scaling functions for finite-size corrections in evs zoltán rácz institute for theoretical physics...
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Scaling functions for finite-size corrections in EVS
Zoltán Rácz
Institute for Theoretical PhysicsEötvös UniversityE-mail: [email protected]: cgl.elte.hu/~racz
Collaborators:
G. Györgyi N. Moloney K. Ozogány
I. Jánosi I. Bartos
IdeaIdea: : 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 annual maximum temperatures.
MotivaMotivation:tion:
ProblemsProblems: : Slow convergence to limit-distributions. Not much is known about the EVS of correlated variables.
f/1 0
t ? )( max ith
~100 years (data) ~200 years (design)
)( ith
EVS and finite-size scaling
is measured:)(0 yP
y
)( NzP
Nz
NN yyyz ,...,,max 21
Y Nyyy ,...,, 21
)(zGN zzN probability of
)()( 0 yPdyzFz
z
limN
Question:Question: Is there a limit distribution for ?
NN zFzG )]([)(
NN bxaz
)()]([ xGbxaF NNN
)( NNN bxaG
limN
N
ResultResult:: Three possible limit distributions depending on the tail of the parent distribution, .)(0 yP
z
)( ii thy
i.i.d. variables:
Extreme value limit distributions: i.i.d. variables
)(0 yP
y0y
aye
1 y
10 )( yy
Fisher & Tippet (1928)Gnedenko (1941)
Fisher-Tippet-Gumbel (exponential tail)
))exp(exp()( xxGFTG
Fisher-Tippet-Frechet (power law tail)
0 0 0 )exp()(
xxxxGFTF
Weibull (finite cutoff)
0 1 0 ))(exp()(
xxxxGW
Characteristic shapes of probability densities:
FTGFTF W
dxxdGx II /)()(
/)( xx /)( xx/)( xx
12/5
Gaussian signalsf/1
Edwards-Wilkinson
Random walk
Random acceleration
Mullins-Herring
f/1noise
White noise
Single mode,random phase
2
~)(
f fT hf
f eh
Independent, nonidentically distributed Fourier modes
fh f ~2
with singular fluctuations 12122 ~)( LhLhhwf f
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Berman, 1964)
Majumdar-Comtet, 2004
EVS
)(th
0 L
t
h
)(2 x
x
EVS and finite-size scaling
is measured:)(0 yP
y
)( NzP
Nz
NN yyyz ,...,,max 21
Y Nyyy ,...,, 21
)(zGN zzN probability of
)()( 0 yPdyzFz
z
limN
There is a limit distribution for .
NN zFzG )]([)(
NN bxaz
)()( xGbxaG NNN
N
z
)( ii thy
i.i.d. variables:
Question:Question: What is the rate of convergence?
...)( ? N
1 ?...
ln
1 )( )(),( 10
xN
xxGNxPNN
How universal is it?
Slow convergence to the limit distribution (i.i.d., FTG class)
)(0 yP
y0y
2ye The Gaussian results are characteristic for the whole FTG class
yeyP ~)(0
0
except for
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Finite-size correction to the limit distribution de Haan & Resnick, 1996Gomes & de Haan, 1999
)(0 yP
y
)()( 0 yPdyzFz
NNN
N bxaFdx
d
dx
xdGNxP )]([
)(),(
Fix the position and the scale of by ),( NxP
0x 12 x Na Nb
z
, is determined.Nexpand in
substitute
...)(ln
1)(),( 1 x
NxNxP
FTG
2ye
3 1 )( 2
1 )( ) ( 2
02001
00 xaxa eexaaxaxxFTG
/60 a
...577.0
0
Finite-size correction to the limit distribution
3 1 )( 2
1 )( ) ( 2
02001
00 xaxa eexaaxaxxFTG
For Gaussian )(0 yP
Comparison with simulations:
How universal is ? )(1 x Signature of corrections? 2)(ln N
0
Finite-size correction: How universal is ? )(1 x
Determines universality
2~)( zzf
different (known) function
Gauss class
0
)(0 yP )(0 1~)()( zf
z
eyPdyzF
z
2ye
)(p
1-p )( 11 xx pzzf ~)( )(1 x
zzf ~)( )(1 x Exponential class
Exponential class is unstableSzzzf ~)(
Crossovers betweenNN ln
1 and
1
)1ln(~)( zazzf )exp(~)( azzzf
1a
1a Gauss class
Exponential class
Weibull, Fisher-Tippett-Frechet?!
