The Fyodorov-Bouchaud formula and Liouville conformal field theory

Transcription

1 The Fyodorov-Bouchaud formula and Liouville conformal field theory Guillaume Remy École Normale Supérieure February 1, 218 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

2 Introduction Statistical Physics: Log-correlated fields Gaussian multiplicative chaos (GMC) The Fyodorov-Bouchaud formula Liouville Field Theory: Conformal field theory Correlation functions and BPZ equation Integrability: DOZZ formula Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

3 Introduction DKRV 214, link between: Gaussian multiplicative chaos Liouville correlations Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

4 Introduction DKRV 214, link between: Gaussian multiplicative chaos Liouville correlations Consequences: Proof of the DOZZ formula KRV 217 Proof of the Fyodorov-Bouchaud formula R 217 Integrability program for GMC and Liouville theory Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

5 Outline 1 Definitions, main result and applications 2 Proof of the Fyodorov-Bouchaud formula 3 The Liouville conformal field theory 4 Outlook and perspectives Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

6 Gaussian Free Field (GFF) Gaussian free field X on the unit circle D E[X (e iθ )X (e iθ )] = 2 ln 1 e iθ e iθ X (e iθ ) has an infinite variance X lives in the space of distributions Cut-off approximation X ɛ Ex: X ɛ = ρ ɛ X, ρ ɛ = 1 ɛ ρ( ɛ ), with smooth ρ. Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

7 Gaussian multiplicative chaos (GMC) For γ (, 2), define on D the measure e γ 2 X dθ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

8 Gaussian multiplicative chaos (GMC) For γ (, 2), define on D the measure e γ 2 X dθ Cut-off approximation e γ 2 X ɛ dθ E[e γ 2 X ɛ ] = e γ2 8 E[X 2 ɛ ] Renormalized measure: e γ 2 X ɛ γ2 8 E[X 2 ɛ ] dθ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

9 Gaussian multiplicative chaos (GMC) For γ (, 2), define on D the measure e γ 2 X dθ Cut-off approximation e γ 2 X ɛ dθ E[e γ 2 X ɛ ] = e γ2 8 E[X 2 ɛ ] Renormalized measure: e γ 2 X ɛ γ2 8 E[X 2 ɛ ] dθ Proposition (Kahane 1985) The following limit exists in the sense of weak convergence of measures, γ (, 2): e γ 2 X (eiθ) dθ := lim e γ 2 X ɛ(e iθ ) γ 2 8 E[X ɛ 2 (e iθ )] dθ ɛ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

10 Illustrations of the GMC measures Simulations of the measure e γx dx 2 on [, 1] 2. γ = 1 γ = 1.8 Simulations by Tunan Zhu Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

11 Moments of the GMC We introduce: γ (, 2), Y γ := 1 2π Existence of the moments of Y γ : 2π e γ 2 X (eiθ) dθ E[Y p γ ] < + p < 4 γ 2. Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

12 The Fyodorov-Bouchaud formula Theorem (Remy 217) Let γ (, 2) and p (, 4 γ 2 ), then: E[Y p γ ] = We also have a density for Y γ, Γ(1 p γ2 4 ) Γ(1 γ2 4 )p f Yγ (y) = 4β γ 2 (βy) 4 γ 2 1 e (βy) 4 γ 2 1 [, [ (y), where we have set β = Γ(1 γ2 4 ). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

13 Some applications 1 Maximum of the Gaussian free field 2 Random matrix theory 3 Tail expansion for Gaussian multiplicative chaos Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

14 Application 1: maximum of the GFF Derivative martingale: work by Duplantier, Rhodes, Sheffield, Vargas. γ 2 in our GMC measure (Aru, Powell, Sepúlveda): Y := 1 2π X (e iθ )e X (eiθ) 1 dθ = lim 2π γ 2 2 γ Y γ. 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 February 1, / 39

15 Application 1: maximum of the GFF Following an impressive series of works (216): Theorem (Ding, Madaule, Roy, Zeitouni) For a reasonable cut-off X ɛ of the GFF: max X ɛ(e iθ ) 2 ln 1 θ [,2π] ɛ ln ln 1 ɛ G + ln Y + C ɛ where G is a standard Gumbel law and C R. Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

