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
I
Let UN be an N × N random matrix chosen uniformly from
the unitary group.
The Circular Unitary Ensemble
I
I
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’:
Y
P(θ1 , . . . , θN ) ∝
|e iθj − e iθk |2
j<k
The Circular Unitary Ensemble
I
I
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’:
Y
P(θ1 , . . . , θN ) ∝
|e iθj − e iθk |2
j<k
I
The eigenvalues e iθ1 , . . . , e iθN form a determinantal point
process with kernel
KN (θ, φ) =
N−1
X
j=0
where pj (θ) = e ijθ .
pj (θ)pj (φ) =
sin(N(θ − φ)/2)
sin((θ − φ)/2)
The Circular Unitary Ensemble
I
I
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’:
Y
P(θ1 , . . . , θN ) ∝
|e iθj − e iθk |2
j<k
I
The eigenvalues e iθ1 , . . . , e iθN form a determinantal point
process with kernel
KN (θ, φ) =
N−1
X
j=0
I
pj (θ)pj (φ) =
sin(N(θ − φ)/2)
sin((θ − φ)/2)
where pj (θ) = e ijθ .
Problem: limit theorems for PN (θ) = det(UN − e iθ ) as
N → ∞.
Characteristic polynomials
Characteristic polynomials
Characteristic polynomials of (large) random matrices:
Characteristic polynomials
Characteristic polynomials of (large) random matrices:
I
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:
I
A good model of the Riemann zeta function ζ(s) high up on
the critical line s = 1/2 + it (Keating and Snaith ’00).
I
An interesting example of a log-correlated Gaussian field. E.g.
how to compute
MN∗ = max log |PN (θ)| ≡ max log | det(UN − e iθ I )|
θ∈[0,2π]
θ∈[0,2π]
Characteristic polynomials
Characteristic polynomials of (large) random matrices:
I
A good model of the Riemann zeta function ζ(s) high up on
the critical line s = 1/2 + it (Keating and Snaith ’00).
I
An interesting example of a log-correlated Gaussian field. E.g.
how to compute
MN∗ = max log |PN (θ)| ≡ max log | det(UN − e iθ I )|
θ∈[0,2π]
I
θ∈[0,2π]
Using these ideas, it has been conjectured and partially
proved that as N → ∞
MN∗ = log(N) −
3
log(log(N)) + (G1 + G2 )/2 + o(1)
4
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
∞
d
VN (θ) := log |PN (θ)| → V (θ) :=
1 X e iθ
√ Zk + c.c.
2
k
k=1
The logarithm
Theorem (Hughes, Keating and O’Connell ’01)
Let {Zj }∞
j=1 be i.i.d. standard complex Gaussian random variables.
Then
∞
d
VN (θ) := log |PN (θ)| → V (θ) :=
1 X e iθ
√ Zk + c.c.
2
k
k=1
Key properties of V (θ):
I V is Gaussian and mean zero E(V (θ)) = 0.
I Logarithmic correlations:
X
∞
1
e ik(θ−φ)
1
E(V (θ)V (φ)) = Re
= − log |e iθ − e iφ |
2
k
2
j=1
I
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
∞
d
VN (θ) := log |PN (θ)| → V (θ) :=
1 X e iθ
√ Zk + c.c.
2
k
k=1
Key properties of V (θ):
I V is Gaussian and mean zero E(V (θ)) = 0.
I Logarithmic correlations:
X
∞
1
e ik(θ−φ)
1
E(V (θ)V (φ)) = Re
= − log |e iθ − e iφ |
2
k
2
j=1
I
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)| = e VN (θ) converges to
e V (θ) ?
The exponential of the logarithm
Naively we might suppose that |PN (θ)| = e VN (θ) converges to
e V (θ) ?
How to define the exponential of a distribution?
The exponential of the logarithm
Naively we might suppose that |PN (θ)| = e VN (θ) converges to
e V (θ) ?
How to define the exponential of a distribution?
Consider measures formally defined by
Z
γ2
(γ)
µ (D) =
e γV (θ)− 2 Var(V (θ)) dθ
D
The measure µ(γ) is defined by a renormalization procedure
V = V ∗ φ .
The exponential of the logarithm
Naively we might suppose that |PN (θ)| = e VN (θ) converges to
e V (θ) ?
How to define the exponential of a distribution?
Consider measures formally defined by
Z
γ2
(γ)
µ (D) =
e γV (θ)− 2 Var(V (θ)) dθ
D
The measure µ(γ) is defined by a renormalization procedure
V = V ∗ φ .
It was shown by Kahane ’85 that
(γ)
I
µ converges as → 0 to a non-trivial limit if and only if
γ < 2.
I
This limit does not depend on (Kahane’s) cut-off procedures.
The exponential of the logarithm
