Fluctuations and extreme values in multifractal patterns
Лекция- Математика
Our goal is to understand the sample-to-sample fluctuations in disorder-generated multifractal intensity patterns. Arguably the simplest model of that sort is the exponential of an ideal periodic 1/f Gaussian noise. The latter process can be looked at as a onedimensional "projection" of 2D Gaussian Free Field (GFF) and inherits from it the logarithmic covariance structure. It most naturally emerges in the random matrix theory context, but attracted also an independent interest in statistical mechanics of disordered systems. By employing the so-called thermodynamic formalism we can predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level of the noise. We will demonstrate that the distribution is characterized by a powerlaw forward tail of the probability density, with exponent controlled by the level. Such tail results in an important difference between the mean and the typical values of the number of points. This will be further used to determine the threshold of extreme values of 1/f noise and provide a rather compelling explanation for the mechanism behind its universality. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. The presentation will be mainly based on the joint work with Pierre Le Doussal and Alb.