Extreme values: a renormalization group approach Eric Bertin Laboratoire de Physique, ENS Lyon Collaboration with G. Gy¨ orgyi (Budapest), F. Angeletti and P. Abry (ENS Lyon) Conference ”Computation of transition trajectories and rare events in non-equilibrium systems”, ENS Lyon, 11-15 June 2012 Eric Bertin Renormalization approach for extreme statistics
Introduction Many extreme value problems... Reaction path with the lowest barrier in a complex landscape Ground state in a disordered system Problems of pinned interfaces,... In many cases, one needs to find minimum or maximum values among a set of random variables ⇒ statistics? ezard, J. Phys. A (1997) . See, e.g., Bouchaud, M´ Difficulties Presence of strong correlations, multiples scales,... Use of renormalization group could be relevant (but still difficult) What about the simplest extreme value problem, with iid random variables? Eric Bertin Renormalization approach for extreme statistics
Introduction Standard results Variables x 1 , . . . , x n drawn from cumulative distribution µ ( x ) (called parent distribution) Rescaled cumulative distribution of max( x 1 , . . . , x n ) F γ ( y ) = exp [ − (1 + γ y )] − 1 /γ 1 + γ y > 0 γ > 0: Fr´ echet distribution (power-law tail of parent dist.) γ = 0: Gumbel distribution (faster than powel-law tail) γ < 0: Weibull distribution (bounded variables) Fisher, Tippett (1928); Gnedenko (1943); Gumbel (1958) Eric Bertin Renormalization approach for extreme statistics
Introduction Motivation Asymptotic distributions of extreme values of iid random variables known for long, but strong finite-size effects, not always easy to handle with standard probabilistic methods Idea: Use the renormalization language as a convenient tool to analyze fixed points and finite-size corrections, in spite of the absence of correlations Approach initiated in Gy¨ orgyi, Moloney, Ozog´ any, R´ acz, PRL (2008) Gy¨ orgyi, Moloney, Ozog´ any, R´ acz, Droz, PRE (2010) Aim of the present contribution: reformulate the results using a differential representation, which is more convenient Eric Bertin Renormalization approach for extreme statistics
Basic renormalization idea Extreme value statistics N iid random variables, cumulative distribution � x −∞ ρ ( x ′ ) dx ′ µ ( x ) = Cumulative distribution for the maximum value Prob (max( x 1 , . . . , x N ) < x ) = Prob ( ∀ i , x i < x ) = µ N ( x ) Decimation procedure Split the set of sufficiently large N random variables x i into N ′ = N / p blocks of p random variables each y j the maximum value in the j th block max( x 1 , . . . , x N ) = max( y 1 , . . . , y N ′ ) y j are also i.i.d. random variables, with a distribution µ p ( y ) µ p ( y ) = µ p ( y ) Eric Bertin Renormalization approach for extreme statistics
Renormalization operation Raising to a power and rescaling [ˆ R p µ ]( x ) = µ p � � a p x + b p Necessity of scale and shift parameters a p and b p to lift degeneracy of the distribution Conditions to fix a p and b p to be specified later on Parameterization of the flow p considered as continuous rather than discrete change of flow parameter p = e s : distribution µ ( x , s ), parameters a ( s ) and b ( s ) Parent distribution µ ( x ) obtained for s = 0 µ ( x , 0) = µ ( x ) Eric Bertin Renormalization approach for extreme statistics
Renormalization operation Change of function double exponential form µ ( x , s ) = e − e − g ( x , s ) Link to the parent distribution: g ( x , s = 0) = g ( x ) Standardization conditions Conditions to fix the parameters a ( s ) and b ( s ) µ (0 , s ) ≡ e − 1 , ∂ x µ (0 , s ) ≡ e − 1 In terms of the function g ( x , s ) g (0 , s ) ≡ 0 , ∂ x g (0 , s ) ≡ 1 Eric Bertin Renormalization approach for extreme statistics
Renormalization operation Renormalization of µ ( x , s ) R s µ ]( x ) = µ e s � µ ( x , s ) ≡ [ˆ � a ( s ) x + b ( s ) Renormalization of g ( x , s ) = − ln[ − ln µ ( x , s )] � � g ( x , s ) = g a ( s ) x + b ( s ) − s . Very simple transformation: linear change of variable in the argument and global additive shift. However, one needs to determine a ( s ) and b ( s ). Eric Bertin Renormalization approach for extreme statistics
