Pairing of Zeros and Critical Points for Random Polynomials

2.1 MB·

by  Boris Hanin

PAIRING OF ZEROS AND CRITICAL POINTS FOR RANDOM
POLYNOMIALS
BORIS HANIN
6 Abstract. LetpN bearandomdegreeN polynomialinonecomplexvariablewhosezeros
1 are chosen independently from a fixed probability measure µ on the Riemann sphere S2.
0 This article proves that if we condition p to have a zero at some fixed point ξ ∈S2, then,
N
2 with high probability, there will be a critical point w a distance N−1 away from ξ. This
ξ
n N−1 distanceismuchsmallerthantheN−1/2 typicalspacingbetweennearestneighborsfor
a N i.i.d. pointsonS2.Moreover,withthesamehighprobability,theargumentofw relative
J ξ
to ξ is a deterministic function of µ plus fluctuations on the order of N−1.
4
2

] 0. Introduction
R
P Thisarticleconcernsasurprisingrelationshipbetweenzerosandcriticalpointsofarandom
h. polynomialinonecomplexvariable. Tointroduceourresults, considerFigure1, whichshows
t the zeros and critical points of p(z) = z9 −1 and of q(z) = p(z)(z −ξ) for various ξ. While
a
m the zeros and critical points of p are quite far apart, most zeros of q seem to have a unique
Sendov's Conjecture Sendov's Conjecture Sendov's Conjecture
[ nearby critical point. This effect becomes more pronounced for a polynomial whose zeros

are chosen at random as in Figure 2. What accounts for such a pairing? How close is a zero
1
v In[5]:t=oManiiptulsatpeaired critical point? WIn[3]:=hMyanipduloatees the pairing break dowIn[3]:=nManiinpulastoeme places? Why is there
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1 {{n,9},{2,3,4,5,6,7,8,9,15}}, {{n,9},{2,3,4,5,6,7,8,9,15}}, {{n,9},{2,3,4,5,6,7,8,9,15}},
0 {{{r.o1o.t1s,,.{1{.01.}`,,{11..`1},,1{..11}.,`C,o0n}t,ro{l0T.y`p,e.01.N`o}n,e}{1.`,0}}},{{{r.o1o.t1s,,.{1{.01.}`,,{11..`1},,1{..11}.,`C,o0n}t,ro{l0T.y`p,e.01.N`o}n,e}{1.`,0}}},{{{r.o1o.t1s,,.{1{.01.}`,,{11..`1},,1{..11}.,`C,o0n}t,ro{l0T.y`p,e.01.N`o}n,e}{1.`,0}}},
6 Research partially supported by NSF grant DMS-1400822. 
1
v: n 9 n 9 n 9
i
X
r
a
Out[5]= Out[3]= Out[3]=
Figure 1. Figures from left to right display zeros (blue squares) and critical
points (orange disks) of p(z) = z9 −1 and of q(z) = p(z)(z −ξ) for various ξ.
1
2 Sendov_random.cdf 2 Sendov_random.cdf 2 Sendov_random.cdf
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], ], ],
{{n,10},{2,3,4,5,6,7,8,9,10,15,20,25,30,40,50}{}{,n,10},{2,3,4,5,6,7,8,9,10,15,20,25,30,40,50}{}{,n,10},{2,3,4,5,6,7,8,9,10,15,20,25,30,40,50}},
{{roots,{{0.`,1.`},{11.`,0},{0.`,11.`},{1.`,0}}},{{roots,{{0.`,1.`},{11.`,0},{0.`,11.`},{1.`,0}}},{{roots,{{0.`,1.`},{11.`,0},{0.`,11.`},{1.`,0}}},
{12.1,12.1},{2.1,2.1},ControlType3None} {12.1,12.1},{2.1,2.1},ControlType3None} {12.1,12.1},{2.1,2.1},ControlType3None}
  
2 BORIS HANIN
n 10 n 20 n 30
Out[39]= Out[39]= Out[39]=
4 Figure 2. Figures from4 left to right display zeros (4blue squares) and crit-
; ; ;
ical points (orange disks) of a polynomial with 10,20,30 zeros each chosen
CAPTION CAPTION CAPTION
independently from the standard Gaussian measure on C.
