Full text of DOI:10.1007/PL00005580

Commun. Math. Phys. 222, 475 – 493 (2001)
Communications in
Mathematical
Physics
© Springer-Verlag 2001
Asymptotic Statistics of Zeroes for the Lamé Ensemble
Alain Bourget, John A. Tothꢀ
Department of Mathematics and Statistics, McGill University, Montreal, Canada
Received: 17 January 2001 / Accepted: 14 May 2001
Abstract: The Lamé polynomials naturally arise when separating variables in Laplace’s
equation in elliptic coordinates. The products of these polynomials form a class of
spherical harmonics, which are joint eigenfunctions of a quantum completely integrable
(QCI) system of commuting, second-order differential operators P0 = ꢁ, P1, . . . , PN−1
acting on C∞(SN ). These operators naturally depend on parameters and thus constitute
an ensemble. In this paper, we compute the limiting level-spacings distributions for the
zeroes of the Lamé polynomials in various thermodynamic, asymptotic regimes. We
give results both in the mean and pointwise, for an asymptotically full set of values of
the parameters.
1. Introduction
Fix N + 1 distinct, positive real numbers 0 < α0 < · · · < αN . Given Cartesian
coordinates (z1, . . . , zN+1) ∈ RN+1, consider the partial differential operators
ꢁ
ꢂ
2
ꢀ
∂
∂
Pk :=
Skij (α0, . . . , αN ) zi
− zj
;
k = 0, . . . , N − 1
(1)
∂zj
∂zi
i<j
acting on C∞(SN ). Here, Skij denotes the kth elementary symmetric polynomial in the
α parameters with αi and αj deleted. It is easy to check that
[Pi, Pj ] = 0 for all i, j = 0, . . . , N − 1.
Consequently, since the Pj ’s are jointly elliptic, they possess a Hilbert basis of joint
eigenfunctions. Since P0 is just the constant curvature spherical Laplacian, these eigen-
functions form a class of spherical harmonics, the so-called generalized Lamé harmonics
ꢀ
Supported in part by an Alfred P. Sloan Research Fellowship and NSERC grant OGP0170280.

476
A. Bourget, J. A. Toth
[T1]. These systems constitute important examples of quantum completely integrable
systems and they have as complex analogues the Gaudin spin-chains of various types
[HW, K]. The purpose of this note is to derive asymptotic formulas for the level-spacings
distributions of the zeroes of these spherical harmonics. To describe our results in more
detail, we begin by noting that in terms of appropriate (elliptic-spherical [HW, T1, T2])
parametrizing coordinates (u1, . . . , uN ) ∈ (α0, α1) × · · · × (αN−1, αN ) on SN , and up
to constant multiples, the joint eigenfunctions of P0, . . . , PN−1 can be written in the
form:
N
ꢃ
ꢄ
ꢄ
ψ(u1, . . . , uN ) =
Sm( uj − α0, . . . , uj − αN ) · φ(uj ).
j=1
Here, φ is a polynomial, and Sm(x0, . . . , xN ); m = 0, . . . , N + 1 denotes the mth
elementary symmetric function on N + 1 variables. Furthermore, the function ψ(x) :=
√
√
Sm( x − α0, . . . , x − αN ) · φ(x) is a solution of the ODE:
N
N
d2ψ
dx2
1
2
dψ
ꢃ
ꢀ ꢃ
(x − αν)
+
(x − αλ)
+ C(x)ψ = 0,
(2)
dx
ν=0
ν=0 λ=ν
where, C(x) is a polynomial of order N − 1 depending linearly on the joint eigenvalues
(λ0, . . . , λN−1) ∈ Spec (P0, . . . , PN−1). The different species [WW] of harmonics are
indexed by m = 0, . . . , N +1. Although for simplicity, we consider here the case where
m = 0, our main result (Theorem 1.1) can be proved for the other cases corresponding
to m = 1, . . . , N + 1 in a similar fashion. When m = 0, the solutions ψ(x) are called
generalized Lamé polynomials.
Consider
E(k) := {φ1(k), . . . , φ(k) },
j(k)
the set of Lamé polynomials of degree k. By the standard theory of spherical harmonics
[WW] and the fact that the corresponding Lamé harmonics form a Hilbert basis, we
know that j(k) = σ(N, k) := (N+k−1)! . Let θi(,k1) ≤ · · · ≤ θ(k) denote the (real) zeroes
i,k
k!(N−1)!
of the polynomial, φi(k), where i = 1, . . . , σ(N, k).
In our main result (Theorem 1.1), we compute the asymptotic weak limit for the level
spacings distribution averaged over the set, E(k), of kth order Lamé polynomials. More
precisely, consider
σ(N,k)
k−1
ꢅ
ꢆ
ꢀ
ꢀ
1
1
dρLASV (x; N, k, α) :=
δ x − k(θl(,kj)+1(α) − θ(k)(α)) , (3)
l,j
σ(N, k)
k − 1
l=1
j=1
where α ∈ "N and
N
N+1
"
:= {(α0, . . . , αN ) ∈ [0, 1]
; α0 < α1 < · · · < αN−1 < αN }.
(4)
We henceforth put the normalized Lebesgue measure dα := (N + 1)!dα on "N , so
−
that meas ("N ) = 1. In order to state our first result, we will also need to introduce the
integrated, averaged level-spacings distribution:
ꢇ
AV
LS
−
dµLS(x; N, k) :=
dρ (x; N, k, α) dα.
(5)
N
"