16 Application 1: maximum of the GFF The Fyodorov-Bouchaud formula implies: Corollary (Remy 217) For a reasonable cut-off X ɛ of the GFF: max X ɛ(e iθ ) 2 ln 1 θ [,2π] ɛ ln ln 1 ɛ G 1 + G 2 + C ɛ where G 1, G 2 are independent Gumbel laws and C R. Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

17 Application 2: random unitary matrices U N := N N random unitary matrix Its eigenvalues (e iθ 1,..., eiθ n ) follow the distribution: 1 n! n e iθ k e iθ j 2 k<j k=1 dθ k 2π Let p N (θ) = det(1 e iθ U N ) = N k=1 (1 ei(θ k θ) ) Webb (215): α ( 1 2, 2), p N (θ) α E[ p N (θ) α ] dθ e α 2 X (eiθ) dθ N Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

18 Application 2: random unitary matrices Conjecture by Fyodorov, Hiary, Keating (212): max ln p N(θ) ln N + 3 ln ln N θ [,2π] 4 G 1 + G 2 + C. N Chhaibi, Madaule, Najnudel (216), tightness of: max ln p N(θ) ln N + 3 ln ln N. θ [,2π] 4 With our result it is sufficient to show: max ln p N(θ) ln N + 3 ln ln N θ [,2π] 4 G 1 + ln Y + C. N Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

19 Application 3: Tail estimates for GMC E[ X (e iθ ) X (e iθ )] = 2 ln 1 e iθ e iθ + f (eiθ, e iθ ) Proposition (Rhodes, Vargas 217) O D, δ > : P( e γ X 2 (e iθ) dθ > t) = C(γ) + o γ 2 O t 4 where C(γ) = R 1 (γ) O e( 4 γ 2 1)f (eiθ,e iθ) dθ 4 t (t γ 2 δ ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

20 Application 3: Tail estimates for GMC E[ X (e iθ ) X (e iθ )] = 2 ln 1 e iθ e iθ + f (eiθ, e iθ ) Corollary (Remy 217) O D, δ > : P( e γ X 2 (e iθ) dθ > t) = C(γ) + o γ 2 O where C(γ) = (2π) 4 γ 2 1 Γ(1 γ2 4 4 ) γ 2 t 4 4 O e( γ 2 1)f (eiθ,e iθ) dθ 4 t (t γ 2 δ ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

21 Integer moments of the GMC The computation of Fyodorov and Bouchaud Fyodorov Y., 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, (28). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

22 Integer moments of the GMC For n N, n < 4 γ 2 : E[( 1 2π e γ 2 X ɛ(e iθ ) γ2 8 E[X ɛ(e iθ ) 2] dθ) n ] 2π n = 1 (2π) n = 1 (2π) n [,2π] n E[ [,2π] n e γ 4 i=1 2 e γ 2 X ɛ(e iθ i ) γ2 8 E[X ɛ(e iθ i ) 2] ]dθ 1... dθ n i<j E[X ɛ(e iθ i )X ɛ (e iθ j )] dθ 1... dθ n Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

23 Integer moments of the GMC For n N, n < 4 γ 2 : E[Yγ n ] = 1 (2π) n = 1 (2π) n = Γ(1 n γ2 4 ) Γ(1 γ2 4 )n [,2π] n e γ 4 2 [,2π] n i<j i<j E[X (eiθ i )X (e iθ j )] dθ 1... dθ n 1 e iθ i e iθ γ j 2 2 dθ 1... dθ n Question: can we replace n N by a real p < 4 γ 2? Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

24 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, , (1984). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

25 The BPZ differential equation We introduce the following observable for t [, 1]: 2π G(γ, p, t) = E[( t e iθ γ2 γ 2 e 2 X (eiθ) dθ) p ] Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

26 The BPZ differential equation We introduce the following observable for t [, 1]: 2π G(γ, p, t) = E[( t e iθ γ2 γ 2 e 2 X (eiθ) dθ) p ] BPZ equation: (t(1 t 2 ) 2 t 2 +(t2 1) t +2(C (A+B+1)t2 ) 4ABt)G(γ, p, t) = t where: A = γ2 p 4, B = γ2 4, C = γ2 (1 p) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

