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Scaling functions for finite-size corrections in EVS Zoltán Rácz Institute for Theoretical Physics Eötvös University E-mail: [email protected] Homepage: cgl.elte.hu/~racz Motivation: Do witches exist if there were 2 very large hurricanes in a century? Introduction: Extreme value statistics (EVS) for physicists in 10 minutes. Collaborators: G. Gyorgyi N. Moloney K. Ozogany I. Janosi I. Bartos 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 1 / f signals ( 0 ). Improved convergence by using the right scaling variables. Distribution of yearly maximum temperatures. Extreme value statistics Y is measured: y1 , y2 ,..., yN z N maxy1 , y2 ,..., yN yi h(ti ) Question: What is the distribution of the largest number? Aim: Trying to extrapolate to values where no data exist. P0 ( y ) Logics: Assume something about yi E.g. independent, identically distributed y Use limit argument: ( N ) Family of limit distributions (models) is obtained Calibrate the family of models by the measured values of xN P( z N ) zN Extreme value statistics: i.i.d. variables z Y is measured: y1 , y2 ,..., yN F ( z ) dyP0 ( y) P0 ( y ) z N maxy1 , y2 ,..., yN GN (z) probability of z N z y z GN ( z) [F ( z)] N Question: Is there a limit distribution for N ? lim N GN (aN x bN ) lim N [ F (aN x bN )]N G( x) Result: Three possible limit distributions depending on the tail of the parent distribution, P0 ( y ) . P( z N ) z zN z aN x bN Fisher & Tippet (1928) Gnedenko (1941) Extreme value limit distributions: i.i.d. variables P0 ( y ) Fisher-Tippet-Gumbel (exponential tail) GFTG ( x) exp( exp( x)) e ya y ( y0 y) 1 y0 Fisher-Tippet-Frechet 1 GFTF ( x) exp( x ) 0 y GW ( x) exp( ( x) ) 1 x0 x0 W FTF 5/ 2 ( x x ) / x0 x0 Weibull (finite cutoff) Characteristic shapes of probability densities: I ( x) dGI ( x) / dx FTG (power law tail) ( x x ) / 1 ( x x ) / Gaussian 1 / f signals h Independent, nonidentically distributed Fourier modes (hk ) ~ e L k k hk 0 .5 1 1/ f White noise noise EVS ) Berman, 1964 L w2 (h h) 2 L1k hk ~ L 1 2 2 EdwardsWilkinson 0 x 0 hk ~ k with singular fluctuations h(x) 2 2 Random walk MajumdarComtet, 2004 MullinsHerring 4 Random acceleration Single mode, random phase 0 Slow convergence to the limit distribution (i.i.d., FTG class) P0 ( y ) 0 e y2 0 .5 1 y 4 The Gaussian results are characteristic for the whole FTG class except for y0 2 P0 ( y) ~ e y 0 Finite-size correction to the limit distribution de Haan & Resnick, 1996 Gomes & de Haan, 1999 z P0 ( y ) F ( z ) dyP0 ( y) e y2 z P ( x, N ) y dG N ( x) d [ F (a N x bN )] N dx dx substitute Fix the position and the scale of P( x, N ) by x 0 x 2 1 aN , bN P( x, N ) FTG ( x) expand in N is determined. 1 1 ( x) ... ln N 1 1 ( x) FTG ( x) a 0 x a 20 1 3 a 0 x e a 0 x e 2 ( a 0 x ) 2 a0 6 / 0.577... 0 Finite-size correction to the limit distribution For Gaussian P0 ( y ) 1 ( x ) 1 FTG ( x) a 0 x a 20 1 3 a 0 x e a 0 x e 2 ( a 0 x ) 2 Comparison with simulations: How universal is 1 ( x) ? Signature of (ln N )2 corrections? 