Addendum to "Energies of zeros of random sections on Riemann surfaces" [arXiv:0705.2000]. Indiana Univ. Math. J. 57 (2008), no. 4, 1753-1780

Abstract

This is an addendum to the article of Qi Zhong cited above [arXiv:0705.2000].
It outlines how to apply the main result of that article to calculate the
asymptotics of the expected energy of zeros of random polynomials on the
Riemann sphere $S^2$ with respect to the log chordal distance $\log [z, w]$.
The cited article did not calculate the asymptotic energy this way, but by an
ad-hoc method, and the calculation contained some errors. The correct
calculation here agrees (up to the stipulted remainder) with that of Armentano-
Beltran-Shub.

Full text

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arXiv:1009.4239v1 [math.PR] 22 Sep 2010
ADDENDUM TO “ENERGIES OF ZEROS OF RANDOM SECTIONS ON
RIEMANN SURFACES”. INDIANA UNIV. MATH. J. 57 (2008), NO. 4,
1753–1780
QI ZHONG AND STEVE ZELDITCH
The purpose of this note is to resolve an apparent discrepancy between the calculations
in the article [ABS] of Armentano- Beltran-Shub (henceforth ABS) and that in Qi Zhong’s
article [Zh] of the asymptotics of the expected energy of zeros of random polynomials on the
Riemann sphere S2with respect to the log chordal distance log[z,w]. For the sake of brevity,
we do not repeat the statement of the problem but refer to [Zh, ABS] for the background
and notation. We show that the calculation in [Zh] which uses the general method of Green’s
function and correlation functions gives the same answer as in [ABS].
The discrepancy between [ABS] and [Zh] arose because [Zh] actually contains two calcu-
lations of the asymptotic energy with respect to log[z,w], one explicit and one implicit. The
implicit one is indicated in the Remarks after [Zh] Theorem 1.2. There, it is pointed out
that the general Green’s energy method of the paper applies to the log chordal energy on S2
if one adds a constant to log[z,w] to convert it to the Green’s function. The Green’s energy
asymptotics in [Zh] are correct and their application to the log chordal energy does give the
correct answer, as we verify in this note. But the energy asymptotics were not computed by
that method in [Zh]. Rather in Section 5.2 of [Zh], the energy asymptotics were calculated
without converting log[z,w] to the Green’s function, and there are some errors in the calcu-
lation of the integrals that arise. As a result, the asymptotics stated in Theorem 1.3 (2) (see
(1.13)) are incorrect. The correct asymptotics are given here.
Besides correcting the calculation of the log[z,w] energy in [Zh], the purpose of this note is
to clarify what is correct and what is incorrect in [Zh]. Most importantly, the main result of
[Zh] is correct, and proves the correct asymptotics for the Green’s energy of zeros of random
holomorphic sections of powers of positive Hermitian line bundles for all Hermitian metrics
of positive (1,1) curvature over all Riemann surfaces. By comparison, the later result of
[ABS] only give the asymptotics in one special case, the round metric on S2. However, as
pointed out to us by M. Shub, the formula in [ABS] is exact: the o(N) error term in Zhong’s
formula is zero. We do not prove that here, because we derive the asymptotics from a general
formula for metrics on Riemann surfaces, where in general the o(N) term is non-zero.
The discrepancy between the asymptotics stated in Theorem 1.3 (2) of [Zh] and those of
[ABS] was spotted by D. Hardin and E. Saff while Q. Zhong was a postdoc at Vanderbilt.
They also informed the authors of [ABS] about the apparent discrepancy. A reference is
made in [ABS] to an erratum by Zhong. The present note supplants his erratum.
0.1. ABS versus Zhong. The calculation in question concerns the expected logarithmic
energy of zero sets of polynomials of degree N on the Riemann sphere. The first important
point of comparison is the normalizations of the energy in [ABS] and [Zh].
Date: September 23, 2010.
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2QI ZHONG AND STEVE ZELDITCH
In both articles, the Riemann sphere is identified with the sphere S(1
at (0,0,1
2) of radius1
2centered
2). The logarithm chordal energy is then given by
−log[z,w],[z,w] =
|z − w|
?1 + |z|2?1 + |w|2.
For this kernel on S(1
2), the expected log energy calculated in [ABS] is
ABS EEN
ABS∼N2
4
−N logN
4
−N
4.(1)
Qi Zhong calculated the energy not for log[z,w] but for minus the Green’s function of
S(1
the usual Green’s function byˆGg. Thus, Gg= −ˆGg. We denote the Green’s function of S(1
byˆG 1
energy, Zhong prove the general (and correct) asymptotics,
2). Following [Zh], we denote minus the Green’s function of a metric g by Ggand denote
2)
2. Then −G 1
2(z,w) := −1
2πlog[z,w]−C 1
2for a certain constant C 1
2. For the Gg-Green’s
ZHONGEEN
ZH= −1
4πN logN −N
4π− N
?
Fg(z,z)ωh/π + o(N),(2)
where Fgis the Robin constant of g. We denote the Robin constant of S(1
that it is a constant for the round metric.
To compare the two formulae (1)-(2), we need to calculate the constants C 1
C 1
2πlog[z,w], and substitute the value of F 1
detailed in §0.2, we need to multiply by
Lemma asserts that the calculations of [Zh] and [ABS] agree:
2) by F 1
2; note
2and F 1
2, add
2to G 1
2to convert it to −1
2. In addition, as
1
πto convert ABS to ZHONG. Thus, the following
Lemma 1. C 1
2=
1
4πand Fg(z,z) = −1
4π. Hence,
π
?
ZHONG + C 1
2N(N − 1)
?
= ABS.
0.2.
to observe that:
• (i) Zhong defines the energy using −1
• (ii) ABS sums over i < j while Zhong sums over i ?= j.
Zhong assumes that the Riemannian area form is dV = ωhwith?
[Zh]. Since this is the area of the sphere of radius1
To emphasize that the metric quantities pertain to the sphere of radius1
metric quantities by the respective radius, except for the geodesic distance, which we denote
by r.
Normalizations. To make the normalizations in [ABS] and [Zh] consistent we need
2πlog[z,w] − C 1
2while ABS use −log[z,w].
CP1ωh= π; see (2.2) of
2, [Zh] and ABS are working on S(1
2).
2, we subscript all
0.3. Proof of Lemma 1. Both constants depend on the radius we pick for S2, namely
radius1
the round metric of area π, G 1
of the chordal distance [z,w]. Subscripting with the radius, we have
2. To keep track of constants, we denote by A the area of S2in the given metric. For
2(z,w) is a function of the geodesic distance r(z,w) and hence
ˆG 1
2(z,w) =
1
2πlog[z,w] 1
2+ C 1
2.