Asymptotic Statistics of Zeroes for the Lamé Ensemble
477
Theorem 1.1. (i) Fix 0 < $ < 1 and assume that k ∼ N1−$ as N → ∞. Then,
w − lim dµLS(x; N, k) = e−xdx.
N→∞
(ii) Suppose that k(N) satisfies the hypotheses of part (i). Then, for any 0 < δ < $ there
exist a measurable subset JN ⊂ "N with meas (JN ) ≥ 1 − N−δ, such that for any
α ∈ JN ,
AV
w − lim dρ (x; N, k, α) = e−xdx.
LS
N→∞
In both (i) and (ii), the weak-limit is taken in the dual space to C00([a, b]), where
0 ≤ a < b < ∞.
Remarks. (i) In recent work, Bleher, Shiffman and Zelditch [BSZ 1,2] have determined
the asymptotics of various measures associated with the distribution of zeroes of eigen-
sections of Toeplitz operators. The Lamé ensemble together with its complex analogues
(the Gaudin spin chains) can be described in the Toeplitz framework [K]. However, in
[BSZ 1,2] the averaging is carried out over a much larger ensemble: namely, all suitably
normalized bases ofToeplitz eigensections. Most such bases are not quantum completely
integrable and consequently, the situation considered in this paper is quite different from
that in [BSZ 1,2]. The main point here is that we are really averaging over a compara-
tively small ensemble indexed by the parameters α ∈ "N and the elements of which are
all quantum completely integrable.
(ii) There are two asymptotic parameters that enter into our analysis here: k, the degree
of the joint eigenfunctions, and N, the dimension of the base space, which in this case is
just SN . So, Theorem 1.1 above is really a hybrid asymptotic result about the zeroes of the
joint eigenfunctions of the Pj ’s on spheres of increasing dimension where we assume
that the number of zeroes, k, satisfies k(N) ∼ N1−$ as N → ∞. This asymptotic
regime can be thought of as a kind of thermodynamic limit. It would also be of interest
to determine what happens in other asymptotic ranges where the number of zeroes is
permitted to grow at faster rates as N → ∞. In particular, one would like to know what
happens in the purely semiclassical regime, where N is fixed and k → ∞. We hope to
address these points in future work.
(iii) As the referee has pointed out, it would be of considerable interest to determine
how the actual zeroes of the Lamé harmonics are distributed in the sense of a Riemann
measure on SN itself. A natural starting point would be to look at the density of states
measures (see [ShZ, NV]). In light of our results in this paper, the zeroes should, at least
on average, behave like random variables in the asymptotic regime where k(N) ∼ N1−$
.
Consequently, we believe that the density of states should on average tend to uniform
measure on SN , but at present we do not know how to prove this. We plan to address
this question for the Lamé harmonics as well as the more general complex XXX Gaudin
spin chains in an upcoming paper.
2. The Lamé Differential Equation
We now give a brief introduction to the Lamé equation following the classical presenta-
tion in Whittaker and Watson [WW], where this equation is introduced via the theory of
ellipsoidal harmonics. In his treatise on heat conduction in an ellipsoidal body, G. Lamé
was led to consider the class of homogeneous, harmonic polynomials on RN+1 that

478
A. Bourget, J. A. Toth
vanish on a family of confocal quadrics. There is an analogous construction of spherical
harmonics which we will now describe. Pick a set {α0, . . . , αN } of positive real con-
stants, all distinct, and ordered in increasing order. Define, for some real parameter θ, the
ꢈ
ꢉ
diagonal matrix Aθ = diag (θ − α0)−1, . . . , (θ − αN )−1 . The problem then reduces
to finding, for any positive integer k and multi-index β = (β0, . . . , βN ) ∈ {0, 1}N+1, k
real numbers θ1, . . . , θk for which the Niven’s functions
k
ꢃ
fβ(X) = Xβ
(Aθ X, X), X ∈ RN+1
,
(6)
j
j=1
are solutions of Laplace’s equation ꢁ(fβ) = 0. The restrictions of the fβ’s to SN yield
an important class of spherical harmonics: the generalized Lamé harmonics. Clearly, the
functions fβ(X) vanish on a family of confocal cones. Moreover, after substitution of
the ansatz into Laplace’s equation, a straightforward computation shows that the relevant
θj are obtained as solutions of the equations
N
N
ꢀ
ꢀ
1
2βν
+
+
θj (α; l) − αν
θj (α; l) − αν
ν=0
ν=0
(7)
ꢀ
4
= 0 for j = 1, . . . , k.
θj (α; l) − θi(α; l)
i=j
In the literature, these equations are commonly referred to as the “Bethe Ansatz” equa-
tions. Consequently, if we denote the solutions of (7) by θ1, . . . , θk, it is not hard to see
that the functions
N
k
ꢃ
ꢃ
ψ(x) =
(x − αν)β /2
(x − θj ), βν ∈ {0, 1}
(8)
ν
ν
j=1
satisfy the second order differential equation given by
N
N
d2ψ
dx2
1
2
dψ
dx
ꢃ
ꢀ ꢃ
(x − αν)
+
(x − αµ)
+ C(x)ψ = 0,
(9)
ν=0
ν=0 µ=ν
where C(x) is a polynomial of degree N − 1 that can be computed explicitly. This is
exactly Eq. (2) of the introduction, and is known as the generalized Lamé differential
equation. In the special case where the multi-index β = 0, it follows that the kth degree
ꢊ
k
polynomial φ(x) =
(x − θj ) is a solution of the Lamé equation. Following the
terminology adopted prje=vi1ously, we refer to these as Lamé polynomials. Restricting our
attention to the N-sphere SN , we can use elliptic-spherical [HW, T1, T2] coordinates
u = (u1, . . . , uN ) ∈ (α0, α1) × · · · × (αN−1, αN ) to rewrite Niven’s function in the
form
N
N
ꢃ ꢃ
(uj − αν)β /2φ(uj ),
(10)
ν
fβ(u1, . . . , uN ) = c
ν=0 j=1
ꢊ
k
where c is some constant depending only on α0, . . . , αN and φ(u) = j=1(u − θj ) .
The functions ψ(u1, . . . , uN ) in the introduction are simply linear combinations of the

Asymptotic Statistics of Zeroes for the Lamé Ensemble
479
fβ’s and consequently, they are solutions of the eigenvalue problem for the Laplace’s
operator on SN written in terms of elliptic-spherical coordinates; that is
ꢋ
ꢁ
ꢂꢍ
N
ꢌ
ꢌ
ꢀ
4
∂
∂ψ
ꢊ
U(uj )
U(uj )
= −λ0ψ,
i=j (uj − ui)
∂uj
∂uj
j=1
ꢊ
N
where U(x) = ν=0(x − αν). The separated equations have the form


N
N
N−1
d2ψ
dx2
1
2
dψ
dx
1
4
ꢃ
ꢀ ꢃ
ꢀ
j


(x − αν)
+
(x − αµ)
+
λ
N−j−1x ψ = 0, (11)
ν
ν=0
µ=ν
j=0
where the separation constants λ1, . . . , λN−1 are the joint eigenvalues of the partial
differential operators P0, . . . , PN−1 defined in (1). Equation (11) is exactly the Lamé
differential equation (2) considered in the introduction.
3. The Heine–Stieltjes Theorem
One of the key steps in the proof of Theorem 1.1 is based on a result originally obtained
by M. Heine [H]. Shortly afterwards, Stieltjes [S] improved Heine’s result in the special
case of differential equations of Lamé’s type, the case that we consider here. For a
complete proof, we refer the reader to Szegö [Sz].
Theorem 3.1 (Heine–Stieltjes). Let A(x) be the polynomial of degree N + 1 given by
A(x) = (x − α0) · (x − α1) · · · (x − αN ),
where 0 < α0 < α1 < · · · < αN and B(x) is a polynomial of degree N satisfying the
condition
B(x)
A(x)
ρ0
ρN
=
+ · · · +
,
x − α0
x − αN
(N+k−1)!
k! (N−1)!
for given numbers ρν > 0, ν = 0, . . . , N. Then, there are exactly σ(N, k) =
polynomials C(x) of degree N − 1 for which the differential equation
d2φ
dx2
dφ
dx
A(x)
+ 2B(x)
+ C(x)φ = 0
(12)
has a polynomial solution of degree k > 0. In addition, for each of the σ(N, k) solutions,
φ(x), the zeroes are simple and uniquely determined by their distribution in the intervals
(α0, α1), . . . , (αN−1, αN ).
Note that the Lamé equation (11) is a particular case of the differential equation
(12) appearing in the statement of the theorem: Indeed, in the Lamé case, we have that
ρν = 1/4 for ν = 0, . . . , N. Taking into account the Heine–Stieltjes result, we denote
the zeroes of φ(x) by θ1(α; l) ≤ · · · ≤ θk(α; l), where α := (α0, . . . , αN ), whereas
l = (l1, . . . , lk); 1 ≤ l1 ≤ · · · ≤ lk ≤ N denotes the configuration of the zeroes. By
this we mean that θ1(α; l) is the smallest zero lying in the interval (αl −1, αl ), the next
1
1
zero θ2(α; l) is contained in the interval (αl −1, αl ) and so on.
Although we will not explicitly use the 2follow2ing result in this paper, it is of inde-
pendent interest and so we include it here for future reference:

480
A. Bourget, J. A. Toth
Lemma 3.2. For any given configuration l = (l1, . . . , lk), the zeroes θ1(α; l ), . . .,
θk(α; l) are differentiable functions of α ∈ "N .
Proof. The proof is based on the argument given in [Sh]. Differentiating the BetheAnsatz
equations (7) with respect to the θ variables, we form the Jacobian matrix B = (bij )
given by
ꢎ
ꢏ
ꢏ
N
ν=0
(θj −θi)2
ρν
(θj −αν)2
1
−
−
if i = j
if i = j.
2
m=i
(θj −θm
)
bij
=
1
By a standard result in matrix theory (Gersˆgorin’s Theorem) it follows that all the
eigenvalues of B are strictly negative since if λ is an eigenvalue of B, then for some
j ∈ {1, 2, . . . , k},
N
ꢀ
ꢀ
ρν
(θj − αν)2
λ ≤ bjj
+
|bij | = −
< 0.
i=j
ν=0
Therefore, the determinant of B is nonzero, so we can apply the implicit function theorem
to conclude the proof. ꢁꢂ
4. Level Spacings Distribution
Consider the Lamé system consisting of N + 1 particles located at α0, . . . , αN with
0 < α0 < · · · < αN < 1 together with the k zeroes θ1(α; l) ≤ · · · ≤ θk(α; l) of
the polynomial solution φ(x) of the differential equation (12). Recall that, we use the
notation θj (α; l) to designate the jth zero in configuration l = (l1, . . . , lk) with respect
to parameters α = (α0, . . . , αN ). As a consequence of Heine–Stieltjes Theorem, we
have that:
ꢀ
1
1
dρLASV (x; N, k, α) =
σ(N, k) 1≤l ≤···≤l ≤N k − 1
1
k
(13)
k−1
ꢀ
ꢈ
ꢈ
ꢉꢉ
·
δ x − k θj+1(α; l) − θj (α; l)
,
j=1
and so,
dµLS(x; N, k) :=
ꢇ
AV
LS
−
dρ (x; N, k, α) dα
N
"
ꢇ
k−1
ꢀ
ꢀ
1
1
−
=
δ(x − k(θj+1(α; l) − θj (α; l))) d α. (14)
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
"
j=1
1
k
4.1. Proof of part (i) of Theorem 1.1. The proof is somewhat long and computational,
so we divide it into several steps. For notational simplicity, we assume here that [a, b] =
[0, 1] and φ ∈ C01([0, 1]). The argument for more general non-negative intervals [a, b]
follows in the same way.

Asymptotic Statistics of Zeroes for the Lamé Ensemble
481
Step 1. We first show that dµLS(x; N, k) can be asymptotically approximated by fairly
simple integrals that do not depend on the explicit formulas for the zeroes
θ1(α; l ), . . . , θ (α; l). More precisely, we claim that:
k
dµLS(x; N, k)(φ)
ꢁ
ꢂ
ꢇ
k−1
ꢀ
ꢀ
1
1
k
−
=
φ(k(αl − αl )) dα + O
. (15)
j+1
j
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
N
"
j=1
1
k
To obtain the estimate in (15), we make a first-order Taylor expansion in (14) around
k(αl − αl ) to get:
j+1
j
dµLS(x; N, k)(φ) =
ꢇ
k−1
ꢀ
ꢀ
1
1
−
φ(k(αl − αl )) dα + E1(N, k, φ), (16)
j+1
j
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
"
j=1
1
k
where the error term E1(N, k, φ) is given by,
ꢀ
1
k
E1(N, k, φ) =
σ(N, k) 1≤l ≤···≤l ≤N k − 1
1
k
ꢇ
k−1
ꢀ
ꢐꢈ
ꢉ
ꢈ
ꢉꢑ
−
dα,
·
φꢃ(kξj (α))
θ
j+1(α; l) − αl
− θj (α; l) − αl
j+1
j
N
"
j=1
ꢈ
ꢉ
k
N
with ξj (α) ∈ (0, 1). We need to show that E1(N, k, φ) = O
.
First, we start with a simple calculus lemma:
Lemma 4.1. For any 0 ≤ i ≤ j ≤ N and multi-indices β = (β1, β2) ∈ N2 \ {(0, 0)},
we have
ꢊ
ꢊ
ꢇ
β
β
2
1
l=1(i + l) l=1(β1 + j + l)
β
β
2
1
−
αi αj dα =
,
ꢊ|β|
N
"
l=1(N + 1 + l)
ꢊ
0
where we define products of the form l=1 to be equal to 1 and |β| := β1 + β2.