27 Solutions of the BPZ equation BPZ equation in t hypergeometric equation in t 2 Two bases of solutions: G(γ, p, t) = C 1 F 1 (t 2 ) + C 2 t γ2 2 (p 1) F 2 (t 2 ) G(γ, p, t) = B 1 F 1 (1 t 2 ) + B 2 (1 t 2 γ2 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 February 1, / 39

28 Link between C 1, C 2, B 1 and B 2 Goal: identify C 1, C 2, B 1, B 2 G(γ, p, t) = E[( 2π t e iθ γ 2 2 e γ 2 X (θ) dθ) p ] Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

29 Link between C 1, C 2, B 1 and B 2 Goal: identify C 1, C 2, B 1, B 2 G(γ, p, t) = E[( 2π t e iθ γ 2 2 e γ 2 X (θ) dθ) p ] C 1 = G(γ, p, ) = (2π) p E[Y p γ ] Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

30 Link between C 1, C 2, B 1 and B 2 Goal: identify C 1, C 2, B 1, B 2 G(γ, p, t) = E[( 2π t e iθ γ 2 2 e γ 2 X (θ) dθ) p ] C 1 = G(γ, p, ) = (2π) p E[Y p γ ] B 1 = G(γ, p, 1) = E[( 2π 1 e iθ γ 2 2 e γ 2 X (θ) dθ) p ] Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

31 Link between C 1, C 2, B 1 and B 2 Goal: identify C 1, C 2, B 1, B 2 G(γ, p, t) = E[( 2π t e iθ γ 2 2 e γ 2 X (θ) dθ) p ] C 1 = G(γ, p, ) = (2π) p E[Y p γ ] B 1 = G(γ, p, 1) = E[( 2π 1 e iθ γ 2 2 e γ 2 X (θ) dθ) p ] G(γ, p, t) G(γ, p, ) = O(t 2 ) C 2 = Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

32 Link between C 1, C 2, B 1 and B 2 Goal: identify C 1, C 2, B 1, B 2 G(γ, p, t) = E[( 2π t e iθ γ 2 2 e γ 2 X (θ) dθ) p ] C 1 = G(γ, p, ) = (2π) p E[Y p γ ] B 1 = G(γ, p, 1) = E[( 2π 1 e iθ γ 2 2 e γ 2 X (θ) dθ) p ] G(γ, p, t) G(γ, p, ) = O(t 2 ) C 2 = B 2 =?? Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

33 The value of B 2 Let h u (t) = t e iu γ2 2 G(γ, p, t) G(γ, p, 1) = p 2π du(h u (t) h u (1))c(e iu ) + R(t) with c(e iu ) = E[( 2π 1 e iθ γ2 2 e iu e iθ γ2 2 e γ 2 X (θ) dθ) p 1 ] Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

34 The value of B 2 Let h u (t) = t e iu γ2 2 G(γ, p, t) G(γ, p, 1) = p 2π du(h u (t) h u (1))c(e iu ) + R(t) with c(e iu ) = E[( 2π 1 2π 2π 1 e iθ γ2 2 e iu e iθ γ2 2 e γ 2 X (θ) dθ) p 1 ] du(h u (t) h u (1)) = ˆF 1 (t) + Γ( γ 2 2 1) Γ( γ2 4 )2 (1 t2 ) 1+ γ 2 2 ˆF2 (t) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

35 The value of B 2 Let h u (t) = t e iu γ2 2 G(γ, p, t) G(γ, p, 1) = p 2π du(h u (t) h u (1))c(e iu ) + R(t) with c(e iu ) = E[( 2π 1 2π 2π 1 2π 2π 1 e iθ γ2 2 e iu e iθ γ2 2 e γ 2 X (θ) dθ) p 1 ] du(h u (t) h u (1)) = ˆF 1 (t) + Γ( γ du(h u (t) h u (1))c(e iu ) = A(1 t) + Γ( γ 2 2 1) Γ( γ2 4 )2 (1 t2 ) ) Γ( γ2 4 )2 (1 t2 ) 1+ γ 2 2 ˆF2 (t) γ 2 γ c(1) + o((1 t) 2 ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