...)(1
)(),( 1 xN
xNxPFTG
Convergence to the limit distribution .0
Size dependence of the skewness:
10
Maximum relative height distribution ( ) Majumdar & Comtet, 2004
)(th
0 L
t
mhh
2
)(),( 00 xLhPh mm
Lwh Lm 20 12
Connection to the PDF of the area under Brownian excursion over the unit interval
26~ xe
2/5~ xaex
mhmaximum height measured from the average height
?),( LhP m
Result: Airy distribution
Choice of scaling
0/ mm hhx
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Finite-size scaling :
2 Schehr & Majumdar (2005)
Solid-on-solid models:pL
ii hhKH 1 1
...)(2
1)(),( 000 x
LxLhPh mm
0/ mm hhx
)(0 x
)(0 x
...
2
1100
Lhch mm
x
Lwh Lm 20 12
Finite-size scaling : Derivation of …
2
Assumption: carries all the first order finite size correction.
)(2
1)(),( 000 x
LxLhPh mm
Lhm 0
212 !
)()(exp)(
mm
mdk
L cm
ikcyikyP
)()( 00 xhPhx mLmL
Lmh
...1)1(1
)0(11 LcLchc Lm
Cumulant generating function
...)0(2
mmm LccScaling with xhy m 0
0/ mhqk
2
)0( )1(1
)0(12 !
)()(exp)( 1
mm
mdq
L cm
iqLccxiqx
)()()( 0
)1(1
0 1x
L
cxxL
Expanding in :1 L
Shape relaxes faster than the position0 mh
2
1
2
1
Finite-size scaling : Scaling with the average1
Assumption: carries all the first order finite size correction(shape relaxes faster than the position).
mhc1
212 !
)()(exp)(
mm
mdk
L cm
ikcyikyP
)()( xhPhx mLmL
Lmh
...1)1(1
)0(11 LcLchc Lm
Cumulant generating function
...)0(2
mmm Lcc
Scaling with xhy m
mhqk /
2)1(
1)0(
1
)0(
2 ][!
)()1(exp)(
1m
mm
mdq
L Lcc
c
m
iqxiqx
)]()()1[()()( 00)0(1
)1(1
0 1xxx
Lc
cxxL
Expanding in :1L
Finite-size scaling : Scaling with the standard deviation1
Assumption: relaxes faster thanany other .
2c
212 !
)()(exp)(
mm
mdk
L cm
ikcyikyP
)()( 1 xcPx LL
2c
...)( 22)1(2
2)0(2
22 LcLchhc Lmm
Cumulant generating function
...)0(3
mmm Lcc
Scaling with xcy
1 /qk
)]()()([2
)()( 000)0(2
)1(2
0 2xxxx
Lc
cxxL
Expanding in :2L
2mc
Faster convergence
2
1
2
1
Finite-size scaling: Comparison of scaling with and .
Much faster convergence
mh
mh
scaling
scaling
1
1L
2
1L
Possible reason for the fast convergence for ( )
)(xh
0 L
x
mhh
1
12122 ~)( LhLhhwk k
mhw ~2
Width distributions Antal et al. (2001, 2002)
Cumulants of 2w
...~1
~1
)1(
1
)1()( 2
mm
mm
L
nm
mwm L
baL
nLc
...1~12
2)1()(2
2
L
bLc w
...1~1
1)1()(1
2
L
bLc w
4 , 2 31 , LL 73 , LL
Finite-size scaling of the cumulants
)/()/(),( 100 chhhLhPh mmmmm
1c
2c
3c
2/)1(1 ~ Lchm
2/)1(~ nnm Lch
Györgyi et al. cond-mat/0610
signals as problems of generalized random-acceleration.
f/1
Path-integral formulation:
Scaling:
Cumulants:
Extreme statistics of Mullins-Herring interfaces ( )4and of random-acceleration generated paths
mm hhx / mm hhx /
)(0 x
)(0 x
/)( mm hhx/)( mm hhx
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Skewness, kurtosis
)( maxTP
Distribution of the daily maximal temperature
0max T
12max
2max TT
Scale for comparability
Calculate skewness and kurtosis
Put it on the map
skewness s
kurtosis ...1.1FTGs 4.2FTG
Reference values:
Yearly maximum temperatures
Corrections to scalingDistribution in scaling
01.01 01.01