482
A. Bourget, J. A. Toth
Proof of Lemma 4.1. A direct computation with iterated integrals gives
ꢇ
ꢇ
β
β
β
β
2
1
2
−
1
−
αi αj dα =
αi αj dα
N
"
0<α <···<αN <1
0
ꢇ
ꢇ
ꢇ
1
αN
α
1
β
β
1
2
= (N + 1)!
· · ·
αj αi dα0 · · · dαN
0
0
0
ꢇ
ꢇ
ꢇ
1
αN
αi+1
(N + 1)!
β
β +i
1
2
=
· · ·
αj αi
dαi · · · dαN
i!
(N + 01)!
0
0
ꢇ
ꢇ
ꢇ
1
αN
αj+1
αjβ +β +j dαj · · · dαN
2
1
=
· · ·
i!(β1 + i + 1) . . . (β1 + j)
0
0
0
(N + 1)!
=
=
i! (β1 + i + 1) . . . (β1 + j) (β1 + β2 + j + 1) . . . (β1 + β2 + N + 1)
ꢊ
ꢊ
β
β
2
1
l=1(i + l) l=1(β1 + j + l)
.
ꢁꢂ
ꢊ|β|
l=1(N + 1 + l)
As a consequence of Lemma 4.1, we see that the integrals of consecutive monomials
over the truncated positive Weyl chamber "N are asymptotically equal as N → ∞.
Combined with the Heine–Stieltjes result, this fact leads to the following simple corollary
of Lemma 4.1:
Corollary 4.2. For any configuration l = (l1, . . . , lk) and integer j satisfying 1 ≤ j ≤
k, we have that
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
−
θj (α; l) − αl dα = O(N−1
)
(17)
j
N
"
uniformly in k.
Proof of Corollary 4.2. As a consequence of the Heine–Stieltjes theorem, we know that
given a configuration l, the jth zero necessarily lies in the interval (αl −1, αl ); that is,
j
j
αl −1 ≤ θj (α; l) ≤ αl .
j
j
ꢓ
j+1
. Thus,
−
On the other hand, by Lemma 4.1,
αj dα =
N
"
N+2
ꢇ
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
ꢈ
ꢉ
−
−
θj (α; l) − αl dα ≤
αl − αl −1 dα
j
j
j
N
N
"
"
lj + 1
lj
=
−
N + 2 N + 2
= O(N−1).
ꢁꢂ

Asymptotic Statistics of Zeroes for the Lamé Ensemble
483
Given Corollary 4.2, we can now estimate the error term, E1(N, k, φ), in (11) as follows:
ꢀ
1
k
E1(N, k, φ) ≤
σ(N, k) 1≤l ≤···≤l ≤N k − 1
1
k
ꢁ
ꢂ
ꢇ
ꢇ
k−1
ꢀ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
−
−
ꢒ
·
θ
j+1(α; l) − αl
dα+
θj (α; l) − αl dα ꢄφ ꢄC1
j+1
j
N
N
"
"
j=1
ꢁ
ꢂ
k
= O
.
N
This proves the identity in (15) and so Step 1 is complete.
Step 2. The next step involves computing the first term on the RHS of (15) explicitly.
We claim that:
ꢇ
k−1
ꢀ
ꢀ
ꢈ ꢈ
ꢉꢉ
1
1
−
dα
φ k αl − αl
j+1
j
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
"
j=1
1
k
k
=
φ(0)
(18)
N + k − 1
N + 1
ꢇ
N−2
1
ꢀ
+
σ(N − m − 1, k − 1)
φ(kx)binom(N, m; x) dx
σ(N, k)
0
m=0
N!
where binom(N, m; x) := m!(N−m)! xm(1 − x)N−m for x ∈ [0, 1]. In order to prove the
identity in (18), we start with a simple lemma which involves a successive application
of the Fubini theorem.
Lemma 4.3. For any integers i, j with 0 ≤ i < j ≤ N, we have that
ꢇ
ꢇ
1
ꢈ ꢈ
ꢉꢉ
−
φ k αj − αi dα = (N + 1)
φ(kx)binom(N, j − i − 1; x) dx.
(19)
N
"
0
Proof of Lemma 4.3. Given the definition of "N , it is clear that
ꢇ
ꢇ
ꢇ
ꢇ
1
αN
α
1
ꢈ ꢈ
ꢉꢉ
ꢈ ꢈ
ꢉꢉ
−
φ k αj − αi dα = (N + 1)!
· · ·
φ k αj − αi dα0 · · · dαN .
N
"
0
0
0
By repeated application of Fubini’s theorem, we can ensure that the iterated integrals
with respect to αi and αj are carried out last. More precisely, we apply Fubini’s theorem
on the double integral with respect to αj and αj+1 to reverse the order of integration.
We then repeat the same procedure for the double integral with respect to αj and αj+2
and so on, until we bring the last integration with respect to αj . This gives
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
1
1
αN
αj+2
αj
αj
α
1
dα =
· · ·
· · ·
dα0 . . . dαj−1dαj+1 . . . dαN−1dαN dαj .
"
N
0
αj αj
0
0

