Multifractality and extreme value statistics O. Giraud LPTMS - CNRS - PowerPoint PPT Presentation

Multifractality and extreme value statistics O. Giraud LPTMS - CNRS and Université Paris Sud, Orsay Luchon, March 18, 2015

Outline • Multifractality • Logarithmically correlated random fields • Disorder-generated multifractals • Critical random matrix ensembles [Y. V. Fyodorov and O. Giraud, Chaos, Solitons and Fractals 74 , 15 (2015)]

Multifractals ◮ d -dimensional lattice ◮ linear size L , lattice spacing a ◮ M = ( L/a ) d ≫ 1 lattice sites with intensities h i > 0 h i ∼ M x i f ( x ) ✻ Multifractal Ansatz : 1 q � ln h i M � � ρ M ( x ) = δ ln M − x i =1 √ ln M M f ( x ) , ≈ c M ( x ) ( M ≫ 1) f ( x ) singularity spectrum ✲ 0 x x − x 0 x +

Multifractality is characterized by : ◮ Power-law correlation of intensities � y q,s � | r 1 − r 2 | � − z q,s � L E { h q ( r 1 ) h s ( r 2 ) } ∝ , a ≪ | r 1 − r 2 | ≪ L a a ◮ Spatial homogeneity � � � d ( ζ q − 1) 1 � L E { h q ( r ) } = E � h q ( r ) ∝ M a r If • intensities do not vary much over the scale a E { h q ( r 1 ) h s ( r 2 ) } ∼ E h q + s ( r 1 ) � � | r 1 − r 2 | ∼ a • intensities are uncorrelated at scale L E { h q ( r 1 ) h s ( r 2 ) } ∼ E { h q ( r 1 ) } E { h s ( r 2 ) } | r 1 − r 2 | ∼ L then y q,s = d ( ζ q + s − 1) , z q,s = d ( ζ q + s − ζ q − ζ s + 1) ⇒ multifractal pattern characterized by ζ q

Large deviations Saddle-point approximation for partition function : � ∞ M c M ( y ∗ ) � h q M qy ρ M ( y ) dy ≈ M ζ q , Z q = i = M ≫ 1 � | f ′′ ( y ∗ ) | −∞ i =1 with f ′ ( y ∗ ) = − q and ζ q = f ( y ∗ ) + q y ∗ (Recall multifractal Ansatz : � ln h i M � √ � ln M M f ( x ) , ρ M ( x ) = ≈ c M ( x ) M ≫ 1) δ ln M − x i =1 Counting function � ∞ c M ( x ) M f ( x ) N M ( x ) = ρ M ( y ) dy ≈ √ | f ′ ( x ) | ln M x Statistics of extreme values of h = M x ⇔ N M ( x ) ∼ 1

Log-correlated fields Logarithm of the multifractal field : V ( r ) = ln h ( r ) − E { ln h ( r ) } With d dsh s | s =0 = ln h one gets 0 ln | r 1 − r 2 | E { V ( r 1 ) V ( r 2 ) } = − d ζ ′′ L ( ζ ′′ 0 : second derivative of ζ q taken at q = 0 ). i.e. multifractal pattern ⇔ log-correlated random field

Gaussian 1 /f noises ∞ 1 v n e int + v n e − int � � √ n � V ( t ) = , t ∈ [0 , 2 π ) n =1 v n , v n complex normal i.i.d. variables with mean zero and variance 1 Then E { V ( t 1 ) V ( t 2 ) } = − 2 ln | 2 sin t 1 − t 2 | , t 1 � = t 2 2 � t = 2 π � Discrete version : M ≫ 1 , V k ≡ V M k random variables with covariance matrix C km = E { V k V m } given by � � 2 sin π ( k − m ) � V 2 � � � � E { V k V m } = − 2 ln � , C kk = E > 2 ln M � � k M h i = e V i Z q = � h q � ∞ i and N M ( x ) = x ρ M ( y ) dy can be obtained analytically

Moment distribution Discrete periodic Gaussian 1 /f noise � 1+ 1 � 1 M 1+ q 2 1 � Z e q 2 � Ze q 2 − P ( Z q ) = e , Z e = Zq q 2 Z e Γ(1 − q 2 ) Z q for Z q < M 2 and | q | < 1 [Fyodorov Bouchaud (2008)] c M ( y ∗ ) c M ( x ) M ζ q , M f ( x ) √ Z q ≈ N M ( x ) ≈ � | f ′′ ( y ∗ ) | | f ′ ( x ) | ln M ⇒ distribution of N M ( x ) via c M Power-law tail P ( z ) ∼ z − 1 − 1 q 2 for the scaled variable z = Z q /Z e

