Multiplicative chaos measures for a random model of
the Riemann zeta function
Eero Saksman
(University of Helsinki)
Based on joint work with C. Webb
Bristol, May 11, 2016
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
1 / 11
The statistical model
• Recall the classical approximation (we have seen it many times during
this week)
X
log(ζ(1/2 + it)) =
p −1/2−it + 0 smallish0 error, t ∈ [0, T ].
p<T
(one should add some smoothing).
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
2 / 11
The statistical model
• Recall the classical approximation (we have seen it many times during
this week)
X
log(ζ(1/2 + it)) =
p −1/2−it + 0 smallish0 error, t ∈ [0, T ].
p<T
(one should add some smoothing). We want to consider ’typical’ statistical
behaviour on an interval of length one’ .
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
2 / 11
The statistical model
• Recall the classical approximation (we have seen it many times during
this week)
X
log(ζ(1/2 + it)) =
p −1/2−it + 0 smallish0 error, t ∈ [0, T ].
p<T
(one should add some smoothing). We want to consider ’typical’ statistical
behaviour on an interval of length one’ . Let x ∈ [0, 1] and consider
X 1
√ p −ix p −iωT ,
p
p≤T
where ω is random, distributes uniformly on [0, 1], and T >> 1.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
2 / 11
The statistical model
• Recall the classical approximation (we have seen it many times during
this week)
X
log(ζ(1/2 + it)) =
p −1/2−it + 0 smallish0 error, t ∈ [0, T ].
p<T
(one should add some smoothing). We want to consider ’typical’ statistical
behaviour on an interval of length one’ . Let x ∈ [0, 1] and consider
X 1
√ p −ix p −iωT ,
p
p≤T
where ω is random, distributes uniformly on [0, 1], and T >> 1. In the
limit T → ∞ the quantities
p −iωT ,
p = 2, 3, 5, 7, . . .
become independent uniform distributions on T = {|z| = 1}.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
2 / 11
The statistical model
One ends up with the statistical model of the field log(ζ(1/2 + ix + ·))
log(ζ(1/2 + ix + ·)) ∼
X
p
p −ix
e iθp √ ,
p
where (θp )p are i.i.d. random variables, uniformly distributed on [0, 2π]
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
3 / 11
The statistical model
One ends up with the statistical model of the field log(ζ(1/2 + ix + ·))
log(ζ(1/2 + ix + ·)) ∼
X
p
p −ix
e iθp √ ,
p
where (θp )p are i.i.d. random variables, uniformly distributed on [0, 2π], or
log |ζ(1/2 + it + ·)|
∞
X
1
cos(x log pj ) cos θpj + sin(x log pj ) sin θpj ,
∼ X (x) :=
√
pj
j=1
where pj :s are primes listed in an increasing order.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
3 / 11
The statistical model
One ends up with the statistical model of the field log(ζ(1/2 + ix + ·))
log(ζ(1/2 + ix + ·)) ∼
X
p
p −ix
e iθp √ ,
p
where (θp )p are i.i.d. random variables, uniformly distributed on [0, 2π], or
log |ζ(1/2 + it + ·)|
∞
X
1
cos(x log pj ) cos θpj + sin(x log pj ) sin θpj ,
∼ X (x) :=
√
pj
j=1
where pj :s are primes listed in an increasing order. We get a well-behaved
approximation (our basic object) by considering the partial sum
N
X
1
XN (x) =
cos(x log pj ) cos θpj + sin(x log pj ) sin θpj .
√
pj
j=1
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
3 / 11
Properties of the field
• This is the randomised Euler product model, whose maxima was
considered in [Arguin, Belius, Harper 2015] (recall Adam’s talk on
Monday!)
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
4 / 11
Properties of the field
• This is the randomised Euler product model, whose maxima was
considered in [Arguin, Belius, Harper 2015] (recall Adam’s talk on
Monday!)
• The random field X (x) can be thought as a random generalized
function on on (0, 1).