484
A. Bourget, J. A. Toth
We proceed in a similar manner for αi to finally obtain:
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
1
αj+2
αj
αj
αi+2
αi
α
1
dα =
1 · · ·
· · ·
· · ·
dα0 . . .
N
"
0
αj
αj
0
αi
αi
0
0
. . . dαi−1dαi+1 . . . dαj−1dαidαj+1 . . . dαN dαj
ꢇ
ꢇ
ꢇ
ꢔ
ꢇ
ꢇ
ꢇ
ꢇ
ꢇ
1
αj
αj+2
αj
αj
αi+2
αi
αi
α
1
ꢔ
dα0 . . . dαi . . .
=
1 · · ·
· · ·
· · ·
0
0
αj
αi
0
0
. . . dαj . . . dαN dαidαj ,
ꢔ
ꢔ
where dαi, dαj means that these variables are omitted in the product measure
dα0 · · · dαN . We then carry out the iterated integration over the first N − 2 variables
α0 < α1 < · · · < αi−1 < αi+1 < · · · < αj−1 < αj+1 < · · · < αN to get
ꢇ
ꢇ
ꢇ
αii
(αj − αi)j−i−1 (1 − αj )N−j
1
αj
−
dα = (N + 1)!
dαidαj .
N
i! (j − i − 1)!
(N − j)!
"
0
0
Finally, we make the change of variables x = αj − αi, y = αj and integrate by parts i
times with respect to y. It follows that
ꢇ
ꢇ
1
xj−i−1
(1 − x)N−(j−i−1)
dx
ꢈ ꢈ
ꢉꢉ
−
φ k αj − αi dα =(N + 1)!
φ(kx)
N
(j − i − 1)! (N − (j − i − 1))!
"
0
ꢇ
1
= (N + 1)
φ(kx)binom(N, j − i − 1; x)dx.
0
This completes the proof of Lemma 4.3. ꢁꢂ
To complete Step 2, we need to compute the asymptotic averages (18) of the integrals
ꢓ
−
φ(k(αl − αl )) dα. First, we start with a simple combinatorial lemma. In order
N
j+1
j
"
to state the lemma, it is useful to introduce some notation at this point: We denote by
Sj (m) the set of all configurations l = (l1, . . . , lk) for which lj+1 − lj = m. As the
following lemma shows, the cardinality of Sj (m) is independant of j.
Lemma 4.4. For each m = 0, . . . , N −1 and each j = 1, . . . , k, the number of k-tuples
(l1, . . . , lk) with 1 ≤ l1 ≤ · · · ≤ lk ≤ N and satisfying lj+1 − lj = m is given by
(N − m + k − 2)!
σ(N − m, k − 1) =
.
(k − 1)! · (N − m − 1)!
Proof of Lemma 4.4. By identifying the jth and the j + 1st zeroes, we are reduced to the
problem of distributing k − 1 zeroes amongst the remaining N − m slots. Clearly, this
number is given by σ(N − m, k − 1). ꢁꢂ
As a consequence of Lemma 4.4,
ꢇ
k−1
ꢀ
ꢀ
ꢈ ꢈ
ꢉꢉ
1
1
−
dα
φ k αl − αl
j+1
j
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
"
j=1
1
k
ꢇ
N−1 k−1
ꢀ ꢀ ꢀ
ꢈ ꢈ
ꢉꢉ
1
1
−
dα.
=
φ k αl − αl
j+1
j
N
(k − 1) σ(N, k)
"
m=0 j=1
l∈Sj (m)

Asymptotic Statistics of Zeroes for the Lamé Ensemble
485
In order to apply Lemma 4.3, we need to treat the two cases where m = 0 and m > 0
separately. Since by Lemma 4.4, we know that Sj (0) = σ(N, k − 1), it is clear that
ꢇ
k−1
ꢀ
ꢀ
ꢈ ꢈ
ꢉꢉ
1
1
−
φ k αl − αl
dα
(20)
j+1
j
N
σ(N, k) 1≤l ≤···≤l ≤N k − 1
"
j=1
1
k
ꢇ
N−1 k−1
ꢀ ꢀ ꢀ
(k − 1) σ(N, k)
m=1 j=1
ꢈ ꢈ
ꢉꢉ
σ(N, k − 1)
1
1
−
=
φ(0) +
φ k αl − αl
dα.
j+1
j
N
σ(N, k)
"
l∈Sj (m)
On one hand, we can apply Lemma 4.3 to the RHS of (20) to get
σ(N, k − 1)
φ(0)
σ(N, k)
ꢇ
ꢇ
N−1 k−1
ꢀ ꢀ ꢀ
ꢈ ꢈ
ꢉꢉ
1
1
k
−
+
φ k αl − αl
dα =
φ(0)
j+1
j
N
(k − 1) σ(N, k)
N + k − 1
"
m=1 j=1
N−1 k−1
l∈Sj (m)
1
ꢀ ꢀ ꢀ
N + 1
+
φ(kx)binom(N, lj+1 − lj − 1; x) dx. (21)
(k − 1)σ(N, k)
0
m=1 j=1
l∈Sj (m)
On the other hand, an application of Lemma 4.4 allow us to remove the summations
over j and l in (21), so that
ꢇ
N−1 k−1
1
ꢀ ꢀ ꢀ
N + 1
φ(kx)binom(N, lj+1 − lj − 1; x) dx
(k − 1)σ(N, k)
0
m=1 j=1
l∈Sj (m)
ꢇ
N−2
1
ꢀ
N + 1
=
σ(N − m − 1, k − 1)
φ(kx)binom(N, m; x)dx. (22)
σ(N, k)
0
m=0
The identity in (18) follows from (21) and (22) and so, Step 2 is complete.
Step 3. Summing up, as a result of Steps 1 and 2 we have shown that:
dµLS(x; N, K)(φ) =
ꢁ
ꢂ
ꢇ
N−2
1
ꢀ
N + 1
k
σ(N − m − 1, k − 1)
φ(kx)binom(N, m; x) dx + O
. (23)
σ(N, k)
N
0
m=0
Our next task is to further simplify the expression on the RHS of (23) by appealing
to the theory of Bernstein approximations [D]. First, we need to estimate the quotient
σ(N−m−1,k−1)
appearing on the RHS of (23). For this, it is convenient to consider two
caseσs:(N,k)

