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

SLIDE 1

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

SLIDE 2

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

SLIDE 3

Gaussian Free Field (GFF)

Gaussian free field X on the unit circle ∂D E[X(eiθ)X(eiθ′)] = 2 ln 1 |eiθ − eiθ′| X(eiθ) has an infinite variance X lives in the space of distributions Cut-off approximation Xǫ Ex: Xǫ = ρǫ ∗ X, ρǫ = 1

ǫρ( · ǫ), with smooth ρ.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 3 / 28

SLIDE 4

Gaussian multiplicative chaos (GMC)

For γ ∈ (0, 2), define on ∂D the measure e

γ 2 Xdθ

Cut-off approximation e

γ 2 Xǫdθ

E[e

γ 2 Xǫ] = e γ2 8 E[X 2 ǫ ]

Renormalized measure: e

γ 2 Xǫ− γ2 8 E[X 2 ǫ ]dθ

Proposition

The following limit holds in probability, for any continuous test function f , ∀γ ∈ (0, 2): 2π e

γ 2 X(eiθ)f (θ)dθ = lim

ǫ→0

2π e

γ 2 Xǫ(eiθ)− γ2 8 E[X 2 ǫ (eiθ)]f (θ)dθ

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 4 / 28

SLIDE 5

Moments of the GMC

We introduce: ∀γ ∈ (0, 2), Yγ := 1 2π 2π e

γ 2 X(eiθ)dθ

Existence of the moments of Yγ: E[Y p

γ ] < +∞ ⇐

⇒ p < 4 γ2.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 5 / 28

SLIDE 6

The Fyodorov-Bouchaud formula

Theorem (R. 2017)

Let γ ∈ (0, 2) and p ∈ (−∞, 4

γ2), then:

E[Y p

γ ] = Γ(1 − p γ2 4 )

Γ(1 − γ2

4 )p

We also have a density for Yγ, fYγ(y) = 4β γ2 (βy)− 4

γ2 −1e−(βy) − 4 γ2 1[0,∞[(y),

where β = Γ(1 − γ2

4 ). Equivalently Yγ law

= 1

βExp(1)− γ2

4 .

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 6 / 28

SLIDE 7

Application 1: maximum of the GFF

Derivative martingale: work by Duplantier, Rhodes, Sheffield, Vargas. γ → 2 in our GMC measure (Aru, Powell, Sep´ ulveda): Y ′ := lim

γ→2

1 2 − γYγ. ln Y ′ has the following density: fln Y ′(y) = e−ye−e−y ln Y ′ ∼ G where G follows a standard Gumbel law

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 7 / 28

SLIDE 8

Application 1: maximum of the GFF

Following an impressive series of works (2016):

Theorem (Ding, Madaule, Roy, Zeitouni)

For a reasonable cut-off Xǫ of the GFF: max

θ∈[0,2π] Xǫ(eiθ) − 2 ln 1

ǫ + 3 2 ln ln 1 ǫ →

ǫ→0 G + ln Y ′ + C

where G is a standard Gumbel law and C ∈ R.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 8 / 28

SLIDE 9

Application 1: maximum of the GFF

The Fyodorov-Bouchaud formula implies:

Corollary (R 2017)

For a reasonable cut-off Xǫ of the GFF: max

θ∈[0,2π] Xǫ(eiθ) − 2 ln 1

ǫ + 3 2 ln ln 1 ǫ →

ǫ→0 G1 + G2 + C

where G1, G2 are independent Gumbel laws and C ∈ R.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 9 / 28

SLIDE 10

Application 2: random unitary matrices

UN := N × N random unitary matrix Its eigenvalues (eiθ1, . . . , eiθn) follow the distribution: 1 n!

k<j

|eiθk − eiθj|2

n

k=1

dθk 2π Let pN(θ) = det(1 − e−iθUN) = N

k=1(1 − ei(θk−θ))

Webb (2015): ∀α ∈ (−1

2,

√ 2), |pN(θ)|α E[|pN(θ)|α]dθ →

N→∞ e

α 2 X(eiθ)dθ

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 10 / 28

SLIDE 11

Application 2: random unitary matrices

Conjecture by Fyodorov, Hiary, Keating (2012): max

θ∈[0,2π] ln |pN(θ)| − ln N + 3

4 ln ln N →

N→∞ G1 + G2 + C.

