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.

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.

Order statistics of periodic, Gaussian noise with 1/f(α) power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d(k) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), known for independent, identically distributed variables, remains valid for 0 ≤ α < 1. Nontrivial, α-dependent scaling exponents, d(k) k((α-3)/2), emerge for 1 < α < 5, and, finally, α-independent scaling, d(k) ~ k, is obtained for α > 5. The spectra of average ordered values ε(k) =(x(1) - x(k))~ k(β) is also examined. The exponent β is derived from the gap scaling as well as by relating ε(k) to the density of near-extreme states. Known results for the density of near-extreme states combined with scaling suggest that β(α = 2) = 1/2, β(4) = 3/2, and β(∞) = 2 are exact values. We also show that parallels can be drawn between ε(k) and the quantum mechanical spectra of a particle in power-law potentials.

Gregarious animals need to make collective decisions in order to keep their cohesiveness. Several species of them live in multilevel societies, and form herds composed of smaller communities. We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected subgroups (e.g. harems) by combining self organization and social dynamics. It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision. Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortobágy, Hungary). We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal.