Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/fα power spectrum. For 0≤α<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 α>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 α→ ∞ and, for large α, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for α=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary α>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.
|Journal||Physical Review E - Statistical, Nonlinear, and Soft Matter Physics|
|Publication status||Published - Feb 28 2007|
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics