Random matrices and Gaussian multiplicative chaos Nick Simm - - PowerPoint PPT Presentation

Random matrices and Gaussian multiplicative chaos

Nick Simm

Mathematics Institute, University of Warwick

Joint work with Gaultier Lambert and Dmitry Ostrovsky.

Optimal Point Configurations and Orthogonal Polynomials April 2017, CIEM

Research supported by Leverhulme fellowship ECF-2014-309

The Circular Unitary Ensemble

The Circular Unitary Ensemble

◮ Let UN be an N × N random matrix chosen uniformly from

the unitary group.

The Circular Unitary Ensemble

◮ Let UN be an N × N random matrix chosen uniformly from

the unitary group.

◮ The joint distribution of points is an example of a (1d)

‘Coulomb gas’: P(θ1, . . . , θN) ∝

• j<k

|eiθj − eiθk|2

The Circular Unitary Ensemble

◮ Let UN be an N × N random matrix chosen uniformly from

the unitary group.

◮ The joint distribution of points is an example of a (1d)

‘Coulomb gas’: P(θ1, . . . , θN) ∝

• j<k

|eiθj − eiθk|2

◮ The eigenvalues eiθ1, . . . , eiθN form a determinantal point

process with kernel KN(θ, φ) =

N−1

• j=0

pj(θ)pj(φ) = sin(N(θ − φ)/2) sin((θ − φ)/2) where pj(θ) = eijθ.

The Circular Unitary Ensemble

◮ Let UN be an N × N random matrix chosen uniformly from

the unitary group.

◮ The joint distribution of points is an example of a (1d)

‘Coulomb gas’: P(θ1, . . . , θN) ∝

• j<k

|eiθj − eiθk|2

◮ The eigenvalues eiθ1, . . . , eiθN form a determinantal point

process with kernel KN(θ, φ) =

N−1

• j=0

pj(θ)pj(φ) = sin(N(θ − φ)/2) sin((θ − φ)/2) where pj(θ) = eijθ.

◮ Problem: limit theorems for PN(θ) = det(UN − eiθ) as

N → ∞.

Characteristic polynomials

Characteristic polynomials

Characteristic polynomials of (large) random matrices:

Characteristic polynomials

Characteristic polynomials of (large) random matrices:

◮ A good model of the Riemann zeta function ζ(s) high up on

the critical line s = 1/2 + it (Keating and Snaith ’00).

Characteristic polynomials

Characteristic polynomials of (large) random matrices:

◮ A good model of the Riemann zeta function ζ(s) high up on

the critical line s = 1/2 + it (Keating and Snaith ’00).

◮ An interesting example of a log-correlated Gaussian field. E.g.

how to compute M∗

N =

max

θ∈[0,2π] log |PN(θ)| ≡

max

θ∈[0,2π] log | det(UN − eiθI)|

Characteristic polynomials

Characteristic polynomials of (large) random matrices:

◮ A good model of the Riemann zeta function ζ(s) high up on

the critical line s = 1/2 + it (Keating and Snaith ’00).

◮ An interesting example of a log-correlated Gaussian field. E.g.

how to compute M∗

N =

max

θ∈[0,2π] log |PN(θ)| ≡

max

θ∈[0,2π] log | det(UN − eiθI)| ◮ Using these ideas, it has been conjectured and partially

proved that as N → ∞ M∗

N = log(N) − 3

4 log(log(N)) + (G1 + G2)/2 + o(1) where G1,2 are standard independent Gumbel variables.

(Fyodorov and Keating ’12, Arguin, Belius, Bourgade ’15, Paquette and Zeitouni ’16, Chaibbi, Madaule and Najnudel ’16)

The logarithm

The logarithm

Theorem (Hughes, Keating and O’Connell ’01)

Let {Zj}∞

j=1 be i.i.d. standard complex Gaussian random variables.

Then

VN(θ) := log |PN(θ)|

d

→ V (θ) := 1 2

∞

• k=1

eiθ √ k Zk + c.c.

The logarithm

Theorem (Hughes, Keating and O’Connell ’01)

Let {Zj}∞

j=1 be i.i.d. standard complex Gaussian random variables.

Then

VN(θ) := log |PN(θ)|

d

→ V (θ) := 1 2

∞

• k=1

eiθ √ k Zk + c.c.