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ADDENDUM TO “ENERGIES OF ZEROS OF RANDOM SECTIONS ON RIEMANN SURFACES”3
0.4. The constant C 1
for all z, which becomes
2. The constant is determined by the fact that?
CP1G 1
2(z,w)ωw= 0
A(S(1
2))C 1
2
= −1
2π
?2π
0log(1
0
? π
2sinϕ)(1
2
0log[0,r] 1
2(1
2sin2rdr)dθ = −? π
2
0log[0,r] 1
2(1
2sin2rdr)
= −?π
= −(1
4(sinϕ)dϕ)
4
?π
?π
0log(sinϕ)(sinϕ)dϕ) − (1
4(log1
2)?π
0(sinϕ)dϕ).
= −(1
4
0(log?2(1 − cosϕ))(sinϕ)dϕ) − (1
4log4
4(log1
2)?π
4(log1
0(sinϕ)dϕ)
= −1
e− (1
4(log1
2)?π
e− (1
0(sinϕ)dϕ) = −1
4log4
e− (1
2)2.
We conclude that
C 1
2= −1
4πlog4
4π(log1
2)2 =
1
4π.
(3)
Then Zhong’s minus Green’s function is given by,
G 1
2(z,w) = −1
2πlog[z,w] +
1
4πlog4
e+ (1
4π(log1
2)2 = −1
2πlog[z,w] −
1
4π.
(4)
It follows that,
−
1
2πlog[z,w] = G 1
2(z,w) +
1
4π.
(5)
0.5. Robin constant. We further show that
F 1
2(z,z) = −1
4π.
(6)
Here, F 1
2(z,z) is the constant in the expansion
G 1
2(z,w) = −1
2πlogr + F 1
2(z,z) + O(r),r → 0.
In fact, it is the same constant that we just calculated.
Indeed,ˆG 1
2=
1
2πlog(1
2sin2r) +
1
4π, and1
2sin2r = rf(r),f(0) = 1. So logf(0) = 0 and
F 1
2= −1
4π,
as desired.
0.6. Conclusion. Adding
totics (2). We then substitute (6) to find that (2) equals
1
4πN(N − 1) −
Finally, we multiply by π, to get
1
4πto G 1
2in (5) results in adding
1
4πN(N −1) to Zhong’s asymp-
1
4πN logN −N
4π− N(−1
4π) + o(N).
N2
4
−N logN
4
−N
4+ o(N),
which is the same as (1).
calculations of [ABS] and [Zh] agree.
Finally, we thank B. Shiffman for checking over the calculations.
This completes the proof of Lemma 1, and proves that the

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4 QI ZHONG AND STEVE ZELDITCH
References
[ABS]D. Armentano, C. Beltran and M. Shub, Minimizing the discrete logarithmic energy function on
the sphere: the role of random polynomials, Trans. AMS (to appear).
Q. Zhong, Energies of zeros of random sections on Riemann surfaces. Indiana Univ. Math. J. 57
(2008), no. 4, 1753–1780.
[Zh]
Department of Mathematics, Northwestern University, Evanston, IL, USA
E-mail address: zelditch@math.northwestern.edu