisible by others. Using extensive simulation studies and finite-size scaling method, we find the primitive set size fluctuations in the division model to show power spectral density of the form 1/f^{α} in the frequency regime 1/M≪f≪1/2 with α≈2 (different from α≈1.80(1) as reported previously) along with an additional scaling in terms of the system size ∼M^{b}. We also show similar power spectra properties for a class of random walks with a power-law distributed jump size (Lévy flights).