There is a vast literature on the distribution of zeros of random polynomials, and we will
not attempt to survey it here. We simply mention that they have been studied from the
pointofviewofrandomanalyticfunctions(cf[15]); randommatrixtheory, whereeigenvalues
are zeros of characteristic polynomials of random matrices (cf [2]); and determinantal point
processes (cf [9]). Previous results specifically relating zeros and critical points of random
polynomials are much more limited, however. This is somewhat surprising because there are
many interesting deterministic theorems that restrict the possible locations of critical points
of a polynomial in terms of the locations of its zeros. We recall two such results, and refer
the reader to Marden’s book [11] for many more.
Theorem (Gauss-Lucas). The critical points of a polynomial in one complex variable lie
inside the convex hull of its zeros.
Theorem (Theorem 3.55 in [17]). Let f be a non-zero holomorphic function on a simply
connected domain Ω ⊆ C, and take Γ to be a smooth closed connected component of the level
set {|f(z)| = t} for some t. Write U = {|f(z)| < t} for the open domain bounded by Γ. Then
(cid:18) (cid:19)
d
#{f(z) = 0}∩U = #{ f(w) = 0}∩U +1.
dw
We are aware of only three previous works concerning the sort of a pairing between zeros
and critical points discussed here. From the math literature, there are the author’s two
articles [7, 8] which study a large class of Gaussian random polynomials called Hermitian
Gaussian Ensembles (HGEs). The simplest HGE is the SU(2) or Kostlan ensemble:
(cid:115)
N (cid:18) (cid:19)
(cid:88) N
(0.1) pSU(2)(z) := a zj, a ∼ N(0,1) i.i.d.
N j j j C
j=0
Even for the SU(2) ensemble, the pairing of zeros and critical points was a new result (see
Figure 2). The proofs in [7, 8] are not elementary, however, because the distribution of
zeros and critical points for HGEs is highly non-trivial and is written in terms of so-called
Bergman kernels. Moreover, most of the theorems in [7, 8] do not discuss the angular
dependence between a zero and its paired critical point. The present article, in constrast,
studies the simplest possible ensembles of random polynomials from the point of view of the
PAIRING OF ZEROS AND CRITICAL POINTS 3
joint distribution of the zeros. Namely, we fix some some number of zeros and choose the
othersuniformlyand independently fromafixedprobabilitymeasureonS2.Inthis situation,
we are able to give the first completely elementary proof of the pairing of zeros and critical
points for a random polynomial.
The other article we are aware of is the heuristic work of Dennis-Hannay [5] from the
physics literature. They give an electrostatic explanation for why, for certain special kinds
of random polynomials, zeros with a large modulus should be paired to a critical point. In
§2 we give a somewhat different and more flexible electrostatic argument that explains the
pairing of zeros and critical points. Our reasoning also predicts the distance between from a
zero to its paired critical point as well as the existence of regions where such a pairing breaks
down.
To conclude, let us mention the works of Kabluchko [10], Pemantle-Rivlin [13], and Sub-
ramanian [16], which study the empirical measure of critical points for ensembles of random
polynomials similar to the ones we consider here. We also point the reader to the work of
Nazarov-Sodin-Volberg [12] and the recent article of Feng [6], which both concern the critical
points of random holomorphic functions with respect to a smooth connection.
Figure 3. Zeros (black disks) and critical points (blue squares) for a degree
50 SU(2) polynomial p (defined in (0.1)) displayed on the left in coordinates
C ∼= S2\{∞} and on the right directly on S2. The colored lines are gradient
flow lines for −|p(z)|2. Lines of the same color terminate at the same point.
The origin is denoted by a red asterisk in both figures.
1. Acknowledgements
I am grateful to Leonid Hanin and Steve Zelditch for many useful comments on earlier
drafts of this article. I am also indebted to Ron Peled who shared with me the Matlab code
(originally written by Manjunath Krishnapur) that I modified to create Figures 3 and 4.
Finally, I would like to acknowledge Bruce Torrence and Paul Abbott, whose Mathematica
demonstration [1] I modified to create Figures 1 and 2.
