The Fyodorov-Bouchaud formula and Liouville conformal field theory - PowerPoint PPT Presentation

The Fyodorov-Bouchaud formula and Liouville conformal field theory Guillaume Remy ´ Ecole Normale Sup´ erieure June 22, 2018 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 1 / 28

Introduction Two fields of physics: Log-correlated fields, Gaussian multiplicative chaos (GMC) Liouville conformal field theory (LCFT) DKRV 2014: link between GMC and LCFT Why is this link interesting ? GMC theory ⇒ Rigorous definition of Liouville CFT CFT techniques ⇒ Exact formulas on GMC DOZZ formula / Fyodorov-Bouchaud formula Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 2 / 28

Gaussian Free Field (GFF) Gaussian free field X on the unit circle ∂ D 1 E [ X ( e i θ ) X ( e i θ ′ )] = 2 ln | e i θ − e i θ ′ | X ( e i θ ) has an infinite variance X lives in the space of distributions Cut-off approximation X ǫ Ex: X ǫ = ρ ǫ ∗ X , ρ ǫ = 1 ǫ ρ ( · ǫ ) , with smooth ρ . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 3 / 28

Gaussian multiplicative chaos (GMC) γ 2 X d θ For γ ∈ (0 , 2), define on ∂ D the measure e γ 2 X ǫ d θ Cut-off approximation e γ 2 γ 8 E [ X 2 2 X ǫ ] = e ǫ ] E [ e 2 X ǫ − γ 2 γ 8 E [ X 2 ǫ ] d θ Renormalized measure: e Proposition The following limit holds in probability, for any continuous test function f , ∀ γ ∈ (0 , 2): � 2 π � 2 π 2 X ǫ ( e i θ ) − γ 2 γ 2 X ( e i θ ) f ( θ ) d θ = lim γ 8 E [ X 2 ǫ ( e i θ )] f ( θ ) d θ e e ǫ → 0 0 0 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 4 / 28

Moments of the GMC We introduce: � 2 π ∀ γ ∈ (0 , 2) , Y γ := 1 γ 2 X ( e i θ ) d θ e 2 π 0 Existence of the moments of Y γ : ⇒ p < 4 E [ Y p γ ] < + ∞ ⇐ γ 2 . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 5 / 28

The Fyodorov-Bouchaud formula Theorem (R. 2017) Let γ ∈ (0 , 2) and p ∈ ( −∞ , 4 γ 2 ), then: γ ] = Γ(1 − p γ 2 4 ) E [ Y p Γ(1 − γ 2 4 ) p We also have a density for Y γ , − 4 f Y γ ( y ) = 4 β γ 2 1 [0 , ∞ [ ( y ) , γ 2 ( β y ) − 4 γ 2 − 1 e − ( β y ) β Exp(1) − γ 2 where β = Γ(1 − γ 2 law = 1 4 . 4 ). Equivalently Y γ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 6 / 28

Application 1: maximum of the GFF Derivative martingale: work by Duplantier, Rhodes, Sheffield, Vargas. γ → 2 in our GMC measure (Aru, Powell, Sep´ ulveda): 1 Y ′ := lim 2 − γ Y γ . γ → 2 ln Y ′ has the following density: f ln Y ′ ( y ) = e − y e − e − y ln Y ′ ∼ G where G follows a standard Gumbel law Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 7 / 28

Application 1: maximum of the GFF Following an impressive series of works (2016): Theorem (Ding, Madaule, Roy, Zeitouni) For a reasonable cut-off X ǫ of the GFF: θ ∈ [0 , 2 π ] X ǫ ( e i θ ) − 2 ln 1 ǫ + 3 2 ln ln 1 ǫ → 0 G + ln Y ′ + C max ǫ → where G is a standard Gumbel law and C ∈ R . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 8 / 28

Application 1: maximum of the GFF The Fyodorov-Bouchaud formula implies: Corollary (R 2017) For a reasonable cut-off X ǫ of the GFF: θ ∈ [0 , 2 π ] X ǫ ( e i θ ) − 2 ln 1 ǫ + 3 2 ln ln 1 max ǫ → ǫ → 0 G 1 + G 2 + C where G 1 , G 2 are independent Gumbel laws and C ∈ R . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 9 / 28

Application 2: random unitary matrices U N := N × N random unitary matrix Its eigenvalues ( e i θ 1 , . . . , e i θ n ) follow the distribution: n 1 d θ k | e i θ k − e i θ j | 2 � � n ! 2 π k < j k =1 Let p N ( θ ) = det(1 − e − i θ U N ) = � N k =1 (1 − e i ( θ k − θ ) ) √ Webb (2015): ∀ α ∈ ( − 1 2 , 2), | p N ( θ ) | α α 2 X ( e i θ ) d θ E [ | p N ( θ ) | α ] d θ → N →∞ e Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 10 / 28

Application 2: random unitary matrices Conjecture by Fyodorov, Hiary, Keating (2012): θ ∈ [0 , 2 π ] ln | p N ( θ ) | − ln N + 3 max 4 ln ln N N →∞ G 1 + G 2 + C . → Chhaibi, Madaule, Najnudel (2016), tightness of: θ ∈ [0 , 2 π ] ln | p N ( θ ) | − ln N + 3 max 4 ln ln N . With our result it is sufficient to show: θ ∈ [0 , 2 π ] ln | p N ( θ ) | − ln N + 3 N →∞ G 1 + ln Y ′ + C . max 4 ln ln N → Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 11 / 28

Integer moments of the GMC The computation of Fyodorov and Bouchaud Fyodorov Y.V., Bouchaud J.P.: Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential, Journal of Physics A: Mathematical and Theoretical , Volume 41, Number 37, (2008). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 12 / 28

