Maximal height statistics for1∕fαsignals

Abstract

Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.

Universal fluctuations and extreme statistics of avalanches near the depinning transition

“…Indeed, the EVS of time records with long-term persistence of Gaussian distributed fluctuations with a < 1 converges to the Fisher-Tippett distribution [48]. The same asymptotic law also determines the distribution of maximum heights of periodic, Gaussian 1/f -noise [43,49], where the maximum is measured relative to the mean.…”

“…On the other hand, the extreme value statistics of a correlated Gaussian process with a > 1 typically has a simple scaling form with the duration, with a scaling function strongly dependent on several parameters, such as the boundary conditions, the value from which the extremum is measured, as well as other ordering constraints on the time evolution [42,43]. For example, different scaling functions are obtained for the maximum heights of periodic Gaussian interfaces: if the maximum is measured relative to the spatially averaged height, the corresponding EVS is determined by the so-called Airy distribution function [41][42][43][44], whereas measuring the maximum relative to the boundary value leads to the Rayleigh distribution [50,51].…”

“…In physics, determining the likelihood of these rare events is important to a variety of problems. For example, the low-temperature thermodynamics of spin glasses is governed by the statistics of low-energy states [36]; the velocity of traveling fronts is selected by the extreme value for a wide class of initial conditions similar to random bisections that occur during fragmentation processes [37]; the level density of a noninteracting Bose gas has an asymptotic behavior governed by the three limit laws of classical EVS [38] due to a deeper connection between the sum of correlated random variables and extreme values [39]; much recent progress on EVS in systems with strong correlation has been achieved in the context of Gaussian height fluctuations of kinetically growing interfaces [40][41][42][43][44]; extreme statistics applies to the largest/smallest eigenvalue of random matrices, also corresponding to the extreme positions occupied in a Coulomb gas with logarithmic interactions [45].…”

“…For example, different scaling functions are obtained for the maximum heights of periodic Gaussian interfaces: if the maximum is measured relative to the spatially averaged height, the corresponding EVS is determined by the so-called Airy distribution function [41][42][43][44], whereas measuring the maximum relative to the boundary value leads to the Rayleigh distribution [50,51]. We will discuss the extreme value distribution for an avalanche in a mean field theory of interface depinning known as the Alessandro-BeatriceBertotti-Montorsi (ABBM) model [52] and will show that this avalanche signal can be viewed as a sequence of strongly-correlated, non-identically distributed, nonGaussian variables.…”

A KPZ Cocktail-Shaken, not Stirred...

“…Hence, for =32, our KPZ Euler MRH patch-PDF possesses (s, k)=(1.05,1.57), which increases further to (1.07,1.62) for =16, clearly heading towards calculated F-Airy values. We find these vanishing differences rather suggestive and note, in passing, that the Airy PDF appropriate to 1+1 KPZ/EW with periodic boundary conditions [121,122] has (s, k)=(0.701,0.560), certainly quite distinct from Majumdar-Comtet F-Airy, relevant for WBC, which we illustrate in Figure 3 and see in the LiqXtal experiment.…”