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```Zeros of random analytic functions and
extreme value theory
Zakhar Kabluchko
University of Ulm
November 9, 2012
A random equation
Statement of the problem
We consider an algebraic equation with random coefficients, for
example
z 2000 − z 1999 + z 1998 + z 1997 − z 1996 − . . . + z 3 + z 2 − z + 1 = 0
What is the distribution of the solutions of this equation in the
complex plane?
2
3
Solutions of the equation
Zeros of a random polynomial of degree n = 2000
Random Polynomials (Kac Ensemble)
Statement of the problem
Let ξ0 , ξ1 , . . . be i.i.d. random variables.
Consider the equation
Pn (z) := ξ0 + ξ1 z + ξ2 z 2 + . . . + ξn z n = 0.
The equation has n complex roots z1 , . . . , zn .
What is the distribution of roots in the complex plane?
Consider the empirical measure
n
1X
δ(zk ).
µn =
n k=1
Problem: Find limn→∞ µn .
Distribution of roots
Theorem (Ibragimov, Zaporozhets, 2011)
The following conditions are equivalent:
1
With probability 1, the sequence µn converges weakly to
the uniform distribution on the unit circle.
2
E log(1 + |ξ0 |) < ∞.
Remark
P∞
k
The series
converges a.s. in the unit circle iff
k=0 ξk z
E log(1 + |ξ0 |) < ∞.
History
Hammersley (1954), Shparo und Shur (1962), Arnold (1966),
Shepp und Vanderbei (1995), ...
5
Logarithmic tails
Problem
What happens if the coefficients ξk are heavy-tailed?
Logarithmic tails
We consider coefficients satisfying
P[|ξ0 | > t] ∼
L(log t)
, t → +∞,
(log t)α
where α > 0 and L is a slowly varying function.
Remark
(
infinite, α < 1,
E log(1 + |ξ0 |) is
finite,
α > 1.
7
Example: Coefficients with logarithmic tails
The coefficients ξk are such that P[|ξk | > t] = 1/(log t)2 and
the degree is n = 2000.
Distribution of roots
Assume for simplicity that
P[|ξ0 | > t] ∼
1
(log t)α
as t → +∞.
(1)
Theorem (Kabluchko, Zaporozhets, 2011)
For coefficients with logarithmic tails, the following weak convergence of random probability measures holds:
n
1
1X
w
hn1− α i
δ(zk
) −→ Πα .
n→∞
n k=1
The limiting random probability measure Πα is a.s. a convex
combination of at most countably many uniform measures concentrated on circles centered at the origin.
9
Light and logarithmically tailed coefficients
Distribution of roots
Example: α = 1
If P[|ξ0 | > t] ∼
1
log t
as t → +∞, then we have
n
1X
w
δ(zk ) −→ Π1 .
n→∞
n k=1
No normalization of roots is needed.
Example: α > 1 (E log |ξ0 | < ∞)
The roots approach the unit circle. The distance between the
1
roots and the unit circle is of order n α −1 .
Example: α < 1 (E log |ξ0 | = ∞)
The roots diverge to ∞ and 0. The absolute values of the roots
1 −1
are of order O(1)n α .
Extremal order statistics of the coefficients
Theorem
Let ξ0 , ξ1 , . . . be i.i.d. random variables with
P[log |ξk | > t] ∼ t −α as t → +∞.
Then, we have the following weak convergence of point processes on [0, 1] × (0, ∞):
n
X
k log |ξk |
w
δ
, 1/α
−→ PPP αv −(α+1) dudv .
n→∞
n n
k=0
11
12
Extremal order statistics of the coefficients
Newton polygon
Idea of the proof
For large n consider the equation
±e nx1 ± . . . ± e nxd = 0,
where xi > 0. The most easy way for this to be true is the
following:
1
Two terms, say e nxk and e nxl , cancel each other.
2
All other terms are much smaller than these two.
Newton polygon
Idea of the proof
Apply this to the equation ξ0 + ξ1 z + . . . + ξn z n = 0.
1
Two terms cancel each other: ξk z k + ξl z l = 0.
2
All other terms are much smaller than these two.
Geometrically: the points (k, log |ξk |) and (l, log |ξl |) are neighboring vertices of the least concave majorant of the set
{(0, log |ξ0 |), . . . , (n, log |ξn |)}.