486
A. Bourget, J. A. Toth
Case 1. (m << (N/k)1+β); 0 < β < 1. Under this assumption,
ꢁ
ꢂ
k−2
ꢃ
σ(N − m − 1, k − 1)
k
m + 1
=
=
=
=
1 −
σ(N, k)
N + k − 1
N + j
j=0


ꢁ
ꢕ
ꢂ
k−2
ꢀ
k
m + 1
N + j


exp
log 1 −
N + k − 1
j=0




ꢖ
k−2
ꢀ
k
m + 1
1
exp
exp
log 1 −
·
j
N
N + k − 1
N
1 +
j=0



ꢁ
ꢂ
k−2
k
m + 1
1
km2
N2
ꢀ

−
·
+ O
N
.
j
N + k − 1
N
1 +
j=0
(24)
After some further simplification involving Taylor expansions, we get that
σ(N − m − 1, k − 1)
σ(N, k)




ꢁ
ꢁ
ꢂꢂ
mk2
N2
ꢁ
ꢂ
km2
N2
k−2
km2
N2
ꢁ
ꢀ
k
m + 1
j
=
exp
−
1 + O
+ O
N + k − 1
N
N
j=0
ꢋ
ꢁ
ꢂ
ꢂꢍ
k
(k − 2)(m + 1)
=
=
exp −
+ O
mk2
+ O
km2
.
N + k − 1
k
N
ꢁ
ꢂ ꢁ
ꢁ
ꢂ
ꢁ
ꢂꢂ
−mk
exp
1 + O
+ O
.
N + k − 1
N
N2
N2
Finally, using the fact that xpe−x = Op(1); for all x ≥ 0, we get
ꢁ
ꢂ
ꢁ
ꢂ
ꢁ
ꢂ
σ(N − m − 1, k − 1)
k
−mk
k2
N2
1
=
exp
+ O
+ O
.
(25)
σ(N, k)
N + k − 1
N
N
1−$
Case 2. (m >> (N/k)1+β). In this case, we can choose 0 < β <
appropriate constants C1, C2 > 0,
so that with
$
ꢁ
ꢂ
k−2
ꢃ
σ(N − m − 1, k − 1)
k
m + 1
N + j
=
1 −
σ(N, k)
N + k − 1
j=0
ꢁ
ꢂ
k−2
ꢃ
(26)
k
Nβ
k1+β
≤
1 − C1
N + k − 1
j=0
ꢅ
ꢆ
β
= O e−C (N/k)
.
2
Substituting the estimates (25) and (26) into (23) and using the fact that
ꢇ
N
1
ꢀ
binom(N, m; x) = 1 and
φ(kx)dx = O(k−1
)
0
m=0

Asymptotic Statistics of Zeroes for the Lamé Ensemble
487
gives:
N−2
ꢀ
mk
N
k(N + 1)
dµLS(x; N, K)(φ) =
e−
N + k − 1
m=0
ꢁ
ꢁ
ꢂ
ꢂ
ꢁ ꢂ
ꢇ
1
k
1
·
φ(kx)binom(N, m; x) dx + O
+ O
+ O
N
k
0
N
ꢀ
−mk
k(N + 1)
N
=
e
N + k − 1
m=0
ꢁ ꢂ
ꢇ
1
k
1
·
φ(kx)binom(N, m; x) dx + O
(27)
N
k
0
since the terms for m = N − 1 and m = N are bounded by 1/N. Recall that for a
function f (x) defined on [0, 1], the Nth degree Bernstein polynomial of f (x) is defined
to be [D]:
N
ꢅ
ꢆ
ꢀ
m
N
BN (f ; x) =
f
binom(N, m; x).
m=0
It is easy to see that in the special case where exp−k(x) := e−kx, there is a concise
closed-form expression for BN (exp−k; x); indeed,
ꢅ
ꢆ
N
k
N
BN (exp−k; x) = xe− + (1 − x)
.
(28)
(29)
From the identity in (28) we easily derive the following:
Lemma 4.5. For x ≥ 0, we have that
ꢁ
ꢂ
k
BN (exp−k; x) = e−kx + O
.
N
k
N
Proof of Lemma 4.5. Expand e− in a second-orderTaylor series and use the identity (28)
directly to get
ꢋ
ꢁ
ꢁ
ꢂꢂꢍ
N
ꢗ
ꢅ
ꢆꢘ
N
k
N
kx
N
k
BN (exp−k; x) = 1 + x e− − 1
= 1 −
1 + O
.
N
From the inequality
ꢅ
ꢆ
x2e−x
N
N
x
0 ≤ e−x − 1 −
≤
,
N
and the fact that xpe−x = Op(1) for all x ≥ 0, it follows that
ꢁ
ꢁ
ꢂꢂ
ꢁ
ꢂ
k2x
k
BN (e−kx; x) = e−kx 1 + O
+ O(N−1) = e−kx + O
.
N
N
This completes the proof of the lemma. ꢁꢂ

