Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal $1/f$ noise

TL;DR

This paper analyzes fluctuations and extreme value thresholds in multifractal patterns generated by ideal $1/f$ noise, revealing universal scaling laws and providing exact formulas for the statistics of high values.

Contribution

It introduces a detailed analytical and numerical study of extreme values in ideal $1/f$ noise, uncovering universal scaling behaviors and deriving formulas for the maximum intensity and counting function.

Findings

The typical threshold for extreme values scales as $2 - c\ln\ln M/\ln M$ with $c=3/2$.

The distribution of high values exhibits a power-law tail affecting mean and typical counts.

Exact formulas for the mean and variance of the counting function are derived for $1/f$ noise.

Abstract

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = \alpha_{-}\ln{M} + \frac{3}{2f'(\alpha_{-})}\ln{\ln{M}} + O(1)$, where $f(\alpha)$ is the corresponding singularity spectrum vanishing at $\alpha=\alpha_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.