Maximal height statistics for 1/f^alpha signals

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<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.

arXiv comment17 pages, 8 figures, RevTex4

References in corpus (19)

• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• 1/f Noise and Extreme Value Statistics
• Long term persistence in the sea surface temperature fluctuations
• Generalised extreme value statistics and sum of correlated variables
• Maximal Height Scaling of Kinetically Growing Surfaces
• Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence
• Variety and Volatility in Financial Markets
• Spatial Persistence of Fluctuating Interfaces
• Global fluctuations and Gumbel statistics
• Roughness distributions for 1/f^alpha signals
• Extreme Value Statistics and Traveling Fronts: An Application to Computer Science
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Universal Statistics of the Critical Depinning Force of Elastic Systems in Random Media
• Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
• Statistics of extremal intensities for Gaussian interfaces
• Extreme Fluctuations in Small-Worlds with Relaxational Dynamics
• Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces
• First passage time exponent for higher-order random walks:Using Levy flights

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