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

TL;DR

This paper investigates the scaling behavior of the number theoretic division model of self-organized criticality, revealing a $1/f^2$ power spectral density and finite-size scaling, with implications for related Lévy flight processes.

Contribution

The study provides new insights into the spectral properties and finite-size scaling of the division model, correcting previous estimates of the spectral exponent.

Findings

Primitive set size fluctuations exhibit $1/f^2$ spectral density.

Power spectra show finite-size scaling with system size $M^b$.

Similar spectral properties are observed in Lévy flights with power-law jump distributions.

Abstract

We revisit the number theoretic division model of self-organized criticality [Phys. Rev. Lett. 101, 158702 (2008)]. The model consists of a pool of $M-1$ ordered integers $\{2, 3, \cdots, M\}$, and the aim is to dynamically form a primitive set of integers, where no number can be divided or divisible by others. Using intensive 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 $1/f^{\alpha}$ in the frequency regime $1/M\ll f \ll 1/2$ with $\alpha \approx 2$ (different from $\alpha \approx 1.80(1)$ as reported previously) along with an additional scaling in terms of the system size $\sim M^b$. We also show similar power spectra properties for a class of random walks with a power-law distributed jump size (L\'evy flights).