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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. Motiva tion:. ~100 years (data). ~200 years (design). Problems :

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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 Motivation: ~100 years (data) ~200 years (design) Problems: Slow convergence to limit-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. Collaborators: G. Györgyi N. Moloney K. Ozogány I. Jánosi I. Bartos 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 annual maximum temperatures.

  2. EVS and finite-size scaling is measured: probability of i.i.d. variables: Question: Is there a limit distribution for ? lim lim Result: Three possible limit distributions depending on the tail of the parent distribution, .

  3. 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:

  4. 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

  5. EVS and finite-size scaling is measured: probability of i.i.d. variables: There is a limit distribution for . lim Question: What is the rate of convergence? How universal is it?

  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. Finite-size correction: How universal is ? Determines universality Gauss class Exponential class different (known) function Exponential class is unstable Crossovers between Exponential class Gauss class Weibull, Fisher-Tippett-Frechet?!

  10. Convergence to the limit distribution . Size dependence of the skewness:

  11. 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

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

  13. 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

  14. 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 :

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

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

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

  18. signals as problems of generalized random-acceleration. Finite-size scaling of the cumulants Györgyi et al. cond-mat/0610 Path-integral formulation: Scaling: Cumulants:

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

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

  21. Distribution in scaling Yearly maximum temperatures Corrections to scaling

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