4 BORIS HANIN
2. Electrostatic Interepretation of Zeros and Critical Points
The idea that zeros and critical points of a complex polynomials have an electrostatic
interpretation goes back to Gauss [11, Preface and §2]. And the observation that zeros
with a large modulus should be paired to a critical point in certain special kinds of random
polynomials (and for certain random entire functions) was stated by Dennis and Hanny in
[5]. They give a heuristic explanation for this pairing, which is similar in spirit to ours, but
does not use that polynomials have a high multiplicity pole at infinity.
We begin by explaining Gauss’s proof of the Gauss-Lucas Theorem. Suppose p is a
N
degree N polynomial and ξ , j = 0,...,N − 1 are its zeros. The critical points of p are
j N
solutions to
N−1
d (cid:88) 1
p (w) = 0 ←→ ∂log|p (w)|2 = = 0.
N N
dw w−ξ
j
j=0
(cid:0) (cid:1)−1
A basic observation is that z −ξ is the electric field at z from a +1 charge at ξ . The
j j
sum ∂log|p (z)|2 is therefore the complex conjugate of the total electric field E (z) at z
N N
from positive point charges placed at each ξ . This means
j
d
p (w) = 0 ←→ electric field satisfies E (w) = 0.
N N
dw
Since all charges have the same sign, E cannot vanish outside of their convex hull. Hence
N
the Gauss-Lucas Theorem.
The preceeding argument relied very much on the particular coordinates on S2 (the con-
vex hull is not a coordinate-free notion). However, the electrostatic interpretation of critical
points can itself be done in a coordinate invariant way. We start by viewing p as a mero-
N
morphic function on S2. Then
(cid:88)
∆log|p |2 = Div(p ) = −Nδ + δ ,
N N ∞ ξ
pN(ξ)=0
where we’ve written ∆ = i ∂∂ for the Laplacian, and the equality is in the sense of distri-
2π
butions. This means that
E (w) = ∂log|p (w)|2
N N
is a one-form that at any w ∈ S2 gives the (complex conjugate of the) electric field at w from
charge distributed according to Div(p ). Gauss’s choice of coordinates makes ∞ infinitely
N
far way and and so the contribution to E from the −N charges at infinity was zero. To
N
get a true electric field, we need to covert E to a vector field using a metric. However, the
N
points where E vanishes (i.e. the critical point of p ) are independent of such a choice.
N N
Thispointofviewiscloselyrelatedtotheclassicalnotionofapolarderivativeofacomplex
polynomial [11, §3], which can be used to prove some coordinate free versions of the Gauss-
Lucas Theorem such as Laguerre’s Theorem [11, p. 49]. That it should imply a pairing of
zeros and critical points never seems to have been observed, however.
It is precisely the charge of size N at ∞ that clarifies why zeros and critical points come in
pairs. To see this, let us consider the case of a degree N polynomial p that has a zero at a
N,ξ
fixed point ξ ∈ S2, while its remaining zeros ξ ,...,ξ are chosen independently from the
1 N−1
uniform measure µ on S2. We must explain why, with high probability, there is a point w
ξ
PAIRING OF ZEROS AND CRITICAL POINTS 5
near ξ at which the electric field E (w ) vanishes. In the holomorphic coordinate w centered
N ξ
at ∞, we have
N−1
1 (cid:89)
(2.1) p (w) = (w−ξ) (w−ξ ) ξ ∼ µ i.i.d.
N,ξ wN j j
j=1
Thus,
N−1
N 1 (cid:88) 1
(2.2) E (w) = − + + ,
N
w w−ξ w−ξ
j
j=1
The first term in (2.2) is the contribution from the −N charges at ∞, while the second comes
from the +1 charge at ξ, which is also of order N if |w−ξ| ≈ N−1. The third term is a sum
of iid random variables. It is equal to zero on average (cf (4.1)). Hence, heuristically, it is
should be on the order of N1/2 by the central limit theorem. Therefore, to leading order in
N, the electric field near ξ is very close to its average
N 1
(2.3) E (w) ≈ E[E (w)] = − + .
