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

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

3. Extreme value statistics: i.i.d. variables is measured : probability of lim Question: Question: Is there a limit distribution for ? lim Result: 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 Edwards- Wilkinson Random walk Random acceleration Mullins- Herring noise White noise Single mode, random phase Independent, nonidentically distributed Fourier modes with singular fluctuations Berman, 1964 Majumdar- Comtet, 2004 EVS

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

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

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

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

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

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

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

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

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

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. Extreme statistics for large. Only the mode remains

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

20. Yearly maximum temperatures Corrections to scaling Distribution in scaling