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Scaling functions for finite-size corrections in EVS Zoltán Rácz

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Institute for Theoretical Physics

Eötvös University

E-mail: [email protected]

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.

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

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, .

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

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

Slow convergence to the limit distribution (i.i.d., FTG class)

The Gaussian results are characteristic for the

whole FTG class

except for

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.

For Gaussian

How universal is ?

Signature of corrections?

Finite-size correction to the limit distribution

Comparison with simulations:

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?!

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

Finite-size scaling :

Schehr & Majumdar (2005)

Solid-on-solid models:

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

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 :

Finite-size scaling : Scaling with the fluctuations

Cumulant generating function

Assumption: relaxes faster than

any other .

Scaling with

Expanding in :

Faster convergence

Finite-size scaling: Comparison of scaling with and .

scaling

scaling

Much faster convergence

Possible reason for the fast convergence for ( )

Width distributions

Antal et al. (2001, 2002)

Cumulants of

Extreme statistics of Mullins-Herring interfaces ( )

and of random-acceleration generated paths

Only the mode remains

Extreme statistics for large .

Skewness, kurtosis

Distribution of the daily

maximal temperature

Scale for comparability

Calculate skewness

and kurtosis

Put it on the map

Reference values:

Distribution in scaling

Yearly maximum temperatures

Corrections to scaling