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Scaling functions for finite-size corrections in EVS Zoltán Rácz

Institute for Theoretical Physics Eötvös University E-mail: racz@general.elte.hu Homepage: c gl.elte.hu/~racz. Scaling functions for finite-size corrections in EVS Zoltán Rácz. Collaborators: G. Gyorgyi N. Moloney K. Ozogany I. Janosi I. Bartos.

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Scaling functions for finite-size corrections in EVS Zoltán Rácz

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  1. Institute for Theoretical Physics Eötvös University E-mail: racz@general.elte.hu Homepage: cgl.elte.hu/~racz Scaling functions for finite-size corrections in EVS ZoltánRácz Collaborators: G. Gyorgyi N. Moloney K. Ozogany I. Janosi I. Bartos Motivation: Do witches exist if there were 2 very large hurricanes in a century? Introduction: Extreme value statistics (EVS) for physicists in 10 minutes. Problems: Slow convergence to limiting distributions. Not much is known about the EVS of correlated variables. Idea: EVS looks like a finite-size scaling problem of critical phenomena – try to use the methods learned there. Results: Finite size corrections to limiting distributions (i.i.d. variables). Numerics for the EVS of signals ( ). Improved convergence by using the right scaling variables. Distribution of yearly maximum temperatures.

  2. Extreme value statistics Question: What is the distribution of the largest number? is measured: Aim: Trying to extrapolate to values where no data exist. Logics: Assume something about E.g. independent, identically distributed Use limit argument: Family of limit distributions (models) is obtained Calibrate the family of models by the measured values of

  3. Extreme value statistics: i.i.d. variables is measured: probability of Question: Is there a limit distribution for ? lim lim Result: Three possible limit distributions depending on the tail of the parent distribution, .

  4. Extreme value limit distributions: i.i.d. variables Fisher & Tippet (1928) Gnedenko (1941) Fisher-Tippet-Gumbel (exponential tail) Fisher-Tippet-Frechet (power law tail) Weibull (finite cutoff) Characteristic shapes of probability densities:

  5. Gaussian signals Independent, nonidentically distributed Fourier modes with singular fluctuations Edwards- Wilkinson Mullins- Herring White noise Random walk Random acceleration Single mode, random phase noise EVS Majumdar- Comtet, 2004 Berman, 1964

  6. Slow convergence to the limit distribution (i.i.d., FTG class) The Gaussian results are characteristic for the whole FTG class except for

  7. Fix the position and the scale of by Finite-size correction to the limit distribution de Haan & Resnick, 1996 Gomes & de Haan, 1999 substitute expand in , is determined.

  8. For Gaussian How universal is ? Signature of corrections? Finite-size correction to the limit distribution Comparison with simulations:

  9. Gauss class eves for Finite-size correction: How universal is ? Determines universality Gauss class Exponential class different (known) function Exponential class is unstable Exponential class Gauss class Weibull, Fisher-Tippet-Frechet?!

  10. Maximum relative height distribution ( ) Majumdar & Comtet, 2004 maximum height measured from the average height Connection to the PDF of the area under Brownian excursion over the unit interval Choice of scaling Result: Airy distribution

  11. Finite-size scaling : Schehr & Majumdar (2005) Solid-on-solid models:

  12. Finite-size scaling : Derivation of … Cumulant generating function Assumption: carries all the first order finite size correction. Scaling with Expanding in : Shape relaxes faster than the position

  13. Finite-size scaling : Scaling with the average Cumulant generating function Assumption: carries all the first order finite size correction (shape relaxes faster than the position). Scaling with Expanding in :

  14. Finite-size scaling : Scaling with the fluctuations Cumulant generating function Assumption: relaxes faster than any other . Scaling with Expanding in : Faster convergence

  15. Finite-size scaling: Comparison of scaling with and . scaling scaling Much faster convergence

  16. Possible reason for the fast convergence for ( ) Width distributions Antal et al. (2001, 2002) Cumulants of

  17. Extreme statistics of Mullins-Herring interfaces ( ) and of random-acceleration generated paths

  18. Only the mode remains Extreme statistics for large .

  19. Skewness, kurtosis Distribution of the daily maximal temperature Scale for comparability Calculate skewness and kurtosis Put it on the map Reference values:

  20. Distribution in scaling Yearly maximum temperatures Corrections to scaling

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