0 Finite-size correction: How universal is 1 ( x) ? z P0 ( y ) F ( z ) dyP0 ( y) ~ 1 e f ( z ) e y2 Determines universality z f ( z) ~ z 2 1 ( x) Gauss class f ( z) ~ z 1 ( x) Exponential class f ( z) ~ z p p -1 1 ( x) 1 ( x) p different (known) function f ( z) ~ z exp(az) Exponential class is unstable f ( z) ~ z z S f ( z ) ~ z a ln(1 z ) a 1 Exponential class a 1 Gauss class Gauss class eves for 0 s 1 Weibull, Fisher-Tippet-Frechet?! Maximum relative height distribution ( 2 ) 0 .5 0 hm h h(x) hm x L 0 Majumdar & Comtet, 2004 1 2 4 maximum height measured from the average height P(hm , L) ? Connection to the PDF of the area under Brownian excursion over the unit interval Choice of scaling x hm /hm 0 ~e hm 0 12wL ~ L 6 x 2 Result: Airy distribution ~ x 5e a / x 2 hm 0 P(hm , L) 0 ( x) Finite-size scaling 2 : Schehr & Majumdar (2005) Solid-on-solid models: H K 1 hi hi 1 hm 0 P(hm , L) 0 ( x) 0 ( x) L 1 0 ( x) ... 2L hm 0 12wL ~ L x hm /hm 0 p 1 hm c0 hm 0 1 ... 2L 0 ( x) x Finite-size scaling 2 : Derivation of hm 0 P(hm , L) 0 ( x) Cumulant generating function PL ( y) dk 2 Scaling with (ik ) m exp ik ( y c1 ) cm m ! m 2 hm 0 L y hm 0 x L ( x) hm 0 PL (hm 0 x) L ( x) dq 2 Expanding in (iq) m ( 0) (0) (1) 1 exp iq( x c1 c1 L ) cm m ! m2 L1 : c1(1) L ( x) 0 ( x) 1 0 ( x) L hm 0 Shape relaxes faster than the position Assumption: hm L carries all the first order finite size correction. c1 hm L c1(0) L c1(1) L 1 ... cm2 cm(0) Lm ... k q / hm 0 1 0 ( x) … 2L Finite-size scaling 1 : Scaling with the average Cumulant generating function PL ( y) dk 2 Scaling with (ik ) m exp ik ( y c1 ) cm m 2 m! c1 hm y hm x k q / hm L ( x) hm PL (hm x) L ( x) dq 2 Expanding in cm( 0) (iq) m exp iq( x 1) ( 0) (1) 1 m ] m 2 m! [c1 c1 L L1 : c1(1) L ( x) 0 ( x) ( 0) 1 [(x 1)0 ( x) 0 ( x)] c1 L Assumption: hm L carries all the first order finite size correction (shape relaxes faster than the position). c1 hm L c1(0) L c1(1) L 1 ... cm2 cm(0) Lm ... Finite-size scaling 1 : Scaling with the fluctuations Assumption: c2 relaxes faster than any other cm 2 . Cumulant generating function PL ( y) dk 2 Scaling with L ( x) dq 2 (ik ) m exp ik ( y c1 ) cm m 2 m! c2 y c1 x k q / L ( x) PL (c1 x) c2 (hm hm )2 L c2(0) L2 c2(1) L2 2 ... cm3 cm(0) Lm ... cm( 0) q 2 (iq) m exp iq( x 1) 2 m3 m! [c2( 0) c2(1) L 2]m / 2 Expanding in L 2 : c2(1) L ( x) 0 ( x) ( 0) 2 [0( x) x0 ( x) 0 ( x)] 2c2 L Faster convergence Finite-size scaling: Comparison of scaling with hm and hm scaling scaling Much faster convergence . Possible reason for the fast convergence for ( 1 ) Width distributions Antal et al. (2001, 2002) hm h w2 L P( w2 ) x w 2 ~ hm h(x) x L 0 w2 (h h) 2 L1k hk ~ L 1 2 w2 Cumulants of ( w2 ) m c ~L ( w2 ) 1 c m ( 1) ( 1) ~L 2,4 bm 1 m ( 1) ~ L a ... m m m 1 L n 1 n x w2 / w2 L b1 1 1 ... L L 1 , L 3 c ( w2 ) 2 ( 1) ~L b2 1 2 1 ... L L 3 , L 7 Extreme statistics of Mullins-Herring interfaces ( 4 ) and of random-acceleration generated paths 0 0 .5 1 0 ( x) x hm / hm x hm / hm 0 ( x) x (hm hm ) / x (hm hm ) / 2 4 Extreme statistics for large . 0 0 .5 1 2 (hk ) ~ e 0 ( x) 4 L k k hk 2 k 2 2 k1 Only the k1 mode remains x hm / hm hm hk1 0 ( x) x exp x 2 2 4 hm 0 (hmax ) ~ e L hk1 2 | hk1| d | hk1 | Skewness, kurtosis Distribution of the daily maximal temperature P(Tmax ) Scale for comparability Tmax 0 s skewness 2 Tmax Tmax 2 1 Calculate skewness and kurtosis Put it on the map Reference values: sFTG 1.1... FTG 2.4 curtosis Yearly maximum temperatures Distribution in scaling Corrections to scaling