Counting Function Fluctuations and Extreme Value Threshold in Multifractal Patterns: The Case Study of an Ideal 1/f Noise

Motivated by the general problem of studying sample-to-sample fluctuations in disorder-generated multifractal patterns we attempt to investigate analytically as well as numerically the statistics of high values of the simplest model??the ideal periodic 1/f Gaussian noise. Our main object of interest is the number of points $\mathcal{N}_{M}(x)$ above a level $\frac{x}{2}V_{m}$ , with V _m =2lnM standing for the leading-order typical value of the absolute maximum for the sample of M points. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of $\mathcal{N}_{M}(x)$ for 0<x<2. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level x, results in an important difference between the mean and the typical values of $\mathcal{N}_{M}(x)$ . 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}^{(\mathit{typ})}=2-c\ln\ln M /\ln M $ with $c=\frac{3}{2}$ . 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 $-\ln p_{\mathit{max}}=\alpha_{-}\ln M +\frac{3}{2f'(\alpha_{-})}\ln\ln M+O(1)$ , where f(??) is the corresponding singularity spectrum positive in the interval ????(?? _?,?? ₊) and vanishing at ??=?? _?>0. For the 1/f noise case we further study asymptotic values of the prefactors in scaling laws for the moments of the counting function. Our numerics shows however that one needs prohibitively large sample sizes to reach such asymptotics even with a moderate precision. This motivates us to 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.