14
The least concave majorant
Remark
The number of segments is finite a.s. if and only if α < 1.
15
Limiting empirical measure of the roots
Limiting measure Πα
Consider the least concave majorant of the Poisson process with
intensity αv −(α+1) dudv .
1
Radii of circles = exponentials of the slopes of the majorant.
2
Number of roots on a circle = length of the linearity interval.
16
Very heavy tails: α = 0
Let the coefficients be such that
P[|ξ0 | > t] ∼ L(log t) as t → +∞,
where L is a slowly varying function.
The roots concentrate on two circles, one with a small radius,
one with a large radius. The proportion of the roots lying on
the small circle is uniform on [0, 1].
17
Weyl Polynomials
Let ξ0 , ξ1 , . . . be i.i.d. random variables.
Consider the Weyl Polynomials
Pn (z) =
n
X
zk
ξk √ .
k!
k=0
Let z1 , . . . , zn be the zeros of Pn .
Theorem (Kabluchko, Zaporozhets, 2012)
The following conditions are equivalent:
P
1
The sequence of probability measures n1 k=1 δ( √zkn ) converges a.s. to the uniform distribution on the unit disk
{|z| ≤ 1}.
2
E log(1 + |ξ0 |) < ∞.
18
19
Weyl Polynomials
Zeros of a Weyl polynomial: Normally distributed coefficients
20
Weyl Polynomials
Zeros of a Weyl polynomial: Logarithmic tails
Littlewood–Offord Polynomials (1939)
Let ξ0 , ξ1 , . . . be i.i.d. random variables with E log(1 +
|ξ0 |) < ∞.
Consider the Littlewood–Offord polynomials
Pn (z) =
n
X
ξk
k=0
zk
.
(k!)α
Let z1 , . . . , zn be the zeros of Pn .
Theorem (Kabluchko, Zaporozhets, 2012)
With
1, the sequence of random measures
Pn probability
zk
1
δ(
)
converges
to the probability measure with the
k=1
n
nα
density
1
1
|z| α −2 , |z| < 1.
2πα
21
Littlewood–Offord Polynomials
Zeros of the Littlewood–Offord polynomials: Normal
coefficients
Littlewood–Offord Polynomials
Zeros of the Littlewood–Offord polynomials: Logarithmic
coefficients
Szegö Polynomials
Szegö Polynomials: sn (z) =
zk
k=0 k! .
Pn
Theorem (Szegö, 1924)
The zeros of sn (nz) cluster along the curve |ze 1−z | = 1, |z| < 1.
Zeros in the Random Energy Model
Random Energy Model (Derrida, 1981)
A system has N states.
√
The energy of the system in state i is log Nξi .
ξ1 , . . . , ξN are i.i.d. standard Gaussian random variables.
Consider the partition function
ZN (β) =
N
X
eβ
√
log Nξk
, β ∈ C.
k=1
Other motivations
ZN is an empirical Laplace transform.
ZN is a nice random analytic function.
Zeros in the Random Energy Model
Complex zeros of ZN . Source: C. Moukarzel und N. Parga:
Physica A 177 (1991).
26
Zeros in the Random Energy Model
Theorem (Derrida, 1991)
For β = σ + iτ ∈ C it holds that

1 + 1 (σ 2 − τ 2 ), β ∈ B 1 ,
log |ZN (β)| √ 2
lim
=
2|σ|,
β ∈ B 2,
N→∞

log N
1
+ σ2,
β ∈ B 3,
2
where
B1 = C\B2 ∪ B3 ,
B2 = {β ∈ C : 2σ 2 > 1, |σ| + |τ | >
2
2
2
√
2},
B3 = {β ∈ C : 2σ < 1, σ + τ > 1}.
27
Zeros in the Random Energy Model
Theorem (Derrida, 1991)
The random measure
2π
log N
X
δ(β)
β:ZN (β)=0
converges weakly (as N → ∞) to the deterministic measure
Ξ = 2Ξ3 + Ξ12 + Ξ13 . Here,
Ξ3 is the Lebesgue measure on B3 .
Ξ13 is the one-dimensional Lebesgue measure on the
boundary between B1 and B3 .
√
Ξ12 is the measure with density 2|τ | on the boundary
between B1 and B2 .
Rigorous proof, further results: Kabluchko und Klimovsky, 2012.
```
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