36 The value of B 2 Let h u (t) = t e iu γ2 2 G(γ, p, t) G(γ, p, 1) = p 2π du(h u (t) h u (1))c(e iu ) + R(t) with c(e iu ) = E[( 2π 1 2π 2π 1 2π 2π 1 e iθ γ2 2 e iu e iθ γ2 2 e γ 2 X (θ) dθ) p 1 ] du(h u (t) h u (1)) = ˆF 1 (t) + Γ( γ du(h u (t) h u (1))c(e iu ) = A(1 t) + Γ( γ 2 2 1) Γ( γ2 4 )2 (1 t2 ) ) Γ( γ2 4 )2 (1 t2 ) 1+ γ2 = B 2 = (2π) p Γ( 2 p 1) Γ( γ2 ) E[Yγ p 1 ] 4 γ 2 2 ˆF2 (t) γ 2 γ c(1) + o((1 t) 2 ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

37 The shift relation Change of basis: ( ) ( ) ( ) B1 m11 m = 12 C1 B 2 m 21 m 22 C 2 and C 2 = Therefore: B 2 = E[Y p γ ] = γ2 Γ( 1 2 )Γ(γ2 4 (1 p) + 1) Γ( γ2 p 4 )Γ( γ2 4 ) C 1 Γ(1 γ2 4 Γ(1 p γ2 4 ) )Γ(1 (p 1)γ2 4 )E[Y p 1 γ ]. Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

38 Negative moments of GMC The shift relation gives all the negative moments: E[Y n γ We check: λ R, ] = Γ(1 + nγ2 γ2 )Γ(1 4 4 )n, n N. n= λ n n! nγ2 γ2 Γ(1 + )Γ(1 4 4 )n < + Negative moments determine the law of Y γ! Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

39 Explicit probability densities Probability densities for Y 1 γ and Y γ f 1 (y) = 4 Yγ βγ 2(y β ) 4 γ 2 1 e ( y 4 β ) γ 2 1 [, [ (y) f Yγ (y) = 4β γ 2 (βy) 4 γ 2 1 e (βy) 4 γ 2 1 [, [ (y) where γ (, 2) and β = Γ(1 γ2 4 ). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

40 Generalizations to log-singularities 1) Degenerate insertions γ 2 and 2 γ E[( 1 2π 2π 1 e iθ γ2 γ2 γ 2 e 2 X (eiθ) dθ) p Γ(1 4 ] = p) Γ(1 + Γ(1 γ2 4 )p γ 2 2 ) γ 2 4 ) Γ(1 + (1 p) Γ(1 + γ2 4 ) Γ(1 + (2 p) γ2 4 ) E[( 1 2π 2π 1 e iθ 2 e γ 2 X (eiθ) dθ) p ] = Γ(1 γ 2 4 p) Γ(1 + 8 γ ) Γ( 4 2 γ p + 1) 2 Γ(1 γ2 4 )p Γ(1 + 4 γ ) Γ( 8 2 γ p + 1) 2 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

41 Generalizations to log-singularities 1) Degenerate insertions γ 2 and 2 γ 1 2π 1 e iθ γ2 γ 2 e 2π 2 X (eiθ) dθ law = Y γ X γ π 2π 1 e iθ 2 e γ 2 X (eiθ) dθ law = Y γ X 1 2 Y γ, X 1, X 2 independent, X 1 B(1 + γ2 4, γ2 4 ), X 2 B(1 + 4 γ 2, 4 γ 2 ). f B(α,β) (x) = Γ(α+β) Γ(α)Γ(β) x α 1 (1 x) β 1 1 [,1] (x) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

42 Generalizations to log-singularities 2) Arbitrary log-singularity, α < Q, p < 4 γ 2 2 γ (Q α), ln T γ (x) = E[( 1 2π 2π 1 1 e iθ αγ e γ 2 X (eiθ) dθ) p ] = Γ(1 γ2 p 4 ) T γ (α + Γ(1 γ2 4 )p dt t and Q = γ γ. ( e Qt 2 e (Q x)t (1 e γt 2 )(1 e 2t γ ) T γ (2α + γp Work in progress with Tunan Zhu. γp 2 )2 T γ (2α)T γ () 2 )T γ(α) 2 T γ ( γp 2 ) ) e t ( Q 2 2 x)2 Q 2x 2t Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