Key properties of V (θ):

◮ V is Gaussian and mean zero E(V (θ)) = 0. ◮ Logarithmic correlations:

E(V (θ)V (φ)) = 1 2Re ∞

• j=1

eik(θ−φ) k

• = −1

2 log |eiθ − eiφ|

◮ What about θ = φ? Implies

Var(V (θ)) = ∞

The logarithm

Theorem (Hughes, Keating and O’Connell ’01)

Let {Zj}∞

j=1 be i.i.d. standard complex Gaussian random variables.

Then

VN(θ) := log |PN(θ)|

d

→ V (θ) := 1 2

∞

• k=1

eiθ √ k Zk + c.c.

Key properties of V (θ):

◮ V is Gaussian and mean zero E(V (θ)) = 0. ◮ Logarithmic correlations:

E(V (θ)V (φ)) = 1 2Re ∞

• j=1

eik(θ−φ) k

• = −1

2 log |eiθ − eiφ|

◮ What about θ = φ? Implies

Var(V (θ)) = ∞ Conclusion: Limit V (θ) is a distribution valued object.

The exponential of the logarithm

The exponential of the logarithm

Naively we might suppose that |PN(θ)| = eVN(θ) converges to eV (θ)?

The exponential of the logarithm

Naively we might suppose that |PN(θ)| = eVN(θ) converges to eV (θ)? How to define the exponential of a distribution?

The exponential of the logarithm

Naively we might suppose that |PN(θ)| = eVN(θ) converges to eV (θ)? How to define the exponential of a distribution? Consider measures formally defined by µ(γ)(D) =

• D

eγV (θ)− γ2

2 Var(V (θ)) dθ

The measure µ(γ) is defined by a renormalization procedure Vǫ = V ∗ φǫ.

The exponential of the logarithm

Naively we might suppose that |PN(θ)| = eVN(θ) converges to eV (θ)? How to define the exponential of a distribution? Consider measures formally defined by µ(γ)(D) =

• D

eγV (θ)− γ2

2 Var(V (θ)) dθ

The measure µ(γ) is defined by a renormalization procedure Vǫ = V ∗ φǫ. It was shown by Kahane ’85 that

◮ µ(γ) ǫ

converges as ǫ → 0 to a non-trivial limit if and only if γ < 2.

◮ This limit does not depend on (Kahane’s) cut-off procedures.

The exponential of the logarithm

Naively we might suppose that |PN(θ)| = eVN(θ) converges to eV (θ)? How to define the exponential of a distribution? Consider measures formally defined by µ(γ)(D) =

• D

eγV (θ)− γ2

2 Var(V (θ)) dθ

The measure µ(γ) is defined by a renormalization procedure Vǫ = V ∗ φǫ. It was shown by Kahane ’85 that

◮ µ(γ) ǫ

converges as ǫ → 0 to a non-trivial limit if and only if γ < 2.

◮ This limit does not depend on (Kahane’s) cut-off procedures.

This limit defines the measure µ(γ) which is called Gaussian multiplicative chaos (GMC).

Properties of measures µ(γ)

Properties of measures µ(γ)

◮ Power law spectrum (multifractality): For 0 ≤ qγ2 < 2 we

have E(µ(γ)[0, r]q) = Cqrξ(q) where ξ(q) = (1 + γ2

2 )q − γ2 2 q2.

Properties of measures µ(γ)

◮ Power law spectrum (multifractality): For 0 ≤ qγ2 < 2 we

have E(µ(γ)[0, r]q) = Cqrξ(q) where ξ(q) = (1 + γ2

2 )q − γ2 2 q2. ◮ In two dimensions, V is essentially the Gaussian free field, a

fundamental object of mathematical physics.

Properties of measures µ(γ)

◮ Power law spectrum (multifractality): For 0 ≤ qγ2 < 2 we

have E(µ(γ)[0, r]q) = Cqrξ(q) where ξ(q) = (1 + γ2

2 )q − γ2 2 q2. ◮ In two dimensions, V is essentially the Gaussian free field, a

fundamental object of mathematical physics.