N N
w w−ξ
As long as ξ (cid:54)∈ {0,∞} (here 0 is the antipodal point on S2 to ∞) there will be a unique
solution
(cid:18) 1 (cid:19)−1
(2.4) w = ξ 1−
N,ξ
N
to
E[E (w)] = 0
N
in the regime where the approximation (2.3) is valid. Note that the distance from w to w is
ξ
on the order of N−1. The true critical point of p will, by Rouch´e’s Theorem, therefore be
N
a small perturbation of w . Note that w is a bit farther away from ∞ than ξ and hence
N,ξ ξ
is closer to 0 as shown in Figures 1-4. The condition that ξ (cid:54)∈ {0,∞} is not an artifact of
our reasoning. Figures 1-4 clearly show the existence of regions where the pairing of critical
points breaks down: at ξ = 0, the contribution to E from the charges at ∞ precisely
N
cancels by symmetry. Near 0 the field E is controlled by the nearby zeros whose statistical
N
fluctuations cannot be ignored.
Now let us suppose that the zeros ξ ,...,ξ are still uniformly distributed but now
1 N−1
according to an arbitrary probability measure µ on S2. There will still be a pairing of zeros
and critical points, but the pairs will no longer necessarily align with ∞. Indeed, the main
difference in this case is that instead of (2.3), the electric field near ξ will have a non-zero
contribution from the average of the third term in (2.2). To leading order in N, we have
(cid:18) (cid:19)
1 1
(2.5) E (w) ≈ E[E (w)] ≈ N − +φ (w) + ,
N N µ
w w−ξ
where
(cid:90)
dµ(ζ)
(2.6) φ (w) =
µ
w−ζ
C
is the (complex conjugate of the) average electric field at w from a zero distributed according
to µ. To ensure a unique point w where the right hand side of (2.5) vanishes that is near
N,ξ
6 BORIS HANIN
ξ we must ask that ξ (cid:54)∈ S , where
µ
(cid:26) (cid:12) (cid:27)
(2.7) S := w ∈ C(cid:12)(cid:12)−1 +φ (w) ∈ {0,∞} .
µ µ
(cid:12) w
Expilcitly,
(cid:18) (cid:19)
1 1
(2.8) w = ξ 1− ·
N,ξ
N φ (ξ)·ξ −1
µ
plus an O(N−2) error and
(cid:18) (cid:19)
1
arg(w −ξ) = arg(ξ)−arg −φ (ξ) +O(N−1)
N,ξ µ
ξ
Before stating the rigorous results of this article, we remark that the heuristic argument
given here does not make strong use of the iid nature of the zeros of p . It does crucially
N
rely on the assumption that they are well-spaced and not too correlated, however.
3. Main Result
Our main result, Theorem 1, can be stated loosely as follows. Consider a degree N
polynomial p , viewed as a meromorphic function on S2. Suppose p has a zero at a fixed
N N
point ξ, while its other zeros are randomly and independently selected. Then, we are likely
to observe a critical point w of p a distance about N−1 away from ξ.
ξ N
Note that if we choose N independent points at random from the uniform measure on S2,
then the typical spacing between nearest neighbors is on the order of N−1/2, which is much
larger than the N−1 spacing between a zero and its paired critical point. Observe also that
Theorem 1 is genuinely probabilistic and does not hold for polynomials of the form zN−RN.
Nonetheless, as explained in the Introduction, multiplying z9 − 1 by a single linear factor
already makes the zeros and critical points of the resulting polynomial come in pairs (see
Figure 1).
Let us write P for the space of polynomials of degree at most N in one complex variable.
N
Since the zeros and critical points of p ∈ P are unchanged after multiplication by a non-
N N
zero constant, we study random the zeros and critical points of a random polynomial by
putting a probability measure directly on the projectivization P(P ) as follows.
N
Definition 1. Fix ξ ∈ S2 and a probability measure µ with a bounded density with respect
to the uniform measure on S2. Define [p ] to be a random element in P(P ) with a (de-
N,ξ N
terministic) zero at ξ and N −1 (random) zeros {ξ ,...,ξ } distributed according to the
1 N−1
product measure µ⊗(N−1) on (S2)N−1.
Slightly abusing notatoin, will hencefore write p for any representative of [p ]. We also
N,ξ N,ξ
identify once and for all polynomials with meromorphic functions on S2 that have a pole at
the distinguished point ∞ ∈ S2, and we will write 0 for the antipodal point to ∞. With w
denoting the usual holomorphic coordinate on S2 centered at ∞, p is given by (2.1).