Chhaibi, Madaule, Najnudel (2016), tightness of: max

θ∈[0,2π] ln |pN(θ)| − ln N + 3

4 ln ln N. With our result it is sufficient to show: max

θ∈[0,2π] ln |pN(θ)| − ln N + 3

4 ln ln N →

N→∞ G1 + ln Y ′ + C.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 11 / 28

SLIDE 12

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

SLIDE 13

Integer moments of the GMC

For n ∈ N∗, n < 4

γ2:

E[( 1 2π 2π e

γ 2 Xǫ(eiθ)− γ2 8 E[Xǫ(eiθ)2]dθ)n]

= 1 (2π)n

[0,2π]n E[

n

i=1

e

γ 2 Xǫ(eiθi )− γ2 8 E[Xǫ(eiθi )2]]dθ1 . . . dθn

= 1 (2π)n

[0,2π]n e

γ2 4

i<j E[Xǫ(eiθi )Xǫ(eiθj )]dθ1 . . . dθn

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 13 / 28

SLIDE 14

Integer moments of the GMC

For n ∈ N∗, n < 4

γ2:

E[Y n

γ ] =

1 (2π)n

[0,2π]n e

γ2 4

i<j E[X(eiθi )X(eiθj )]dθ1 . . . dθn

= 1 (2π)n

[0,2π]n
i<j

1 |eiθi − eiθj|

γ2 2

dθ1 . . . dθn = Γ(1 − n γ2

4 )

Γ(1 − γ2

4 )n

Question: can we replace n ∈ N∗ by a real p < 4

γ2 ?

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 14 / 28

SLIDE 15

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

SLIDE 16

The BPZ differential equation

We introduce the following observable for t ∈ [0, 1]: G(γ, p, t) = E[( 2π |t − eiθ|

γ2 2 e γ 2 X(eiθ)dθ)p]

BPZ equation:

(t(1−t2) ∂2 ∂t2+(t2−1) ∂ ∂t +2(C−(A+B+1)t2) ∂ ∂t −4ABt)G(γ, p, t) = 0

where: A = −γ2p 4 , B = −γ2 4 , C = γ2 4 (1 − p) + 1.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 16 / 28

SLIDE 17

Solutions of the BPZ equation

BPZ equation in t → hypergeometric equation in t2 Two bases of solutions: G(γ, p, t) = C1F1(t2) + C2t

γ2 2 (p−1)F2(t2)

G(γ, p, t) = B1 ˜ F1(1 − t2) + B2(1 − t2)1+ γ2

2 ˜

F2(1 − t2) where: C1, C2, B1, B2 ∈ R F1, F2, ˜ F1, ˜ F2 := hypergeometric series depending

n γ and p.

Change of basis: (C1, C2) ↔ (B1, B2).

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 17 / 28

SLIDE 18

The shift relation

By direct asymptotic expansion: C1 = (2π)pE[Y p

γ ]

C2 = 0 B1 = E[( 2π |1 − eiθ|

γ2 2 e γ 2 X(θ)dθ)p]

B2 = (2π)pp Γ(− γ2

2 −1)

Γ(− γ2

4 ) E[Y p−1

γ

] The change of basis implies: E[Y p

γ ] =

Γ(1 − p γ2

4 )

Γ(1 − γ2

4 )Γ(1 − (p − 1)γ2 4 )

E[Y p−1

γ

].

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 18 / 28

SLIDE 19

Negative moments of GMC

The shift relation gives all the negative moments: E[Y −n

γ ] = Γ(1 + nγ2

4 )Γ(1 − γ2 4 )n, ∀n ∈ N. We check: ∀λ ∈ R,

∞

n=0

λn n! Γ(1 + nγ2 4 )Γ(1 − γ2 4 )n < +∞ Negative moments ⇒ determine the law of Yγ !

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 19 / 28

SLIDE 20

Explicit probability densities

Probability densities for Y −1

γ

and Yγ f 1

Yγ (y) =

4 βγ2(y β)

4 γ2 −1e−( y β ) 4 γ2 1[0,∞[(y)

fYγ(y) = 4β γ2 (βy)− 4

γ2 −1e−(βy) − 4 γ2 1[0,∞[(y)

where γ ∈ (0, 2) and β = Γ(1 − γ2

4 ).

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 20 / 28

SLIDE 21

What is Liouville field theory?

