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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.

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.

We study the convergence and shape correction to the limit distributions of extreme values 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. We find that, for the correlated systems of subcritical percolation and 1/f α stationary (α < 1) noise, the iid shape correction compares favorably to simulations. Furthermore, for the strongly correlated regime (α > 1) of 1/f α noise, the shape correction is obtained in terms of the limit distribution itself. [5,6], and front propagations [7]. Unfortunately, the use of EVS is hampered by the cost of acquiring good quality statistics: EVS is derived from the extremes of subsets of a data set, requiring abundant data for reasonable statistics. Data analysis is further complicated by the fact that, while the EVS limit distribution may be known, the convergence with increasing sample size is slow. Clearly, a detailed finite-size (FS) analysis providing the convergence rate and shape corrections to the limit distribution is much needed. While for iid variables FS studies exist in the mathematical literature [8], for correlated systems the convergence rate and shape corrections are known only in a few cases, such as Brownian motion [9].In this Letter we use analytic and phenomenological approaches, combined with simulations, to investigate FS scaling in EVS. First, we develop a renormalization group (RG) method, in which the limit distribution is a fixed point of the flow in function space of the finitesample EVS distributions. Applied to iid variables, the approach provides an intuitive and accessible summary of the mathematical results for the leading FS correction, including the explicit forms of the shape corrections (scaling functions). Next, we consider two systems with correlated variables, namely percolation and 1/f α signals. We numerically study the distribution of the largest clusters in subcritical percolation. While the limit distribution is known to be an iid problem [10, 11], we find that even the FS correction fits the iid prediction well. In the case of the maximum statistics of 1/f α signals, 0 ≤ α < 1 corresponds to the weakly correlated regime, with an iid limit distribution [12]. Our simulations indicate that the FS properties are very close to the iid case for 0 ≤ α 0.5, but deviations appear for 0.5 α < 1. For α > 1, however, the convergence becomes fast (power law) and we can show that, under a mild assumption, the FS shape correction is given in terms of the limit distribution and, furthermore, the order as well as the shape of the correction strongly depends on the way the distribution is scaled. The paper is concluded by remarks on higher order FS corrections.The case of iid variables has been extensively studied [13], and we begin our FS study by a reinterpretation of the ori...

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.