N,ξ
Let S be defined as in (2.7). For each ξ ∈ S2\S , we define w to be the unique
µ µ ξ,N
solution to the averaged critical point equation
(3.1) E(cid:2)∂ log|p (w)|2(cid:3) = 0
w N,ξ
PAIRING OF ZEROS AND CRITICAL POINTS 7
whose distance from ξ is on the order of N−1. The point w is therefore a point whose
N,ξ
distance from ξ is approximately N−1 where the electric field is expected to vanish. See
(2.8) for an asymptotic formula for w .
ξ,N
Theorem 1 (Pairing of Single Zero and Critical Point). Let µ be any probability measure
on S2 that has a bounded density with respect to the uniform measure, and fix ξ ∈ S2\S ,
µ
where S is defined in (2.7). Consider p as in Definition 1. Fix r > 0, and write Γ for
µ N,ξ N
the geodesic circle of radius rN−1 centered at w . Suppose that ξ (cid:54)∈ Γ for all N. Then,
N,ξ N
for any δ ∈ (0,1), there exists C = C(r,δ) > 0 so that for all N
(cid:18) (cid:19)
d
P ∃! w inside Γ s.t. p (w) = 0 ≥ 1−C ·N−δ.
N N,ξ
dw
Remark 1. The conclusion of Theorem 1 is actually true for any simple closed contour Γ
N,ξ
with winding number 1 around w that does not pass through ξ and satisfies:
ξ,N
(i) There exists c > 0 so that
1
inf (cid:12)(cid:12)E(cid:2)∂log|p (w)|2(cid:3) (cid:12)(cid:12) ≥ c ·N.
N,ξ 1
w∈Γ
N,ξ
(ii) There exists c > 0 so that for all N
2
sup d (w,ξ) ≤ c ·N−1,
S2 2
w∈Γ
N,ξ
where d is the usual distance function on S2.
S2
3.1. Generalizations of Theorem 1. Theorem 1 can generalized in many different ways.
A particularly simple extension concerns the simultaneous pairing of Nα zeros and critical
points for any α ∈ [0,1). To give a exact statement, consider for each N a finite collection
of at most N points Ξ ⊆ S2\S . Define p to be a random degree N polynomial
N µ N,ΞN
that vanishes at each ξ ∈ Ξ and whose other zeros are chosen independently from µ. As
N
above, this defintion actually specifies an equivalence class [p ] in P(P ) all of whose
N,ΞN N
representatives have the same zeros and critical points.
Theorem 2 (Pairing of Nα Zeros and Critical Points). Fix α ∈ [0,1) and δ ∈ (0,1 − α).
For every N ≥ 1, choose an integer n(N) between 1 and Nα and a collection of n(N) points
Ξ ⊆ S2\S satisfying
N µ
(A) There exists (cid:15) ∈ (0,1) so that for every N and every pair of distinct point ξ ,ξ ∈ Ξ
1 2 N
we have |ξ −ξ | > N−1/2+(cid:15)/2.
1 2
Define p as above, and for each N and every ξ ∈ Ξ , let Γ be the geodesic circle of
N,ΞN N N,ξ
radius rN−1 centered at w . If ξ (cid:54)∈ Γ for all N, then there exists C = C(α,δ,(cid:15),r) > 0
N,ξ N,r,ξ
so that
(cid:18) (cid:19)
d
P ∀ξ ∈ Ξ ∃! w inside Γ s.t. p (w) = 0 ≥ 1−C ·N−δ.
N N,ξ dw N,ΞN
Remark 2. The conclusion of Theorem 2 is satisfied if Γ is any contour satisfying (A)
N,ξ
and conditions (i),(ii) from Remark 1.
There are other directions in which Theorem 1 can be extended. We indicate some of
them here.
8 BORIS HANIN
(1) The assumption that µ has a bounded density with respect to Haar measure ensures
that typical spacings between the random zeros are like N−1/2. The intuitive electro-
static argument in §2 for why zeros and critical points are paired used only that zeros
are weakly correlated and spaced more than N−1 apart, however. This suggests that
perhaps µ can be taken to be any measure satisfying a finite energy condition
(cid:90) (cid:90)
log(d (z,w))dµ(w)dµ(z) < ∞,
S2
S2 S2
which rules out µ having an atom or being supported on a curve.
