Fine structure and complex exponents in power law distributions from random maps

TL;DR

This paper provides new evidence of log-periodic oscillations in power law distributions from affine random maps with noise, highlighting their robustness and significance in revealing underlying physical processes.

Contribution

It demonstrates the presence of log-periodic structures in probability distributions of affine random maps with noise and analyzes their robustness and smoothing with increasing randomness.

Findings

Log-periodic oscillations are observed in the distributions.

Robustness of log-periodicity persists despite increased randomness.

Smoothing of structures occurs as randomness increases.

Abstract

Discrete scale invariance (DSI) has recently been documented in time-to-failure rupture, earthquake processes and financial crashes, in the fractal geometry of growth processes and in random systems. The main signature of DSI is the presence of log-periodic oscillations correcting the usual power laws, corresponding to complex exponents. Log-periodic structures are important because they reveal the presence of preferred scaling ratios of the underlying physical processes. Here, we present new evidence of log-periodicity overlaying the leading power law behavior of probability density distributions of affine random maps with parametric noise. The log-periodicity is due to intermittent amplifying multiplicative events. We quantify precisely the progressive smoothing of the log-periodic structures as the randomness increases and find a large robustness. Our results provide useful markers for the search of log-periodicity in numerical and experimental data.