Typical extreme value Typical counting function N t ( x ) : e E { ln N M ( x ) } ∼ N t ( x ) Scaled counting function n = N M ( x ) / N t ( x ) characterized by P x ( n ) = 4 − 4 x 2 n − ( 1+ 4 x 2 ) , x 2 e − n 0 < x < 2 and M f ( x ) 1 f ( x ) = 1 − x 2 / 4 N t ( x ) = √ Γ(1 − x 2 / 4) , x ln M Threshold for typical value N t ( x ) ∼ 1 x m = 2 − 3 ln ln M + O (1 / ln M ) 2 ln M

Average extreme value E ( n ) = Γ(1 − x 2 / 4) N M ( x ) = n N t ( x ) and E {N M ( x ) } = Γ(1 − x 2 / 4) N t ( x ) ⇒ 1 1 − x 2 / 4 � � Γ ∼ x → 2 2 − x Threshold for average value E {N M ( x ) } ∼ 1 x m = 2 − 1 ln ln M + O (1 / ln M ) 2 ln M Threshold for typical value N t ( x ) ∼ 1 x m = 2 − 3 ln ln M + O (1 / ln M ) 2 ln M

Disorder-generated multifractals | ψ � = � M i =1 ψ i | i � normalized vector : h i = | ψ i | 2 ∼ M − α i , i = 1 , . . . M, multifractal Ansatz ρ M ( α ) ∝ M f ( α ) , ( ρ M ( α ) the density of exponents α i ) or M | ψ i | 2 q ∝ M − τ q , � Z q = τ q = D q ( q − 1) i =1 Inverse participation ratios = exp E { ln Z q } ∼ M − τ typ E { Z q } ∼ M − τ q Z typ q q Scaling � ∞ M � h q M − qα ρ M ( α ) dα Z q = i = 0 i =1

Singularity spectrum τ q f ( α ) ✻ ✻ 1 q ♣ ♣ ♣ ♣ ♣ ♣ ♣ ✲ q q min q 1 q q max − 1 ✲ 0 α α − α 0 ♣♣♣♣♣♣♣♣ α + ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣ � α + M − qα + f typ ( α ) dα ∼ M − τ typ Z typ q ( qα − τ q ) , f ( α ) = min = q q α − Counting function � α N M ( α ) = ρ M ( α ) dα −∞ and scaled variable N M ( α ) ≃ n N t ( α )

Extreme value statistics Distribution of the scaled variable z = Z q /Z typ q P ( z ) ∼ z − 1 − ω q ⇒ Power-law tail [Mirlin-Evers 2000] Tail with ω q → 1 for q → q c = q max ⇒ Divergence of E ( z ) ∼ q → q c | q − q c | − 1 ⇒ Divergence of E ( n ) ∼ α → α − | α − α − | − 1 M f ( x ) M f ( α ) 1 1 N t ( x ) = √ − → N t ( α ) ∝ √ Γ(1 − x 2 / 4) E ( n ) x ln M ln M Threshold for typical value N t ( α ) ∼ 1 α m ≈ α − + 3 1 ln ln M f ′ ( α − ) 2 ln M Extreme value | ψ max | 2 = M − α m

Random matrix ensembles with multifractal eigenvalues One-dimensional N -body models with Hamiltonian H ( p , q ) ◮ equations of motion are equivalent to ˙ L = K L − L K L, K pair of Lax matrices of size M × M ◮ explicit canonical transformation to action-angle variables We choose L ( p , q ) as a random matrix with some measure d L = P ( p , q ) d p d q Canonical transformation d L = P ( λ , φ ) d λ d φ with λ = ( λ 1 , . . . , λ M ) eigenvalues of L . Integration over φ yields P ( λ )