43 What is Liouville field theory? Path integral formalism Σ = {X : D R} Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

44 What is Liouville field theory? Path integral formalism Σ = {X : D R} For X Σ, energy of X := 1 4π D X 2 dx 2 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

45 What is Liouville field theory? Path integral formalism Σ = {X : D R} For X Σ, energy of X := 1 4π D X 2 dx 2 Random field φ G : E[F (φ G )] = F (X )e 1 4π D X 2 dx 2 DX Σ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

46 What is Liouville field theory? Path integral formalism Σ = {X : D R} For X Σ, energy of X := 1 4π D X 2 dx 2 Random field φ G : E[F (φ G )] = F (X )e 1 4π D X 2 dx 2 DX Informally: X 2 dx 2 = X Xdx 2 Σ E[φ G (x)φ G (y)] = 1 (x, y) ln 1 x y φ G is a GFF! Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

47 What is Liouville field theory? Path integral formalism Σ = {X : D R} For X Σ, energy of X := 1 4π D X 2 dx 2 + D e γ 2 X ds Random field φ L : E[F (φ L )] = F (X )e 1 4π D X 2 dx 2 D e γ 2 X ds DX with γ (, 2). φ L is the Liouville field Σ Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

48 Correlations of Liouville theory Liouville theory is a conformal field theory Correlation function of z i D, α i R: N e α i φ L (z i ) D = i=1 Σ DX N i=1 e α i X (z i ) e 1 4π D X 2 dx 2 D e γ 2 X ds Expressed as moments of the Gaussian multiplicative chaos Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

49 Correlations of Liouville theory Degenerate fields: e γ 2 φ L(z) and e 2 γ φ L(z). BPZ equation, for z 1, z D, α R, γ (, 2): (z 1, z) e γ 2 φ L(z) e αφ L(z 1 ) D is solution of a differential equation. e γ 2 φ L(z) e αφ L(z 1 ) D = Ct αγ 2 (1 t 2 ) γ2 8 G(γ, p, t) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

50 BPZ equation on the upper half plane H Proposition (Remy 217) Let γ (, 2) and α > Q + γ 2. Then: ( 4 γ 2 zz + γ 2 (z z) 2 + α (z z 1 ) 2 + α (z z 1 ) z z z + 1 z1 + 1 z1 ) e γ 2 φl(z) e αφ L(z 1 ) H = z z 1 z z 1 where Q = γ γ, α = α 2 (Q α 2 ), γ 2 = γ 4 (Q + γ 4 ). differential equation for G(p, γ, t). Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

51 Outlook and perspectives Work in progress, analogue for the bulk meaure. For γ (, 2), α ( γ 2, Q): 1 E[( (x) dx 2 ) Q α x γαeγx γ ] = γ 2 (π D Γ(γ2 4 ) ))Q α γ Γ(1 γ2 4 cos( α Q γ π) Γ(2α γ 4 γ )Γ( γ 2 2 (α Q)) Γ( α Q γ ) Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

52 Outlook and perspectives Work in progress, analogue for the bulk meaure. For γ (, 2), α ( γ 2, Q): 1 E[( (x) dx 2 ) Q α x γαeγx γ ] = γ 2 (π D Γ(γ2 4 ) ))Q α γ Γ(1 γ2 4 cos( α Q γ π) Γ(2α γ 4 γ )Γ( γ 2 2 (α Q)) Γ( α Q γ ) Liouville theory with action: D ( X 2 + e γx )dx 2 Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

53 Outlook and perspectives Integrability program for GMC and Liouville theory: GMC on the unit interval [, 1] More general Liouville correlations Other geometries Link with planar maps Conformal bootstrap Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39

54 Thank you! Kupiainen A., Rhodes R., Vargas V.: Local conformal structure of Liouville quantum gravity, arxiv: Kupiainen A., Rhodes R., Vargas V.: Integrability of Liouville theory: Proof of the DOZZ formula, arxiv: Remy G.: The Fyodorov-Bouchaud formula and Liouville conformal field theory, arxiv: Guillaume Remy (ENS) The Fyodorov-Bouchaud formula February 1, / 39