Renormalization-group theory for finite-size scaling in extreme statistics.

We present a renormalization-group (RG) approach to explain universal features of extreme statistics applied here to independent identically distributed variables. The outlines of the theory have been described in a previous paper, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.

Finite-size scaling in extreme statistics.

We study the deviations from the limit distributions in extreme value statistics arising due to the finite size (FS) of data sets. A renormalization method is introduced for the case of independent, identically distributed (iid) variables, showing that the iid universality classes are subdivided according to the exponent of the FS convergence, which determines the leading order FS shape correction function as well. It is found that, for the correlated systems of subcritical percolation and 1/f;(alpha) stationary (alpha<1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (alpha>1) of 1/f;(alpha) noise, the shape correction is obtained in terms of the limit distribution itself.

Extreme statistics for time series: distribution of the maximum relative to the initial value.

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Maximal height statistics for 1/f(alpha) signals.

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 01 (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.

Dislocation glasses: aging during relaxation and coarsening.

The dynamics of dislocations is reported to exhibit a range of glassy properties. We study numerically various versions of 2D edge dislocation systems, in the absence of externally applied stress. Two types of glassy behavior are identified (i) dislocations gliding along randomly placed, but fixed, axes exhibit relaxation to their spatially disordered stable state; (ii) if both climb and annihilation are allowed, irregular cellular structures can form on a growing length scale before all dislocations annihilate. In all cases both the correlation function and the diffusion coefficient are found to exhibit aging. Relaxation in case (i) is a slow power law, furthermore, in the transient process (ii) the dynamical exponent z approximately 6, i.e., the cellular structure coarsens relatively slowly.

Debye screening of dislocations.

Debye-like screening by edge dislocations of some externally given stress is studied by means of a variational approach to coarse grained field theory. Explicitly given are the force field and the induced geometrically necessary dislocation (GND) distribution, in the special case of a single glide axis in 2D, for (i) a single edge dislocation and (ii) a dislocation wall. Numerical simulation demonstrates that the correlation in relaxed dislocation configurations is in good agreement with the induced GND in case (i). Furthermore, the result (ii) well predicts the experimentally observed decay length for the GND developing close to grain boundaries.

Statistics of extremal intensities for Gaussian interfaces.

The extremal Fourier intensities are studied for stationary Edwards-Wilkinson-type, Gaussian, interfaces with power-law dispersion. We calculate the probability distribution of the maximal intensity and find that, generically, it does not coincide with the distribution of the integrated power spectrum (i.e., roughness of the surface), nor does it obey any of the known extreme statistics limit distributions. The Fisher-Tippett-Gumbel limit distribution is, however, recovered in three cases: (i) in the nondispersive (white noise) limit, (ii) for high dimensions, and (iii) when only short-wavelength modes are kept. In the last two cases the limit distribution emerges in nonconventional scenarios.

Roughness distributions for 1/f alpha signals.

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f(alpha) noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, 1/f noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions (alpha < or = 1/2, 1/2 < alpha < or = 1, and 1< alpha), distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for alpha < or = 1/2 the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for alpha > 1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha = 2 and alpha-->infinity, in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of 1/f(alpha) processes.

1/f noise and extreme value statistics.

We study finite-size scaling of the roughness of signals in systems displaying Gaussian 1/f power spectra. It is found that one of the extreme value distributions, the Fisher-Tippett-Gumbel (FTG) distribution, emerges as the scaling function when boundary conditions are periodic. We provide a realistic example of periodic 1/f noise, and demonstrate by simulations that the FTG distribution is a good approximation for the case of nonperiodic boundary conditions as well. Experiments on voltage fluctuations in GaAs films are analyzed and excellent agreement is found with the theory.