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
4 / 11
Properties of the field
• This is the randomised Euler product model, whose maxima was
considered in [Arguin, Belius, Harper 2015] (recall Adam’s talk on
Monday!)
• The random field X (x) can be thought as a random generalized
function on on (0, 1).
• X not a Gaussian field. This is so despite of the fact that
(log log N)−1/2 XN (x) has a Gaussian limit as N → ∞ for any fixed x !
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
4 / 11
Properties of the field
• This is the randomised Euler product model, whose maxima was
considered in [Arguin, Belius, Harper 2015] (recall Adam’s talk on
Monday!)
• The random field X (x) can be thought as a random generalized
function on on (0, 1).
• X not a Gaussian field. This is so despite of the fact that
(log log N)−1/2 XN (x) has a Gaussian limit as N → ∞ for any fixed x !
• Thus we are dealing with a random non-gaussian and non-harmonic
Fourier series.
•
However, the field X is log-correlated.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
4 / 11
Properties of the field
• This is the randomised Euler product model, whose maxima was
considered in [Arguin, Belius, Harper 2015] (recall Adam’s talk on
Monday!)
• The random field X (x) can be thought as a random generalized
function on on (0, 1).
• X not a Gaussian field. This is so despite of the fact that
(log log N)−1/2 XN (x) has a Gaussian limit as N → ∞ for any fixed x !
• Thus we are dealing with a random non-gaussian and non-harmonic
Fourier series.
•
However, the field X is log-correlated.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
4 / 11
What are we after?
•
Our main question: does the field X define a multiplicative chaos ?
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
5 / 11
What are we after?
•
Our main question: does the field X define a multiplicative chaos ?
In other words, for given β > 0 is there a non-trivial limiting random
measure µβ on (0, 1):
w∗
µB =limN→∞ µβ,N ,
where
µβ,N (dx) :=
Saksman
exp(βXn (x))
dx
E exp(βXn (x))
Chaos for models of zeta
?
Bristol, May 11, 2016
5 / 11
What are we after?
•
Our main question: does the field X define a multiplicative chaos ?
In other words, for given β > 0 is there a non-trivial limiting random
measure µβ on (0, 1):
w∗
µB =limN→∞ µβ,N ,
where
µβ,N (dx) :=
•
exp(βXn (x))
dx
E exp(βXn (x))
?
µβ could be thought as a toy model of (suitably normalized)
|ζ(1/2 + ix + ·)|β
.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
5 / 11
What are we after?
•
Our main question: does the field X define a multiplicative chaos ?
In other words, for given β > 0 is there a non-trivial limiting random
measure µβ on (0, 1):
w∗
µB =limN→∞ µβ,N ,
where
µβ,N (dx) :=
•
exp(βXn (x))
dx
E exp(βXn (x))
?
µβ could be thought as a toy model of (suitably normalized)
|ζ(1/2 + ix + ·)|β
.
• By martingale property, the existence of the limit is easy! Non-triviality
and properties of the limit are not so obvious.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
5 / 11
Results
THEOREM 1 •
Saksman
The limit µβ (dx) is non-trivial for β ∈ (0, 2).
Chaos for models of zeta
Bristol, May 11, 2016
6 / 11
Results
THEOREM 1 • The limit µβ (dx) is non-trivial for β ∈ (0, 2).
• Actually, there is a Gaussian multiplicative chaos-measure νβ on (0, 1)
such that
µβ = f νβ ,
where the random function f : (0, 1) → R+ is almost surely continuous
and bounded from above and below.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
6 / 11
Results
THEOREM 1 • The limit µβ (dx) is non-trivial for β ∈ (0, 2).
• Actually, there is a Gaussian multiplicative chaos-measure νβ on (0, 1)
such that
µβ = f νβ ,
where the random function f : (0, 1) → R+ is almost surely continuous
and bounded from above and below.
