Scaling functions for finite-size corrections in EVS Zoltán Rácz Institute for Theoretical Physics Eötvös University Homepage: cgl.elte.hu/~racz Collaborators: G. Györgyi N. Moloney K. Ozogány I. Jánosi I. Bartos 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 annual maximum temperatures. Motivation: Problems: Slow convergence to limit-distributions. Not much is known about the EVS of correlated variables. ~100 years (data)~200 years (design)

EVS and finite-size scaling 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,. i.i.d. variables :

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:

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

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

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

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

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

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

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

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

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

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

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 :

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

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

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

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

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

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

Yearly maximum temperatures Corrections to scaling Distribution in scaling