488
A. Bourget, J. A. Toth
Substituting (29) into (27), we finally obtain
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
1
k(N + 1)
k
1
dµLS(x; N, K)(φ) =
φ(kx)BN (exp−k; x) dx + O
+ O
N + k − 1
N
k
0
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
1
k(N + 1)
k
1
=
φ(kx)e−kx dx + O
+ O
N + k − 1
N
k
0
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
1
k
1
=
φ(x)e−x dx + O
+ O
.
(30)
N
k
0
By noting that C01([0, 1]) is dense in C00([0, 1]), this completes the proof of Theo-
rem 1.1 (i). ꢁꢂ
ꢏ
4.2. Proof of part (ii) of Theorem 1.1. For convenience, we henceforth denote by
l,lꢃ
the double sum over the indices 1 ≤ l1 ≤ · · · ≤ lk ≤ N and 1 ≤ lꢃ ≤ · · · ≤ lꢃ ≤ N.
1
k
We also define φk(x) := φ(kx). First, we claim that
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
2
ꢒ
−
dρAV (x; N, k, α)(φ) dα
ꢒ
LS
N
"
ꢁ
ꢂ
ꢇ
k−1
ꢀ
ꢀ
1
1
k
−
=
φk(αl − αl )φk(αlꢃ − αlꢃ ) dα + O
.
j+1
j
i+1
i
σ2(N, k)
(k − 1)2
N
N
"
l,lꢃ
i,j=1
(31)
To obtain (31), we essentially repeat the argument of Step 1 in Sect. 4.1. That is, we
expand each of the functions φk(θj+1(α; l)−θj (α; l)) and φk(θi+1(α; lꢃ)−θi(α; lꢃ)) in
a first-order Taylor series around the points (αl − αl ) and (αlꢃ − αlꢃ ) respectively.
j+1
j
i+1
i
k
N
ꢈ
ꢉ
First, we claim that the terms involving the derivative of φk are all O
. Indeed, the
Heine–Stieltjes Theorem and Lemma 4.1 imply that
ꢇ
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢈ
ꢉ
ꢃ
−
−
θ (α, l) − α θi(α, l ) − α ꢃ dα ≤
αl +1 − αl (αlꢃ +1 − αlꢃ ) dα
ꢒ
j
lj
l
j
j
i
i
i
N
N
"
"
(lꢃ + 2)(lj + 3) − (lꢃ + 1)(lj + 3) − (lꢃ + 2)(lj + 2) + (lꢃ + 1)(lj + 2)
i
i
i
i
=
(N + 2)(N + 3)
1
=
.
(N + 2)(N + 3)
Thus, it follows that
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
ꢃ
−
θ (α, l) − α θi(α, l ) − α dα = O(N−2),
(32)
ꢃ ꢒ
ꢒ
j
lj
li
N
"
uniformly for all lꢃ , lj ∈ {1, . . . , N}. Consequently, the terms involving the derivative
i
ꢈ
ꢉ
k
of φ are all O
as desired. In the integral (32) we have assumed without loss of
generality that lꢃN< lj . The other cases lꢃ = lj and lꢃ > lj can be treated in a similar
i
i
i
fashion. We next prove an L2 estimate for dρLASV (x; N, k, α)(φ) and then derive as an
LS
AV
immediate corollary an estimate for the variance of dρ (x; N, k, α)(φ). In order to

Asymptotic Statistics of Zeroes for the Lamé Ensemble
489
simplify the writing in the next proposition, we introduce the following notation for the
multinomial coefficient:
N!
ꢃ
ꢃ
multi(N − 1, m, mꢃ; x, y) :=
xmym (1 − x − y)N−m−m
,
m!mꢃ!(N − m − mꢃ)!
where m, mꢃ are positive integers satisfying m + mꢃ ≤ N and x, y ∈ [0, 1].
Lemma 4.6. (i) For any x, y ∈ [0, 1],
N−2 N−2−mꢃ
ꢁ
ꢂ
ꢀ
ꢀ
ꢃ
N
km
N
km
k
e− e− multi(N − 1, mꢃ, m; x, y) = e−kx−ky + O
. (33)
N
mꢃ=0
m=0
1
k
(ii) Also, for 0 ≤ x ≤ y ≤
,
ꢁ
ꢂ
N−2
m
ꢀ ꢀ
ꢃ
N
km
N
km
k
e− e− multi(N − 1, mꢃ, N − m; x, 1 − y) = e−kx−ky + O
.
N
ꢃ
m=0
m =0
(34)
Proof of Lemma 4.6. (i) As in the proof of the first part of the theorem, modulo O(N−1
)
errors, we can replace N − 2 by N − 1 in the upper limit of both summations. Define
exp−k(x) := exp(−kx). Then, as a consequence of Lemma 4.5, we have that
N−2 N−2−mꢃ
ꢀ
ꢀ
ꢃ
N
km
N
km
e− e− multi(N − 1, mꢃ, m; x, y)
mꢃ=0
m=0
N−1 N−1−mꢃ
ꢀ
ꢀ
k
N
k
N
−
−1
)
=
multi(N − 1, mꢃ, m; xe− , ye ) + O(N
mꢃ=0
m=0
ꢅ
ꢆ
= 1 − (x + y) + (x + y)e− N−1 + O(N−1
)
k
N
ꢁ
ꢂ
k
= BN (exp−k; x + y) + O
N
ꢁ
ꢂ
k
= exp(−kx − ky) + O
.
N

490
A. Bourget, J. A. Toth
(ii) Once again, we can replace N − 2 by N − 1 in the upper limit of the first sum. We
make successive applications of the binomial theorem to get:
N−2
m
ꢀ ꢀ
ꢃ
N
km
N
km
e− e− multi(N − 1, mꢃ, N − m; x, 1 − y)
ꢃ
m=0
m =0
ꢅ
ꢆ
m−1
k
N
x − y + ye−
= (N − 1)! N−1 e−
+ O(N−1
)
(1 − x)N−m
(N − m)!
ꢀ
km
N
(m − 1)!
m=0
ꢅ
ꢆ
N−1 + O(N−1
)
k
N
k
N
k
N
k
= e−
= e−
= e−kx−ky + O
1 − (x + ye− ) + (x + ye )e
N−1(exp−k; x + e− y) + O(N
−
−
N
k
N
k
N
−1
B
)
ꢁ
ꢂ
k
.
ꢁꢂ
N
We now use the combinatorial identities in Lemma 4.6 to estimate the variance of
the averaged level-spacings measures:
Proposition 4.7. For any φ ∈ C01([0, 1]), we have that
ꢁ
ꢂ
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
ꢇ
−xφ(x) dx 2 + O
+ O
.
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
1
2
k
1
dρAV (x; N, k, α)(φ) dα =
e
−
ꢒ
LS
N
N
k
"
0
(35)
Proof of Proposition 4.7. As a consequence of the estimate in (31), it suffices to show
that
ꢇ
k−1
ꢀ
ꢀ
1
1
−
φk(αl − αl )φk(αlꢃ − αlꢃ ) dα
j+1
j
i+1
i
σ2(N, k)
(k − 1)2
N
"
l,lꢃ
i,j=1
ꢁ
ꢂ
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
−xφ(x)dx 2 + O
+ O
.
1
k
1
=
e
N
k
0
In order to show this, we need to distinguish three different cases corresponding to the
ꢃ
various relative configurations of αl , αl ꢃ , αl and αl
:
j
j+1
i
i+1
ꢃ
ꢃ
ꢃ
ꢃ
).
Case 1. αl < αl
< αl < αl (or equivalently, αl < αl < αl < αl
j
j+1
j
j+1
i
i+1
i
i+1
The argument is essentially the same as in Lemma 4.3. The only difference is that
instead of getting a simple integral, we obtain a double integral at the end of the iterated
integration. More precisely, just as in Step 2, we make repeated applications of Fubini’s
ꢃ
ꢃ
Theorem to ensure that the last four iterated integrals are with respect to αl , αl , αl
j
i
and αl variables. We then integrate by parts in the first N − 4 integrals withir+e1spect
to the jr+e1maining α’s. As before, we make the change of variables x = αl
− αl and
ꢃ
ꢃ
i+1
ꢃ
y = αl − αl and then integrate by parts with respect to αl
and αl . Thie end
j+1
j
j+1
i+1