(2) The 1−C N−δ estimate from Theorem 1 is sharp in the sense that with probability
δ
on the order of N−1 there exists a j so that |ξ −ξ| ≤ N−1. Such a zero will disrupt
j
the local pairing of zeros and critical points. However, if one studies ensembles
of polynomials for which zeros repel one another, then the N−δ estimates can be
improved. For instance, in [7, 8] the author studied such ensembles and showed that
the probability of pairing is like N−3/2. Zeros will always repel for polynomials whose
coefficients relative to a fixed basis are taken to be iid since the change of variables
fromcoefficientstozerosinvolvesaVandermondedeterminant. Itshouldthereforebe
possible to generalize the results in this paper to the case when zeros are distributed
like a Coulomb gas.
(3) Theorem 1 will not be true as stated for polynomials whose zeros tend to be N−1
apart. Consider, for example, the Kac polynomials pKac(z) = (cid:80)N a zj with a ∼
N j=0 j j
N(0,1) iid. The zeros are well-known to approximately equidistribute on the unit
C
circle S1. Nonetheless it is clear from Figure 4 that most zeros are still paired to a
unique critical point. A general family of such ensembles introduced by Shiffman-
Zelditch in [14] and further studied by Bloom in [3] and Bloom-Shiffman in [4].
Figure 4. Zeros (black disks) and critical points (blue squares) for pKac. The
50
colored lines are gradient flow lines for (cid:12)(cid:12)pKac(z)(cid:12)(cid:12)2. Lines of the same color
50
terminate at the same point.
PAIRING OF ZEROS AND CRITICAL POINTS 9
4. Proof of Theorem 1
We prove Theorem 1 when µ is the uniform measure on S2 since the argument for general
µ is identical and only involves carrying along various factors of φ (w). For the uniform
µ
measure, the average electric field at any fixed w from one of the random zeros vanishes. To
see this, we compute in polar coordinates around w :
(cid:90) (cid:90)
dµ(ζ) 1 idζ ∧dζ
φ (w) = = ·
µ S2 w−ζ C ζ 2π(cid:0)1+|ζ|2(cid:1)2
(cid:90) ∞ (cid:90) 2π
(4.1) = (cid:0)1+r2(cid:1)−2dr· e−iθdθ = 0.
0 0
We therefore have
S = {0,∞}.
µ
We work in the holomorphic coordinate w centered at ∞ and fix ξ ∈ S2\S . In our coordi-
µ
nates, p is given by (2.1) and w , which we shall henceforth abbreviate w , is given by
N,ξ ξ,N ξ
(2.4). Write as in §2
N−1
N (cid:88) 1
E (w) = − + ,
N
w w−ξ
j
j=0
and recall that critical points of p are solutions to E = 0. The contour Γ satisfies
N,ξ N N
(i) There exists c = c (r,ξ) > 0 so that
1 1
(cid:12) (cid:12)
(cid:12) N 1 (cid:12)
inf (cid:12)− + (cid:12) ≥ c ·N.
1
w∈ΓN (cid:12) w w−ξ(cid:12)
(ii) There exists c = c (r,ξ) > 0 so that
2 2
sup |w−ξ| ≤ c ·N−1,
2
w∈ΓN
which are precisely the conditions from Remark 1. Write
N−1
(cid:88) 1
E(cid:101) (w) := E (w)−E[E (w)] = ,
N N N
w−ξ
j
j=1
and fix δ ∈ (0,1). We will show that there exists γ = γ(c ,c ,δ) > 0 and C = C (c ,c ,δ)
1 2 3 3 1 2
so that
(cid:18) (cid:19)
(cid:12) (cid:12)
(4.2) P sup (cid:12)E(cid:101) (w)(cid:12) ≤ N1−γ ≥ 1−C ·N−δ.
(cid:12) N (cid:12) 3
w∈ΓN
The relation (4.2) and Rouch´e’s theorem would then imply that p has a unique critical
N,ξ
point inside Γ with probability at least 1 − C · N−δ, as desired. The proof of (4.2) is
N 3
elementary but somewhat technical. Before giving the details we give a brief outline for the
argument.