Path integral formalism Σ = {X : D → R} For X ∈ Σ, energy of X :=

1 4π

D |∂X|2dx2 +
∂D e

γ 2 Xds

Random field φL: E[F(φL)] =

Σ

F(X)e− 1

4π

D |∂X|2dx2−
∂D e

γ 2 XdsDX

with γ ∈ (0, 2). ⇒ φL is the Liouville field

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 21 / 28

SLIDE 22

Correlations of Liouville theory

Correlation function of zi ∈ D, αi ∈ R:

N
i=1

eαiφL(zi)D =

X:D→R

DX

N

i=1

eαiX(zi)e− 1

4π

D |∂X|2dx2−
∂D e

γ 2 X ds

Expressed in terms of Gaussian multiplicative chaos eαφL(0)D = ˜ C1E[Y p−1

γ

] eαφL(0)e− γ

2 φL(t)D = ˜

C2t

αγ 2 (1 − t2)− γ2 8 G(γ, p, t)

with p = 2 − 2α − 4

γ2.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 22 / 28

SLIDE 23

Correlations of Liouville theory

Liouville theory is a conformal field theory Degenerate fields: e− γ

2 φL(z) and e− 2 γ φL(z).

BPZ equation, for z1, z ∈ D, α ∈ R, γ ∈ (0, 2): (z1, z) → eαφL(z1)e− γ

2 φL(z)D

is solution of a differential equation. Use conformal map ψ, ψ(z1) = 0, ψ(z) = t.

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 23 / 28

SLIDE 24

BPZ equation on the upper half plane H

Proposition (R. 2017)

Let γ ∈ (0, 2) and α > Q + γ

2. Then:

( 4 γ2∂zz + ∆− γ

2

(z − z)2 + ∆α (z − z1)2 + ∆α (z − z1)2 + 1 z − z ∂z + 1 z − z1 ∂z1 + 1 z − z1 ∂z1)e− γ

2 φL(z)eαφL(z1)H = 0

where Q = γ

2 + 2 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 June 22, 2018 24 / 28

SLIDE 25

Analogue for the unit interval [0, 1]

Log-correlated field X on [0, 1]: E[X(x)X(y)] = −2 ln |x − y| For γ ∈ (0, 2), and suitable a, b and p, define: M(γ, p, a, b) := E[( 1 xa(1 − x)be

γ 2 X(x)dx)p].

Theorem (R., Zhu 2018)

M(γ, p, a, b) has the following expression,

(2π)p( 2

γ )p γ2 4

Γ(1 − γ2

4 )p

Γγ( 2

γ (a + 1) − (p − 1) γ 2 )Γγ( 2 γ (b + 1) − (p − 1) γ 2 )Γγ( 2 γ (a + b + 2) − (p − 2) γ 2 )Γγ( 2 γ − p γ 2 )

Γγ( 2

γ )Γγ( 2 γ (a + 1) + γ 2 )Γγ( 2 γ (b + 1) + γ 2 )Γγ( 2 γ (a + b + 2) − (2p − 2) γ 2 )

, ln Γγ(x) = ∞ dt t    e−xt − e− Qt

2

(1 − e− γt

2 )(1 − e − 2t γ )

− ( Q

2 − x)2

2 e−t + x − Q

2

t    , Q = γ 2 + 2 γ . Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 25 / 28

SLIDE 26

Analogue for the unit interval [0, 1]

Log-correlated field X on [0, 1]: E[X(x)X(y)] = −2 ln |x − y| Equivalent statement: 1 xa(1 − x)be

γ 2 X(x)dx

law

= ceN(0,γ2 ln 2)YγX −1

1 X −1 2 X −1 3

c := deterministic constant Yγ

law

= 1

βExp(1)− γ2

4

Xi := generalized beta law

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 26 / 28

SLIDE 27

Analogue for the bulk meaure on D

Work in progress, for γ ∈ (0, 2), α ∈ (γ

2, Q):

E[(

D

1 |x|γαeγX(x)dx2)

Q−α γ ] =

γ2(π Γ(γ2

4 )

Γ(1 − γ2

4 )

)

Q−α γ cos(α − Q

γ π) Γ(2α

γ − 4 γ2)Γ(γ 2(α − Q))

Γ(α−Q

γ )

Liouville theory with action:

D(|∂X|2 + eγX)dx2

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 27 / 28

SLIDE 28

Outlook and perspectives

Integrability program for GMC and Liouville theory More general Liouville correlations on D Work in progress to recover the law of the quantum disk of the Duplantier-Miller-Sheffield approach to Liouville Quantum Gravity Other geometries, higher genus Conformal bootstrap

Guillaume Remy (ENS) The Fyodorov-Bouchaud formula June 22, 2018 28 / 28