• For β ≥ 2, µβ,N (dx) converges almost surely to the zero measure.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
6 / 11
Results
THEOREM 1 • The limit µβ (dx) is non-trivial for β ∈ (0, 2).
• Actually, there is a Gaussian multiplicative chaos-measure νβ on (0, 1)
such that
µβ = f νβ ,
where the random function f : (0, 1) → R+ is almost surely continuous
and bounded from above and below.
• For β ≥ 2, µβ,N (dx) converges almost surely to the zero measure.
THEOREM 2 •
In the critical case β = βc := 2, the normalised measure
p
log log Nµβc ,N (dx)
converges in distribution to a non-trivial random measure µc .
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
6 / 11
Results
THEOREM 1 • The limit µβ (dx) is non-trivial for β ∈ (0, 2).
• Actually, there is a Gaussian multiplicative chaos-measure νβ on (0, 1)
such that
µβ = f νβ ,
where the random function f : (0, 1) → R+ is almost surely continuous
and bounded from above and below.
• For β ≥ 2, µβ,N (dx) converges almost surely to the zero measure.
THEOREM 2 •
In the critical case β = βc := 2, the normalised measure
p
log log Nµβc ,N (dx)
converges in distribution to a non-trivial random measure µc .
• Also µc is absolutely continuous with respect to a critical Gaussian
multiplicative chaos measure.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
6 / 11
The Gaussian approximation
The above results are based on showing first that the field X and the
approximations XN are very close to Gaussian ones:
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
7 / 11
The Gaussian approximation
The above results are based on showing first that the field X and the
approximations XN are very close to Gaussian ones:
THEOREM 3 There is the decomposition
XN (x) = GN (x) + EN (x),
where
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
7 / 11
The Gaussian approximation
The above results are based on showing first that the field X and the
approximations XN are very close to Gaussian ones:
THEOREM 3 There is the decomposition
XN (x) = GN (x) + EN (x),
where
• GN (x) is a Gaussian field that has the covariance structure of a
standard approximation to a log-correlated field.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
7 / 11
The Gaussian approximation
The above results are based on showing first that the field X and the
approximations XN are very close to Gaussian ones:
THEOREM 3 There is the decomposition
XN (x) = GN (x) + EN (x),
where
• GN (x) is a Gaussian field that has the covariance structure of a
standard approximation to a log-correlated field.
• EN is smooth and almost surely EN → E uniformly, where E is almost
surely continuous on (0, 1).
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
7 / 11
The Gaussian approximation
The above results are based on showing first that the field X and the
approximations XN are very close to Gaussian ones:
THEOREM 3 There is the decomposition
XN (x) = GN (x) + EN (x),
where
• GN (x) is a Gaussian field that has the covariance structure of a
standard approximation to a log-correlated field.
• EN is smooth and almost surely EN → E uniformly, where E is almost
surely continuous on (0, 1).
• The maximal error in the approximation has finite exponential
moments:
E exp λ
sup
|EN (x)| < ∞ for all λ > 0.
N≥1,x∈[0,1]
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
7 / 11
Some ideas of the proofs
Proof of Thm 3:
m(n)
•
Divide Xn into blocks
Xn (x) ∼
X
Ym (x), where
m=1
rm+1 −1
Ym (x) =
X
k=rm
Saksman
1
√ (cos(x log pk ) cos θpk + sin(x log pk ) sin θpk ) .
pk
Chaos for models of zeta
Bristol, May 11, 2016
8 / 11
Some ideas of the proofs
Proof of Thm 3:
m(n)
•
Divide Xn into blocks
Xn (x) ∼
X
Ym (x), where
m=1
rm+1 −1
Ym (x) =
X
k=rm
1
√ (cos(x log pk ) cos θpk + sin(x log pk ) sin θpk ) .
pk
Approximate the m:th block by ’freezing’ the factors log(pk ):
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
8 / 11
Some ideas of the proofs
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
9 / 11
Some ideas of the proofs
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
• Use 2-dim Gaussian approximation on the sums inside brackets
(carefully quantitative - CLT or simple Barry-Esseen not enough).