Naively we might suppose that |PN (θ)| = e VN (θ) converges to
e V (θ) ?
How to define the exponential of a distribution?
Consider measures formally defined by
Z
γ2
(γ)
µ (D) =
e γV (θ)− 2 Var(V (θ)) dθ
D
The measure µ(γ) is defined by a renormalization procedure
V = V ∗ φ .
It was shown by Kahane ’85 that
(γ)
I
µ converges as → 0 to a non-trivial limit if and only if
γ < 2.
I
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)
I
Power law spectrum (multifractality): For 0 ≤ qγ 2 < 2 we
have
E(µ(γ) [0, r ]q ) = Cq r ξ(q)
where ξ(q) = (1 +
γ2
2 )q
−
γ2 2
2 q .
Properties of measures µ(γ)
I
Power law spectrum (multifractality): For 0 ≤ qγ 2 < 2 we
have
E(µ(γ) [0, r ]q ) = Cq r ξ(q)
where ξ(q) = (1 +
I
γ2
2 )q
−
γ2 2
2 q .
In two dimensions, V is essentially the Gaussian free field, a
fundamental object of mathematical physics.
Properties of measures µ(γ)
I
Power law spectrum (multifractality): For 0 ≤ qγ 2 < 2 we
have
E(µ(γ) [0, r ]q ) = Cq r ξ(q)
where ξ(q) = (1 +
γ2
2 )q
−
γ2 2
2 q .
I
In two dimensions, V is essentially the Gaussian free field, a
fundamental object of mathematical physics.
I
In that context, e γV 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)
I
Power law spectrum (multifractality): For 0 ≤ qγ 2 < 2 we
have
E(µ(γ) [0, r ]q ) = Cq r ξ(q)
where ξ(q) = (1 +
γ2
2 )q
−
γ2 2
2 q .
I
In two dimensions, V is essentially the Gaussian free field, a
fundamental object of mathematical physics.
I
In that context, e γV 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,. . . )
I
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
√
2 is called the L2 -phase. This is because
Z
2
(γ)
2
E(µ (D) ) =
|e iθ − e iφ |−γ /2 dθ dφ < ∞
The range 0 ≤ γ <
D×D
if and only if 0 ≤ γ <
√
2.
The L2 -phase
√
2 is called the L2 -phase. This is because
Z
2
(γ)
2
E(µ (D) ) =
|e iθ − e iφ |−γ /2 dθ dφ < ∞
The range 0 ≤ γ <
D×D
if and only if 0 ≤ γ <
√
2.
Theorem (Webb ’15)
Consider
(γ)
µN (D)
Then for any γ <
√
R
|PN (θ)|γ dθ
= RD
E D |PN (θ)|γ dθ
2 we have
(γ) d
µN → 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
X
j=1
χJ(θ) (N α θj ),
J(θ) = [θ − 1, θ + 1]
Counting statistics in the CUE
Instead of VN (θ), we consider counting statistics
XN (θ) =
N
X
χJ(θ) (N α θj ),
J(θ) = [θ − 1, θ + 1]
j=1
In reality, we consider a slightly smoother version:
XN,N (θ) =
N
X
(χJ(θ) ∗ φ )(N α θj ),
j=1
1
φ (θ) = φ
θ
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
X
χJ(θ) (N α θj ),
J(θ) = [θ − 1, θ + 1]
j=1
In reality, we consider a slightly smoother version:
XN,N (θ) =
N
X
(χJ(θ) ∗ φ )(N α θj ),
j=1
1
φ (θ) = φ
θ
where f ∗ g stands for the convolution of the functions f and g .
We will study the field XN, with mollifying scale → 0 depending
on N.
A realization of the process
A plot of the process X̃N (u) := XN (u) − E(XN (u)) and
N = 3000, α = 0.
A realization of the process
A plot of the process X̃N (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
Z
γ2
(γ)
µN,N (D) =
e γ X̃N,N (θ)− 2 Var(X̃N,N (θ)) dθ
D
where X̃N,N (θ) = XN,N (θ) − E(XN,N (θ)).
The main result: all 0 ≤ γ < 2
From the smoothed counting statistic, we construct a measure
Z
γ2
(γ)
µN,N (D) =
e γ X̃N,N (θ)− 2 Var(X̃N,N (θ)) dθ
D
where X̃N,N (θ) = XN,N (θ) − E(XN,N (θ)).
Theorem (Lambert, Ostrovsky, S’ 2016)
α−1 → 0. Then for
Suppose 0 < α < 1 and N → 0 such that −1
N N
every γ < 2 we have
(γ)
d
µN,N → 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
Z
γ2
(γ)
µN,N (D) =
e γ X̃N,N (θ)− 2 Var(X̃N,N (θ)) dθ
D
where X̃N,N (θ) = XN,N (θ) − E(XN,N (θ)).
Theorem (Lambert, Ostrovsky, S’ 2016)
α−1 → 0. Then for
Suppose 0 < α < 1 and N → 0 such that −1
N N
every γ < 2 we have
(γ)
d
µN,N → 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 lim→0 limN→∞ .
Ideas in the proof
Try to reduce to the case lim→0 limN→∞ . For fixed > 0 there is:
Theorem (Soshnikov ’00)
Let f be a smooth function with rapid decay. Then