Integer moments of the GMC For n ∈ N ∗ , n < 4 γ 2 : � 2 π E [( 1 2 X ǫ ( e i θ ) − γ 2 γ 8 E [ X ǫ ( e i θ ) 2 ] d θ ) n ] e 2 π 0 n 1 � 2 X ǫ ( e i θ i ) − γ 2 γ 8 E [ X ǫ ( e i θ i ) 2 ] ] d θ 1 . . . d θ n � = [0 , 2 π ] n E [ e (2 π ) n i =1 1 � γ 2 i < j E [ X ǫ ( e i θ i ) X ǫ ( e i θ j )] d θ 1 . . . d θ n � = [0 , 2 π ] n e 4 (2 π ) n Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 13 / 28

Integer moments of the GMC For n ∈ N ∗ , n < 4 γ 2 : 1 � γ 2 i < j E [ X ( e i θ i ) X ( e i θ j )] d θ 1 . . . d θ n E [ Y n � γ ] = [0 , 2 π ] n e 4 (2 π ) n 1 1 � � = d θ 1 . . . d θ n (2 π ) n γ 2 | e i θ i − e i θ j | [0 , 2 π ] n 2 i < j = Γ(1 − n γ 2 4 ) Γ(1 − γ 2 4 ) n Question: can we replace n ∈ N ∗ by a real p < 4 γ 2 ? Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 14 / 28

Proof of the Fyodorov-Bouchaud formula Framework of conformal field theory Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear. Physics. , B241, 333-380, (1984). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 15 / 28

The BPZ differential equation We introduce the following observable for t ∈ [0 , 1]: � 2 π γ 2 γ 2 X ( e i θ ) d θ ) p ] | t − e i θ | 2 e G ( γ, p , t ) = E [( 0 BPZ equation: ( t (1 − t 2 ) ∂ 2 ∂ t 2 +( t 2 − 1) ∂ ∂ t +2( C − ( A + B +1) t 2 ) ∂ ∂ t − 4 ABt ) G ( γ, p , t ) = 0 where: A = − γ 2 p 4 , B = − γ 2 4 , C = γ 2 4 (1 − p ) + 1 . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 16 / 28

Solutions of the BPZ equation BPZ equation in t → hypergeometric equation in t 2 Two bases of solutions: γ 2 G ( γ, p , t ) = C 1 F 1 ( t 2 ) + C 2 t 2 ( p − 1) F 2 ( t 2 ) F 1 (1 − t 2 ) + B 2 (1 − t 2 ) 1+ γ 2 G ( γ, p , t ) = B 1 ˜ 2 ˜ F 2 (1 − t 2 ) where: C 1 , C 2 , B 1 , B 2 ∈ R F 1 , F 2 , ˜ F 1 , ˜ F 2 := hypergeometric series depending on γ and p . Change of basis: ( C 1 , C 2 ) ↔ ( B 1 , B 2 ). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 17 / 28

The shift relation By direct asymptotic expansion: C 1 = (2 π ) p E [ Y p γ ] C 2 = 0 � 2 π γ 2 γ 2 X ( θ ) d θ ) p ] | 1 − e i θ | 2 e B 1 = E [( 0 B 2 = (2 π ) p p Γ( − γ 2 2 − 1) 4 ) E [ Y p − 1 ] Γ( − γ 2 γ The change of basis implies: Γ(1 − p γ 2 4 ) E [ Y p − 1 E [ Y p γ ] = ] . γ Γ(1 − γ 2 4 )Γ(1 − ( p − 1) γ 2 4 ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 18 / 28

Negative moments of GMC The shift relation gives all the negative moments: γ ] = Γ(1 + n γ 2 4 )Γ(1 − γ 2 E [ Y − n 4 ) n , ∀ n ∈ N . We check: ∞ λ n n ! Γ(1 + n γ 2 4 )Γ(1 − γ 2 4 ) n < + ∞ � ∀ λ ∈ R , n =0 Negative moments ⇒ determine the law of Y γ ! Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 19 / 28

Explicit probability densities Probability densities for Y − 1 and Y γ γ 4 4 γ 2 1 [0 , ∞ [ ( y ) βγ 2 ( y 4 γ 2 − 1 e − ( y β ) f 1 Y γ ( y ) = β ) − 4 γ 2 1 [0 , ∞ [ ( y ) f Y γ ( y ) = 4 β γ 2 ( β y ) − 4 γ 2 − 1 e − ( β y ) where γ ∈ (0 , 2) and β = Γ(1 − γ 2 4 ). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 20 / 28

What is Liouville field theory? Path integral formalism Σ = { X : D → R } D | ∂ X | 2 dx 2 + γ 1 � � 2 X ds For X ∈ Σ, energy of X := ∂ D e 4 π Random field φ L : � γ F ( X ) e − 1 D | ∂ X | 2 dx 2 − 2 X ds DX � � ∂ D e E [ F ( φ L )] = 4 π Σ with γ ∈ (0 , 2). ⇒ φ L is the Liouville field Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 21 / 28

Correlations of Liouville theory Correlation function of z i ∈ D , α i ∈ R : N N � γ 2 X ds e α i X ( z i ) e − 1 D | ∂ X | 2 dx 2 − � � � � e α i φ L ( z i ) � D = ∂ D e � DX 4 π X : D �→ R i =1 i =1 Expressed in terms of Gaussian multiplicative chaos � e αφ L (0) � D = ˜ C 1 E [ Y p − 1 ] γ 2 (1 − t 2 ) − γ 2 � e αφ L (0) e − γ αγ 2 φ L ( t ) � D = ˜ 8 G ( γ, p , t ) C 2 t with p = 2 − 2 α − 4 γ 2 . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 22 / 28