Ruijsenaars-Schneider model � 1 � 2 sin 2 τ Hamiltonian H ( p , q ) = � j cos( p j ) � 1 − k � = j qj − qk sin 2 [ ] 2 Lax matrix : sin[ q j − q s qk − qj ] sin τ 1 sin[ q k − q s e i[ τ ( N − 1)+ p j + + τ ] − τ ] 2 2 � � 2 2 L jk = sin[ q j − q s sin[ q j − q k sin[ q k − q s 1 ] ] + τ ] 2 2 s � = j 2 2 s � = k For q k = 2 πk/M and τ = πa/M , L jk = e i p j 1 − e 2 π i a 1 − e 2 π i( j − k + a ) /M M p j = independent random variables uniformly distributed in [0 , 2 π ] [Phys. Rev. Lett. 103 , 054103 (2009)]

Multifractality of eigenvectors Eigenvectors of L jk = e i p j 1 − e 2 π i a 1 − e 2 π i( j − k + a ) /M , M 2 1 1 0 typ f( α) τ q , τ q -1 0 -2 -3 -1 0 1 2 3 0 0.5 1 1.5 2 q α a = 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9

Perturbation expansion for RS Fractal dimensions are accessible via perturbation series L mn = e iΦ m 1 − e 2 π i a 1 − e 2 π i( m − n + a ) /M M Perturbation series are possible around all integer points a = κ , a = κ + ǫ � 1 + π i( M − 1) � L mn = L (0) + ǫL (1) mn + O ( ǫ 2 ) ǫ mn M where L (0) e iΦ m δ n, m + κ = mn π e − π i( m − n + κ ) /M L (1) e iΦ m (1 − δ n, m + κ ) = mn M sin( π ( m − n + κ ) /M ) ( δ n, m + κ = 1 when n ≡ m + κ mod M and 0 otherwise)

Fractal dimensions for RS ◮ Strong multifractality (almost localized) : a ≪ 1 , L (0) mn diagonal ◮ Unperturbed eigenfunctions Ψ (0) j ( α ) = δ jα ◮ Unperturbed eigenvalues λ α = e iΦ α At first order in a � � q − 1 Γ 2 √ π Γ( q − 1) τ q = 2 a ◮ Weak multifractality (almost extended) : a = κ + ǫ and κ � = 0 . The unperturbed matrix L (0) mn = e iΦ m δ n, m + κ is the shift matrix and its eigenfunctions are extended. τ q = q − 1 − q ( q − 1)( a − κ ) 2 , | a − κ | ≪ 1 κ 2 [Phys. Rev. Lett. 106 , 044101 (2011)]

Correlations in the Ruijsenaars-Schneider model V i = ln | Ψ i | 2 − E � ln | Ψ i | 2 � Weak multifractality limit a = κ + ǫ with κ � = 0 Expansion to order 2 in ǫ , κ = 1 :   E { V k ( α ) V k + r ( α ) } = π 2 ǫ 2 x (2 r − x − M )) ( x − 2 r )( M − x ) � � +  sin 2 πx sin 2 πx M 3 x<r M M x ≥ r ( E = average over eigenvectors α , phases Φ and position k ) For r = cM , M → ∞ , c fixed, E { V k ( α ) V k + r ( α ) } ∼ − 2 ǫ 2 ln r M , r ≪ M ⇒ hidden logarithmic structure of the RS model. Compare with 0 ln | r 1 − r 2 | E { V ( r 1 ) V ( r 2 ) } = − d ζ ′′ L Here τ q = q − 1 − q ( q − 1) ( a − k ) 2 ⇒ τ ′′ 0 = − 2 ǫ 2 k 2

Correlations in the Ruijsenaars-Schneider model 1 6 4 slope E{V i V j } 0.5 2 0 -2 0 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 -ln(|i-j|/N) a a = 0 . 1 (black), 0.3 (red), 0.5 (green), 0.7 (blue), 0.9 (orange) stars : τ ′′ q at q = 0 circles : slope of the correlator − τ ′′ − τ ′′ q | q =0 = 2(1 − a ) 2 , q | q =0 = 4 a ln 4 , a ≃ 0 , a ≃ 1

Extreme values in RS For a = 0 . 7 , M up to 2 12 1.5 1.5 1 1 P(y) P(y) 0.5 0.5 0 0 -2 -1 0 1 2 0 1 2 3 4 5 6 7 y y 3 y → y − α − ln M − 2 f ′ ( α − ) ln ln M y = − ln h m 1.5 1 P(y) 0.5 0 -2 -1 0 1 2 y