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Scaling functions for finite-size corrections in EVS Zoltán Rácz PowerPoint PPT Presentation


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Institute for Theoretical Physics Eötvös University E-mail: [email protected] 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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Scaling functions for finite size corrections in evs zolt n r cz

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.


Scaling functions for finite size corrections in evs zolt n r cz

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


Scaling functions for finite size corrections in evs zolt n r cz

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, .


Scaling functions for finite size corrections in evs zolt n r cz

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:


Scaling functions for finite size corrections in evs zolt n r cz

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


Scaling functions for finite size corrections in evs zolt n r cz

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

The Gaussian results are characteristic for the

whole FTG class

except for


Scaling functions for finite size corrections in evs zolt n r cz

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.


Scaling functions for finite size corrections in evs zolt n r cz

For Gaussian

How universal is ?

Signature of corrections?

Finite-size correction to the limit distribution

Comparison with simulations:


Scaling functions for finite size corrections in evs zolt n r cz

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?!


Scaling functions for finite size corrections in evs zolt n r cz

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


Scaling functions for finite size corrections in evs zolt n r cz

Finite-size scaling :

Schehr & Majumdar (2005)

Solid-on-solid models:


Scaling functions for finite size corrections in evs zolt n r cz

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


Scaling functions for finite size corrections in evs zolt n r cz

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 :


Scaling functions for finite size corrections in evs zolt n r cz

Finite-size scaling : Scaling with the fluctuations

Cumulant generating function

Assumption: relaxes faster than

any other .

Scaling with

Expanding in :

Faster convergence


Scaling functions for finite size corrections in evs zolt n r cz

Finite-size scaling: Comparison of scaling with and .

scaling

scaling

Much faster convergence


Scaling functions for finite size corrections in evs zolt n r cz

Possible reason for the fast convergence for ( )

Width distributions

Antal et al. (2001, 2002)

Cumulants of


Scaling functions for finite size corrections in evs zolt n r cz

Extreme statistics of Mullins-Herring interfaces ( )

and of random-acceleration generated paths


Scaling functions for finite size corrections in evs zolt n r cz

Only the mode remains

Extreme statistics for large .


Scaling functions for finite size corrections in evs zolt n r cz

Skewness, kurtosis

Distribution of the daily

maximal temperature

Scale for comparability

Calculate skewness

and kurtosis

Put it on the map

Reference values:


Scaling functions for finite size corrections in evs zolt n r cz

Distribution in scaling

Yearly maximum temperatures

Corrections to scaling


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