Step 1. Estimating the supremum of the random function E(cid:101) (w) restricted to Γ is not
N N
simple to do directly. The basic reason is that E(cid:101) (w) fluctuates rather wildly (it does not
N
even have a finite variance at a point). So instead we estimate separately the fluctuations of
E(cid:101) (w)−E(cid:101) (w ) and of E(cid:101) (w ).
N N ξ N ξ
10 BORIS HANIN
Step 2. To estimate E(cid:101) (w) − E(cid:101) (w ) we use that |w−w | ≈ N−1 to throw away the
N N ξ ξ
contribution to E(cid:101) (w)−E(cid:101) (w ) coming from zeros that are far from ξ (and hence from w
N N ξ
and w as well). Specifically, by condition (ii), there exists K = K (c ,δ) so that for all N
ξ 1 1 2
(cid:12) (cid:12)
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:88) 1 1 (cid:12)
sup (cid:12)E(cid:101) (w)−E(cid:101) (w )(cid:12) = sup (cid:12) − (cid:12)
(cid:12) N N ξ (cid:12) (cid:12) w−ξ w −ξ (cid:12)
w∈ΓN w∈ΓN (cid:12)|ξj−ξ|>N−1/2+δ/2 j ξ j (cid:12)
(cid:88) |w −w|
ξ
≤ sup
|w−ξ ||w −ξ |
w∈ΓN |ξj−ξ|>N−1/2+δ/2 j ξ j
≤ K ·N1−δ.
1
Step 3. Writing
(cid:88) 1
E(cid:101) (w,δ) := ,
N
w−ξ
j
|ξ−ξj|≤N−1/2+δ/2
relation (4.2) now follows once we show that there exists γ = γ(c ,c ,δ) > 0 as well as
1 2
K = K (c ,c ,δ) > 0 and K = K (c ,c ,δ) > 0 such that
2 2 1 2 3 3 1 2
(cid:18) (cid:19)
(cid:12) (cid:12)
(4.3) P sup (cid:12)E(cid:101) (w ,δ)−E(cid:101) (w,δ)(cid:12) ≥ N1−γ ≤ K ·N−δ
(cid:12) N ξ N (cid:12) 2
w∈ΓN
and
(cid:16)(cid:12) (cid:12) (cid:17)
(4.4) P (cid:12)E(cid:101) (w )(cid:12) ≥ N1−γ ≤ K ·N−δ.
(cid:12) N ξ (cid:12) 3
Step 4. There are essentially two reasons that the events whose probabilities we seek to
bound in (4.3) and (4.4) occur. First, if |ξ −ξ | ≈ N−1 for some j, then E(cid:101) (w),E(cid:101) (w ) will
j N N ξ
both be on the order of N because of the single term involving ξ . Second, if there are many
j
more than Nδ zeros ξ for which |ξ −ξ| ≤ N−1/2+δ/2, then each term in E(cid:101) (w ,δ) will be
j j N ξ
large enough that their sum could well be on the order of N. However, both of these events
themselves have small probability. To quantify this, we write for ζ ∈ S2
N(ζ,R) := #{j = 1,...,N | d (ξ ,ζ) ≤ R},
S2 j
where as before d is the usual distance function on S2 and prove the following lemma.
S2
Lemma 1. Fix η ∈ (0, 1), κ > 0. There exist K = K(η) > 0 and K(cid:48) = K(cid:48)(κ,δ) > 0 so that
2
(4.5) P(cid:0)N (cid:0)w ,N−1+η(cid:1) ≥ 1(cid:1) ≤ K ·N−1+2η
ξ
and
(cid:16) (cid:16) (cid:17) (cid:17)
P N w ,N−1+δ ≥ Nδ+κ ≤ K(cid:48) ·N−δ−2κ.
ξ 2 2
Step 5. Finally, note that the variance of E(cid:101) (w) is
N
(cid:34) (cid:35)
(cid:20)(cid:12) (cid:12)2(cid:21) N(cid:88)−1 1
E (cid:12)E(cid:101) (w)(cid:12) = E ,
(cid:12) N (cid:12) |w−ξ |2
j=1 j
which is infinite. However, the conditional variance given N(w,N−1+η) = 0 is fairly small
and allows us to get a good estimate on the tail probability in (4.4). This is the content of
the following lemma.