◮ In that context, eγV is used in Liouville quantum gravity to

construct a uniform random metric on the sphere. (see work and recent surveys of e.g. Berestycki, Duplantier, Rhodes, Sheffield, Vargas,. . . )

Properties of measures µ(γ)

◮ Power law spectrum (multifractality): For 0 ≤ qγ2 < 2 we

have E(µ(γ)[0, r]q) = Cqrξ(q) where ξ(q) = (1 + γ2

2 )q − γ2 2 q2. ◮ In two dimensions, V is essentially the Gaussian free field, a

fundamental object of mathematical physics.

◮ In that context, eγV is used in Liouville quantum gravity to

construct a uniform random metric on the sphere. (see work and recent surveys of e.g. Berestycki, Duplantier, Rhodes, Sheffield, Vargas,. . . )

◮ The distribution of µ(γ) near γ = γc is believed to be closely

related to statistics of max|z|=1|PN(z)|.

The L2-phase

The L2-phase

The range 0 ≤ γ < √ 2 is called the L2-phase. This is because E(µ(γ)(D)2) =

• D×D

|eiθ − eiφ|−γ2/2dθ dφ < ∞ if and only if 0 ≤ γ < √ 2.

The L2-phase

The range 0 ≤ γ < √ 2 is called the L2-phase. This is because E(µ(γ)(D)2) =

• D×D

|eiθ − eiφ|−γ2/2dθ dφ < ∞ if and only if 0 ≤ γ < √ 2.

Theorem (Webb ’15)

Consider µ(γ)

N (D) =

• D |PN(θ)|γ dθ

E

• D |PN(θ)|γ dθ

Then for any γ < √ 2 we have µ(γ)

N d

→ µ(γ), N → ∞ where µ(γ) is the same measure constructed from Kahane’s theory.

Counting statistics in the CUE

Counting statistics in the CUE

Instead of VN(θ), we consider counting statistics XN(θ) =

N

• j=1

χJ(θ)(Nαθj), J(θ) = [θ − 1, θ + 1]

Counting statistics in the CUE

Instead of VN(θ), we consider counting statistics XN(θ) =

N

• j=1

χJ(θ)(Nαθj), J(θ) = [θ − 1, θ + 1] In reality, we consider a slightly smoother version: XN,ǫN(θ) =

N

• j=1

(χJ(θ) ∗ φǫ)(Nαθj), φǫ(θ) = 1 ǫ φ θ ǫ

• where f ∗ g stands for the convolution of the functions f and g.

Counting statistics in the CUE

Instead of VN(θ), we consider counting statistics XN(θ) =

N

• j=1

χJ(θ)(Nαθj), J(θ) = [θ − 1, θ + 1] In reality, we consider a slightly smoother version: XN,ǫN(θ) =

N

• j=1

(χJ(θ) ∗ φǫ)(Nαθj), φǫ(θ) = 1 ǫ φ θ ǫ

• where f ∗ g stands for the convolution of the functions f and g.

We will study the field XN,ǫ with mollifying scale ǫ → 0 depending

• n N.

A realization of the process

A plot of the process ˜ XN(u) := XN(u) − E(XN(u)) and N = 3000, α = 0.

A realization of the process

A plot of the process ˜ XN(u) := XN(u) − E(XN(u)), u ∈ [−π, π) and N = 3000, α = 0. (Zoomed in around the origin u ∈ (−0.2, 0.2))

The main result: all 0 ≤ γ < 2

The main result: all 0 ≤ γ < 2

From the smoothed counting statistic, we construct a measure µ(γ)

N,ǫN(D) =

• D

eγ ˜

XN,ǫN (θ)− γ2

2 Var( ˜

XN,ǫN (θ)) dθ

where ˜ XN,ǫN(θ) = XN,ǫN(θ) − E(XN,ǫN(θ)).

The main result: all 0 ≤ γ < 2

From the smoothed counting statistic, we construct a measure µ(γ)

N,ǫN(D) =

• D

eγ ˜

XN,ǫN (θ)− γ2

2 Var( ˜

XN,ǫN (θ)) dθ

where ˜ XN,ǫN(θ) = XN,ǫN(θ) − E(XN,ǫN(θ)).

Theorem (Lambert, Ostrovsky, S’ 2016)

Suppose 0 < α < 1 and ǫN → 0 such that ǫ−1

N Nα−1 → 0. Then for

every γ < 2 we have µ(γ)

N,ǫN d

→ µ(γ), N → ∞ where µ(γ) is the same GMC constructed via Kahane’s theory.

