Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces

We investigate the statistics of the maximal fluctuation of two-dimensional Gaussian interfaces. Its relation to the entropic repulsion between rigid walls and a confined interface is used to derive the average maximal fluctuation $\sim \sqrt{2/(\pi K)} \ln N$ and the asymptotic behavior of the whole distribution $P(m) \sim N^2 e^{-{\rm (const)} N^2 e^{-\sqrt{2\pi K} m} - \sqrt{2\pi K} m}$ for $m$ finite with $N^2$ and $K$ the interface size and tension, respectively. The standardized form of $P(m)$ does not depend on $N$ or $K$, but shows a good agreement with Gumbel's first asymptote distribution with a particular non-integer parameter. The effects of the correlations among individual fluctuations on the extreme value statistics are discussed in our findings.Comment: 4 pages, 4 figures, final version in PR

Crowding at the Front of the Marathon Packs

We study the crowding of near-extreme events in the time gaps between successive finishers in major international marathons. Naively, one might expect these gaps to become progressively larger for better-placing finishers. While such an increase does indeed occur from the middle of the finishing pack down to approximately 20th place, the gaps saturate for the first 10-20 finishers. We give a probabilistic account of this feature. However, the data suggests that the gaps have a weak maximum around the 10th place, a feature that seems to have a sociological origin.Comment: 5 pages, 2 figures; version 2: published manuscript with various changes in response to referee comments and some additional improvement

Mapping of Coulomb gases and sine-Gordon models to statistics of random surfaces

We introduce a new class of sine-Gordon models, for which interaction term is present in a region different from the domain over which quadratic part is defined. We develop a novel non-perturbative approach for calculating partition functions of such models, which relies on mapping them to statistical properties of random surfaces. As a specific application of our method, we consider the problem of calculating the amplitude of interference fringes in experiments with two independent low dimensional Bose gases. We calculate full distribution functions of interference amplitude for 1D and 2D gases with nonzero temperatures.Comment: final published versio

Chemical fracture and distribution of extreme values

When a corrosive solution reaches the limits of a solid sample, a chemical fracture occurs. An analytical theory for the probability of this chemical fracture is proposed and confirmed by extensive numerical experiments on a two dimensional model. This theory follows from the general probability theory of extreme events given by Gumbel. The analytic law differs from the Weibull law commonly used to describe mechanical failures for brittle materials. However a three parameters fit with the Weibull law gives good results, confirming the empirical value of this kind of analysis.Comment: 7 pages, 5 figures, to appear in Europhysics Letter

Record statistics in random vectors and quantum chaos

The record statistics of complex random states are analytically calculated, and shown that the probability of a record intensity is a Bernoulli process. The correlation due to normalization leads to a probability distribution of the records that is non-universal but tends to the Gumbel distribution asymptotically. The quantum standard map is used to study these statistics for the effect of correlations apart from normalization. It is seen that in the mixed phase space regime the number of intensity records is a power law in the dimensionality of the state as opposed to the logarithmic growth for random states.Comment: figures redrawn, discussion adde

Edwards-Wilkinson surface over a spherical substrate: 1/f1/f noise in the height fluctuations

We study the steady state fluctuations of an Edwards-Wilkinson type surface with the substrate taken to be a sphere. We show that the height fluctuations on circles at a given latitude has the effective action of a perfect Gaussian $1/f$ noise, just as in the case of fixed radius circles on an infinite planar substrate. The effective surface tension, which is the overall coefficient of the action, does not depend on the latitude angle of the circles.Comment: 6 page

Density of near-extreme events

We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than, faster than, or as a pure exponential function. We argue that some of these results would remain valid even for certain {\em correlated} cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.Comment: 4 pages, 3 figures, revtex4. Minor corrections, references updated. This is slightly extended version of the Published one (Phys. Rev. Lett.

Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models

We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large $L$ limit, a central limit argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a universal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12} w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using transfer matrix technique, valid for any $L>0$. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a non universal amplitude, and simply given by the derivative of the Airy distribution function $f'(x)$.Comment: 13 pages, 4 figure

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/f^alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRH_I). The exact MRH_I 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 MRH_I 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 MRH_I 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 non-periodic 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 MRH_I distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.Comment: 29 pages, 4 figure

Roughness correction to the Casimir force at short separations: Contact distance and extreme value statistics

So far there has been no reliable method to calculate the Casimir force at separations comparable to the root-mean-square of the height fluctuations of the surfaces. Statistical analysis of rough gold samples has revealed the presence of peaks considerably higher than the root-mean-square roughness. These peaks redefine the minimum separation distance between the bodies and can be described by extreme value statistics. Here we show that the contribution of the high peaks to the Casimir force can be calculated with a pairwise additive summation, while the contribution of asperities with normal height can be evaluated perturbatively. This method provides a reliable estimate of the Casimir force at short distances, and it solves the significant, so far unexplained discrepancy between measurements of the Casimir force between rough surfaces and the results of perturbation theory. Furthermore, we illustrate the importance of our results in a technologically relevant situation.Comment: 29 pages, 11 figures, to appear in Phys. Rev.