1. dia

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Institute for Theoretical Physics Eötvös
University E-mail racz_at_general.elte.hu Homepage
cgl.elte.hu/racz
Scaling functions for finite-size corrections
in EVS Zoltán
Rácz
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.
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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
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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, .
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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
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Gaussian signals
Independent, nonidentically distributed Fourier
modes
with singular fluctuations
Edwards- Wilkinson
Mullins- Herring
Random walk
Random acceleration
White noise
Single mode, random phase
noise
EVS
Majumdar- Comtet, 2004
Berman, 1964
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Slow convergence to the limit distribution
(i.i.d., FTG class)
The Gaussian results are characteristic for the
whole FTG class
except for
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Finite-size correction to the limit distribution
de Haan Resnick, 1996 Gomes de Haan, 1999
substitute
expand in
, is determined.
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Finite-size correction to the limit distribution
Comparison with simulations
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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?!
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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
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Finite-size scaling
Schehr Majumdar (2005)
Solid-on-solid models
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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
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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
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Finite-size scaling Scaling with the
fluctuations
Assumption relaxes faster than any
other .
Cumulant generating function
Scaling with
Expanding in
Faster convergence
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Finite-size scaling Comparison of scaling with
and .
scaling
scaling
Much faster convergence
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Possible reason for the fast convergence for (
)
Width distributions Antal et al. (2001,
2002)
Cumulants of
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Extreme statistics of Mullins-Herring interfaces
( )
and of random-acceleration generated paths
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Extreme statistics for large .
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Skewness, kurtosis
Distribution of the daily maximal temperature
Scale for comparability
Calculate skewness and kurtosis
Put it on the map
Reference values
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Yearly maximum temperatures
Corrections to scaling