The main result: all 0 ≤ γ < 2

From the smoothed counting statistic, we construct a measure µ(γ)

N,ǫN(D) =

• D

eγ ˜

XN,ǫN (θ)− γ2

2 Var( ˜

XN,ǫN (θ)) dθ

where ˜ XN,ǫN(θ) = XN,ǫN(θ) − E(XN,ǫN(θ)).

Theorem (Lambert, Ostrovsky, S’ 2016)

Suppose 0 < α < 1 and ǫN → 0 such that ǫ−1

N Nα−1 → 0. Then for

every γ < 2 we have µ(γ)

N,ǫN d

→ µ(γ), N → ∞ where µ(γ) is the same GMC constructed via Kahane’s theory. Thus we go beyond the L2 bounds γ < √ 2 to establish convergence in the full phase γ < 2.

Ideas in the proof

Ideas in the proof

Try to reduce to the case limǫ→0 limN→∞.

Ideas in the proof

Try to reduce to the case limǫ→0 limN→∞. For fixed ǫ > 0 there is:

Theorem (Soshnikov ’00)

Let f be a smooth function with rapid decay. Then

N

• j=1

f (θjNα) − E  

N

• j=1

f (θjNα)   d → N(0, σ2(f ))

Ideas in the proof

Try to reduce to the case limǫ→0 limN→∞. For fixed ǫ > 0 there is:

Theorem (Soshnikov ’00)

Let f be a smooth function with rapid decay. Then

N

• j=1

f (θjNα) − E  

N

• j=1

f (θjNα)   d → N(0, σ2(f )) where σ2(f ) = f 2

H1/2 =

∞

−∞

|k||ˆ f (k)|2 dk = 1 2π2 ∞

−∞

∞

−∞

f ′(x)f ′(y) log 1 |x − y| dx dy

Ideas in the proof

Try to reduce to the case limǫ→0 limN→∞. For fixed ǫ > 0 there is:

Theorem (Soshnikov ’00)

Let f be a smooth function with rapid decay. Then

N

• j=1

f (θjNα) − E  

N

• j=1

f (θjNα)   d → N(0, σ2(f )) where σ2(f ) = f 2

H1/2 =

∞

−∞

|k||ˆ f (k)|2 dk = 1 2π2 ∞

−∞

∞

−∞

f ′(x)f ′(y) log 1 |x − y| dx dy However, if ǫ ≡ ǫN then Var(XN,ǫN(u)) ∼ 1

2 log(ǫ−1 N ) diverges...

Strong Gaussian approximation

Strong Gaussian approximation

Goal: Show that XN,ǫN is closely approximated by the fixed ǫ and N → ∞ limiting Gaussian field.

Strong Gaussian approximation

Goal: Show that XN,ǫN is closely approximated by the fixed ǫ and N → ∞ limiting Gaussian field.

Lemma

Suppose 0 < α < 1 and ǫN → 0 such that ǫ−1

N Nα−1 → 0. Then for

any finite sequence of points u1, . . . , uq, we have

E  exp  

q

• j=1

αj ˜ XN,ǫN(uj)     = exp  1 2

• q
• j=1

αj(χJ(uj) ∗ φǫN)

• 2

H1/2

  (1 + o(1))

as N → ∞. The error term is uniform in u1, . . . , uq varying in compact subets of R.

Strong Gaussian approximation

Goal: Show that XN,ǫN is closely approximated by the fixed ǫ and N → ∞ limiting Gaussian field.

Lemma

Suppose 0 < α < 1 and ǫN → 0 such that ǫ−1

N Nα−1 → 0. Then for

any finite sequence of points u1, . . . , uq, we have

E  exp  

q

• j=1

αj ˜ XN,ǫN(uj)     = exp  1 2

• q
• j=1

αj(χJ(uj) ∗ φǫN)

• 2

H1/2

  (1 + o(1))

as N → ∞. The error term is uniform in u1, . . . , uq varying in compact subets of R. Case q = 2 quite easily gives the L2-phase: µN,ǫN = µN,ǫ + (µN,ǫN − µN,ǫ) Uniformity above allows precise computation of the second moment of the blue term, provided γ < √ 2.

Getting the estimate

Getting the estimate

The CLT usually proved by the method of moments. Difficult to

• btain good estimates on the Laplace transform.