N
N
X
X
d
f (θj N α ) − E 
f (θj N α ) → N (0, σ 2 (f ))
j=1
j=1
Ideas in the proof
Try to reduce to the case lim→0 limN→∞ . For fixed > 0 there is:
Theorem (Soshnikov ’00)
Let f be a smooth function with rapid decay. Then


N
N
X
X
d
f (θj N α ) − E 
f (θj N α ) → N (0, σ 2 (f ))
j=1
j=1
where
2
σ (f ) =
=
kf k2H 1/2
1
2π 2
Z
=
Z ∞Z
−∞
∞
−∞
∞
−∞
|k||fˆ(k)|2 dk
f 0 (x)f 0 (y ) log
1
dx dy
|x − y |
Ideas in the proof
Try to reduce to the case lim→0 limN→∞ . For fixed > 0 there is:
Theorem (Soshnikov ’00)
Let f be a smooth function with rapid decay. Then


N
N
X
X
d
f (θj N α ) − E 
f (θj N α ) → N (0, σ 2 (f ))
j=1
j=1
where
2
σ (f ) =
=
kf k2H 1/2
1
2π 2
Z
=
Z ∞Z
−∞
∞
|k||fˆ(k)|2 dk
−∞
∞
f 0 (x)f 0 (y ) log
−∞
However, if ≡ N then Var(XN,N (u)) ∼
1
2
1
dx dy
|x − y |
−1
log(N
) diverges...
Strong Gaussian approximation
Strong Gaussian approximation
Goal: Show that XN,N is closely approximated by the fixed and
N → ∞ limiting Gaussian field.
Strong Gaussian approximation
Goal: Show that XN,N is closely approximated by the fixed and
N → ∞ limiting Gaussian field.
Lemma
α−1 → 0. Then for
Suppose 0 < α < 1 and N → 0 such that −1
N N
any finite sequence of points u1 , . . . , uq , we have


E exp 
q
X
j=1



X
2
q
1

αj X̃N,N (uj ) = exp  αj (χJ(uj ) ∗ φN )
1/2 (1 + o(1))
2
H
j=1
as N → ∞. The error term is uniform in u1 , . . . , uq varying in
compact subets of R.
Strong Gaussian approximation
Goal: Show that XN,N is closely approximated by the fixed and
N → ∞ limiting Gaussian field.
Lemma
α−1 → 0. Then for
Suppose 0 < α < 1 and N → 0 such that −1
N N
any finite sequence of points u1 , . . . , uq , we have


E exp 
q
X
j=1



X
2
q
1

αj X̃N,N (uj ) = exp  αj (χJ(uj ) ∗ φN )
1/2 (1 + o(1))
2
H
j=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,N = µN, + (µN,N − µN, )
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
obtain good estimates on the Laplace transform.
Getting the estimate
The CLT usually proved by the method of moments. Difficult to
obtain 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{F̂k−j }k,j=0..N−1 = exp N log
F0 +
∞
X
!
[
|k||log
F k |2
k=1
× det(I − RN H(c)H(c)† RN )
where RN are projections on {N + 1, N + 2, . . .} and
H(c) = {ĉj+k−1 }∞
j,k=1 where
X
!
∞
ikθ
[
c(e ) = exp iIm
log F k e
.
iθ
k=1
Getting the estimate
The CLT usually proved by the method of moments. Difficult to
obtain 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{F̂k−j }k,j=0..N−1 = exp N log
F0 +
∞
X
!
[
|k||log
F k |2
k=1
× det(I − RN H(c)H(c)† RN )
where RN are projections on {N + 1, N + 2, . . .} and
H(c) = {ĉj+k−1 }∞
j,k=1 where
X
!
∞
ikθ
[
c(e ) = exp iIm
log F k e
.
iθ
k=1
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] : X̃N,N (u) > τ log(−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 µγN,N (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] : X̃N,N (u) > τ log(−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 µγN,N (Pc,τ ) converges to
zero in L1 .
c it can be shown that now the L2
On the complement Pc,τ
√
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] : X̃N,N (u) > τ log(−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 µγN,N (Pc,τ ) converges to
zero in L1 .
c it can be shown that now the L2
On the complement Pc,τ
√
techniques work for any γ < 2.
We are guided by Berestycki’s calculation, adapted to our ‘almost
Gaussian’ setting.
Conclusions
Conclusions
Summary:
I
I showed how counting statistics in the CUE are related to
log-correlated Gaussian fields.
Conclusions
Summary:
I
I showed how counting statistics in the CUE are related to
log-correlated Gaussian fields.
I
The corresponding exponentials were constructed in terms of
Gaussian multiplicative chaos.
Conclusions
Summary:
I
I showed how counting statistics in the CUE are related to
log-correlated Gaussian fields.
I
The corresponding exponentials were constructed in terms of
Gaussian multiplicative chaos.
I
We also proved that the same results hold when the CUE is
replaced with the sine process.
Conclusions
Summary:
I
I showed how counting statistics in the CUE are related to
log-correlated Gaussian fields.
I
The corresponding exponentials were constructed in terms of
Gaussian multiplicative chaos.
I
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
I showed how counting statistics in the CUE are related to
log-correlated Gaussian fields.
I
The corresponding exponentials were constructed in terms of
Gaussian multiplicative chaos.
I
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.