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
9 / 11
Some ideas of the proofs
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
• Use 2-dim Gaussian approximation on the sums inside brackets
(carefully quantitative - CLT or simple Barry-Esseen not enough).
• After replacing the sums by the Gaussian equivalents, repeat the steps
backwards to obtain the Gaussian approximation
rm+1 −1
1 X 1
e
Ym (x) ≈ Gm (x) := √
√ (cos(x log pk )g1,pk + sin(x log pk )g2,pk ) .
pk
2 k=r
m
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
9 / 11
Some ideas of the proofs
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
• Use 2-dim Gaussian approximation on the sums inside brackets
(carefully quantitative - CLT or simple Barry-Esseen not enough).
• After replacing the sums by the Gaussian equivalents, repeat the steps
backwards to obtain the Gaussian approximation
rm+1 −1
1 X 1
e
Ym (x) ≈ Gm (x) := √
√ (cos(x log pk )g1,pk + sin(x log pk )g2,pk ) .
pk
2 k=r
m
• The sup-norms of the errors induced are approximated by suitable
Sobolev norms.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
9 / 11
Some ideas of the proofs
em (x) := cos(x log pr )
Ym (x) ≈ Y
m
rm+1
X−1
k=rm
+ sin(x log prm
1
√ cos θpk
pk
rm+1
X−1 1
)
√ sin θpk
pk
k=rm
• Use 2-dim Gaussian approximation on the sums inside brackets
(carefully quantitative - CLT or simple Barry-Esseen not enough).
• After replacing the sums by the Gaussian equivalents, repeat the steps
backwards to obtain the Gaussian approximation
rm+1 −1
1 X 1
e
Ym (x) ≈ Gm (x) := √
√ (cos(x log pk )g1,pk + sin(x log pk )g2,pk ) .
pk
2 k=r
m
• The sup-norms of the errors induced are approximated by suitable
1/3
Sobolev norms. The final choice for the subsequence (rm ) is rm := e m .
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
9 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Uniform integrability follows then from this and the exponential moments
of supN kEN k∞ .
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Uniform integrability follows then from this and the exponential moments
of supN kEN k∞ .
Proof of Thm 2:
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Uniform integrability follows then from this and the exponential moments
of supN kEN k∞ .
Proof of Thm 2:
• Need to modify the Gaussian approximation so that the obtained field
belongs to the class covered by the fundamental result on the existence of
critical continuous chaos in [Duplantier-Rhodes-Scheffield-Vargas 2014].
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Uniform integrability follows then from this and the exponential moments
of supN kEN k∞ .
Proof of Thm 2:
• Need to modify the Gaussian approximation so that the obtained field
belongs to the class covered by the fundamental result on the existence of
critical continuous chaos in [Duplantier-Rhodes-Scheffield-Vargas 2014].
• We verify that a ’scaling invariant’ 1-dim log-correlated field works for
the above.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Some ideas of the proofs
Proof of Thm 1:
• Enough to quarantee uniform integrability.
• Above, the Gaussian approximations GN have a standard log-normal
approximation structure. This leads to Lp -bounds for the Gaussian part.
Uniform integrability follows then from this and the exponential moments
of supN kEN k∞ .
Proof of Thm 2:
• Need to modify the Gaussian approximation so that the obtained field
belongs to the class covered by the fundamental result on the existence of
critical continuous chaos in [Duplantier-Rhodes-Scheffield-Vargas 2014].
• We verify that a ’scaling invariant’ 1-dim log-correlated field works for
the above.
• Finally, to prove the convergence of the Gaussian part one applies a
uniqueness result for critical chaos in [Junnila-S 2015], and the rest is not
difficult.
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
10 / 11
Thanks for your patience!
Saksman
Chaos for models of zeta
Bristol, May 11, 2016
11 / 11