Differential representation Iteration of the RG transformation g ( x , s + ∆ s ) = [ˆ R ∆ s g ]( x , s ) Infinitesimal transformation ∆ s = ds g ( x , s + ds ) = [ˆ R ds g ]( x , s ) More explicitly, with a ( ds ) = 1 + γ ( s ) ds and b ( ds ) = η ( s ) ds : �� � � g ( x , s + ds ) = g 1 + γ ( s ) ds x + η ( s ) ds , s − ds where the functions γ ( s ) and η ( s ) are to be specified Linearizing with respect to ds , we get � � ∂ s g ( x , s ) = γ ( s ) x + η ( s ) ∂ x g ( x , s ) − 1 Eric Bertin Renormalization approach for extreme statistics
Partial differential equation Determination of γ ( s ) and η ( s ) Standardiz. conditions g (0 , s ) ≡ 0 and ∂ x g (0 , s ) ≡ 1 yield η ( s ) ≡ 1 γ ( s ) = − ∂ 2 x g (0 , s ) Partial differential equation of the flow ∂ s g ( x , s ) = (1 + γ ( s ) x ) ∂ x g ( x , s ) − 1 Eric Bertin Renormalization approach for extreme statistics
Fixed points of the flow Stationary solution g ( x , s ) = f ( x ): 0 = (1 + γ x ) f ′ ( x ) − 1 with γ = − f ′′ (0) Using the standardization condition f (0) = 0 � x (1 + γ y ) − 1 dy = 1 f ( x ; γ ) = γ ln(1 + γ x ) 0 Fixed point integrated distribution M ( x ; γ ) = e − e − f ( x ; γ ) = e − (1+ γ x ) − 1 /γ Easy way to recover the well-known generalized extreme value distributions, obtained here as a fixed line of the RG transformation Eric Bertin Renormalization approach for extreme statistics
Perturbations about a fixed point Linear perturbations Perturbation φ ( x , s ) introduced through g ( x , s ) = f ( x ) + f ′ ( x ) φ ( x , s ) Linearized partial differential equation ∂ s φ ( x , s ) = (1 + γ x ) ∂ x φ ( x , s ) − γ φ ( x , s ) − x ∂ 2 x φ (0 , s ) Convergence properties to the fixed point distribution are obtained from the analysis of this PDE Eric Bertin Renormalization approach for extreme statistics
Eigenfunctions Perturbations of the form φ ( x , s ) = e γ ′ s ψ ( x ) Solution for the Weibull and Fr´ echet cases ( γ � = 0 ) ψ ( x ; γ, γ ′ ) = 1 + ( γ ′ + γ ) x − (1 + γ x ) γ ′ /γ +1 γ ′ ( γ ′ + γ ) in the range of x such that 1 + γ x > 0. Solution for the Gumbel case ( γ = 0 ) ψ ( x ; γ ′ ) = 1 � 1 + γ ′ x − e γ ′ x � γ ′ 2 Eric Bertin Renormalization approach for extreme statistics
Eigenfunctions Empirical interpretation N variables in the block ⇒ s = ln N Convergence g ( x , s = ln N ) → f ( x ) Corrections proportional to e γ ′ s ∝ N γ ′ (if γ ′ = 0: logarithmic convergence in N ). Interpretation of γ ′ > 0? Are there unstable solutions? ⇒ Can we look at non-perturbative solutions? Eric Bertin Renormalization approach for extreme statistics
Non-perturbative solutions Motivation Unstable solutions around the fixed point may seem counterintuitive: can we find an example of full RG trajectory starting from an unstable direction? Back to the equations: the Gumbel case Equation to be solved ∂ s g ( x , s ) = (1 + γ ( s ) x ) ∂ x g ( x , s ) − 1 Ansatz for the solution starting from f ( x ) = x x ; γ ′ ( s ) � � g ( x , s ) = x + ǫ ( s ) ψ Same as linear perturbation, except that γ ′ depends on s Eric Bertin Renormalization approach for extreme statistics
Illustration of the flow Parameter space ( ǫ, γ ′ ) 1.5 1.5 1.5 1.5 1.5 II II II II II I I I I I A= 1 A=-1 1 1 1 1 1 e e e e e A= 0 A= ∞ 0.5 0.5 0.5 0.5 0.5 A=- ∞ f f f f f b b b b b 0 0 0 0 0 ε ε ε ε ε a a a a a d d d d d -0.5 -0.5 -0.5 -0.5 -0.5 -1 -1 -1 -1 -1 c c c c c III III III III III IV IV IV IV IV -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1.5 -1 -1 -1 -1 -1 -0.5 -0.5 -0.5 -0.5 -0.5 0 0 0 0 0 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 γ ’ γ ’ γ ’ γ ’ γ ’ Eric Bertin Renormalization approach for extreme statistics
Evolution of the distributions Starting close to the Gumbel distribution ( γ ′ = 2)... and coming back to it (at γ ′ = 0) after an excursion 0.5 0.5 γ ’=2 γ ’=1 1.8 0.8 0.4 0.4 1.6 0.6 1.4 0.4 0.3 1.2 0.3 0.2 ρ (x; γ ’) ρ (x; γ ’) 1 0 0.2 0.2 0.1 0.1 0 0 -2 0 2 4 -2 0 2 4 x x Bertin, Gy¨ orgyi, J. Stat. Mech. (2010) Eric Bertin Renormalization approach for extreme statistics
Generalization of standard extreme statistics Raising the variables to an increasing power Choose iid variables whose statistics depends on the sample size n , for instance by raising x 1 , . . . , x n to a power q n Question: statistics of the quantity max( x q n 1 , . . . , x q n n ) Motivation: link with the Random Energy Model Ben Arous et.al. (2005), Bogachev (2007) Results Emergence of new limit distributions, for q ( n ) ∼ n Q � � 1 / Q � � 1 − Q F γ, Q = exp γ ln(1 + γ x ) − Standard distributions recovered for Q → 0 Angeletti, Bertin, Abry, J. Phys. A (2012) Eric Bertin Renormalization approach for extreme statistics
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