Getting the estimate

The CLT usually proved by the method of moments. Difficult to

• btain good estimates on the Laplace transform. Instead, we

exploit

Theorem (Borodin-Okounkov formula (2000))

Let F be a 2π-periodic function. Then the following identity holds:

TN[F] := det{ ˆ Fk−j}k,j=0..N−1 = exp

• N

log F0 +

∞

• k=1

|k|| log F k|2

• × det(I − RNH(c)H(c)†RN)

where RN are projections on {N + 1, N + 2, . . .} and H(c) = {ˆ cj+k−1}∞

j,k=1 where

c(eiθ) = exp

• iIm

∞

• k=1
• log F keikθ
• .

Getting the estimate

The CLT usually proved by the method of moments. Difficult to

• btain good estimates on the Laplace transform. Instead, we

exploit

Theorem (Borodin-Okounkov formula (2000))

Let F be a 2π-periodic function. Then the following identity holds:

TN[F] := det{ ˆ Fk−j}k,j=0..N−1 = exp

• N

log F0 +

∞

• k=1

|k|| log F k|2

• × det(I − RNH(c)H(c)†RN)

where RN are projections on {N + 1, N + 2, . . .} and H(c) = {ˆ cj+k−1}∞

j,k=1 where

c(eiθ) = exp

• iIm

∞

• k=1
• log F keikθ
• .

We prove that the last determinant is close to 1, uniformly as N → ∞.

The L1-phase 1 < γ < √ 2

The L1-phase 1 < γ < √ 2

Idea is to study γ-thick points (see e.g. Berestycki ’15): Pc,τ := {u ∈ [−c, c] : ˜ XN,ǫN(u) > τ log(ǫ−1

N )}

Note that Pc,τ are the points where the field fluctuates above its maximum. We show that for any τ > γ, the mass µγ

N,ǫN(Pc,τ) converges to

zero in L1.

The L1-phase 1 < γ < √ 2

Idea is to study γ-thick points (see e.g. Berestycki ’15): Pc,τ := {u ∈ [−c, c] : ˜ XN,ǫN(u) > τ log(ǫ−1

N )}

Note that Pc,τ are the points where the field fluctuates above its maximum. We show that for any τ > γ, the mass µγ

N,ǫN(Pc,τ) converges to

zero in L1. On the complement Pc

c,τ it can be shown that now the L2

techniques work for any γ < √ 2.

The L1-phase 1 < γ < √ 2

Idea is to study γ-thick points (see e.g. Berestycki ’15): Pc,τ := {u ∈ [−c, c] : ˜ XN,ǫN(u) > τ log(ǫ−1

N )}

Note that Pc,τ are the points where the field fluctuates above its maximum. We show that for any τ > γ, the mass µγ

N,ǫN(Pc,τ) converges to

zero in L1. On the complement Pc

c,τ it can be shown that now the L2

techniques work for any γ < √ 2. We are guided by Berestycki’s calculation, adapted to our ‘almost Gaussian’ setting.

Conclusions

Conclusions

Summary:

◮ I showed how counting statistics in the CUE are related to

log-correlated Gaussian fields.

Conclusions

Summary:

◮ I showed how counting statistics in the CUE are related to

log-correlated Gaussian fields.

◮ The corresponding exponentials were constructed in terms of

Gaussian multiplicative chaos.

Conclusions

Summary:

◮ I showed how counting statistics in the CUE are related to

log-correlated Gaussian fields.

◮ The corresponding exponentials were constructed in terms of

Gaussian multiplicative chaos.

◮ We also proved that the same results hold when the CUE is

replaced with the sine process.

Conclusions

Summary:

◮ I showed how counting statistics in the CUE are related to

log-correlated Gaussian fields.

◮ The corresponding exponentials were constructed in terms of

Gaussian multiplicative chaos.

◮ We also proved that the same results hold when the CUE is

replaced with the sine process. What about other Coulomb gas type systems or more general point processes? How does the characteristic polynomial behave?

Conclusions

Summary:

◮ I showed how counting statistics in the CUE are related to

log-correlated Gaussian fields.

◮ The corresponding exponentials were constructed in terms of

Gaussian multiplicative chaos.

◮ We also proved that the same results hold when the CUE is

replaced with the sine process. What about other Coulomb gas type systems or more general point processes? How does the characteristic polynomial behave?

Thank you.