Asymptotic Statistics of Zeroes for the Lamé Ensemble
491
result is that:
ꢇ
−
φk(αl − αl )φk(αlꢃ − αlꢃ ) dα
j+1
j
i+1
i
N
"
ꢇ
ꢇ
1
1−y
= (N + 1)N
φk(x)φk(y)
0
0
· multi(N − 1, lꢃi+1 − lꢃ − 1, lj+1 − lj − 1; x, y) dxdy
i
ꢇ
ꢇ
1
1
= (N + 1)N
φk(x)φk(y)
0
0
· multi(N − 1, lꢃi+1 − lꢃ − 1, lj+1 − lj − 1; x, y) dxdy + O(k−1),
i
since 0 ≤ y ≤ 1/k in supp φk(y).
Case 2. αlꢃ < αl < αlꢃ < αl (or equivalently αl < αlꢃ < αl < αlꢃi+1 ).
j
j+1
j
j+1
i
i+1
i
When compared with all possible relative configurations, the proportion of config-
urations satisfying the assumptions of Case 2 are asymptotically small. Indeed, the
proportion of such relative configurations is O(k−1). One can see this as follows: Given
N + 1 positive real numbers 0 < α0 < . . . < αN < 1, we consider the following two
subsets of k elements given by:
αl ≤ · · · ≤ αl and αlꢃ ≤ · · · ≤ αlꢃ .
(36)
1
k
1
k
Foreachofthesubsetsabove, therearek−1pairsoftheform(αl , αl )and(αlꢃ , αlꢃi+1 ).
j
j+1
i
From (36) it follows that for any fixed pair (αl , αl ), there is at most one pair
j
j+1
(αlꢃ , αlꢃi+1 ) for which Case 2 is possible.
i
Case 3. αl < αlꢃ < αlꢃ < αl (or equivalently, αlꢃ < αl < αl < αlꢃi+1 ).
j
j+1
j
j+1
i
i+1
i
This case can be dealt with in a similar fashion to Case 1. That is, we apply the Fubini
Theorem repeatedly to ensure that the last four iterated integrals involve αlꢃ , αl , αl
j
j+1
i
and αlꢃi+1 . Then, we integrate by parts with respect to the remaining α’s. Finally, we
make the change of variables x = αlꢃ −αlꢃ and y = αl −αl and integrate by parts
j+1
j
i+1
i
again with respect to αlꢃ and αl to get:
j+1
i+1
ꢇ
−
φk(αl − αl )φk(αlꢃ − αlꢃ ) dα
j+1
j
i+1
i
N
"
ꢇ
ꢇ
1
1
= (N + 1)N
φk(x)φk(y)
0
x
· multi(N − 1, lꢃi+1 − lꢃ − 1, N − lj+1 − lj + 1; x, 1 − y) dydx
i
ꢇ
ꢇ
1
1
= (N + 1)N
φk(x)φk(y)
0
0
· multi(N − 1, lꢃi+1 − lꢃ − 1, N − lj+1 − lj + 1; x, 1 − y) dydx + O(k−1).
i
As in the proof of part (i) of Theorem 1.1, we make the substitution m = lj+1 −lj −1
and mꢃ = lꢃi+1 − lꢃ − 1 in order to apply Lemma 4.6. From the estimate in (31) and the
i

492
A. Bourget, J. A. Toth
analysis of Cases 1-3 above, we deduce that
ꢇ
ꢒ
ꢒ
ꢒ
ꢒ
2
ꢒ
−
dρAV (x; N, k, α)(φ) dα
ꢒ
LS
N
"
N−2 N−2−mꢃ
ꢇ
ꢇ
ꢗ
1
1
k2(N + 1)N
ꢀ
ꢀ
ꢃ
mk
e− e−
m k
N
N
=
φk(x)φk(y)
(N + k − 1)2
0
0
mꢃ=0
m=0
· multi(N − 1, mꢃ, m; x, y) dxdy
N−2 mꢃ
ꢇ
ꢇ
1
1
ꢀ ꢀ
ꢃ
mk
e− e−
m k
N
N
+
φk(x)φk(y)
0
0
ꢃ
m=0
m =0
ꢘ
· multi(N − 1, mꢃ, N − m; x, 1 − y) dydx
ꢁ
ꢂ
ꢁ ꢂ
k
1
+ O
+ O
.
N
k
By Lemma 4.6, we finally conclude that
ꢁ
ꢂ
ꢇ
ꢇ
2
ꢒ
ꢒ
ꢒ
ꢒ
ꢒ
1
k2(N + 1)N
2
−
dρAV (x; N, k, α)(φ) dα =
φk(x)e−kxdx
ꢒ
LS
(N + k − 1)2
N
"
0
ꢁ
ꢂ
ꢁ ꢂ
k
1
+ O
+ O
N
k
ꢁ
ꢂ
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
φ(x)e−xdx 2 + O
+ O
.
ꢁꢂ
1
k
1
=
N
k
0
Theorem 1.1 (ii) is then an immediate consequence of the Chebyshev inequality and the
following corollary of Proposition 4.7:
Corollary 4.8. For any φ ∈ C01([0, 1]), we have
ꢁ
ꢂ
ꢁ
ꢂ
ꢁ ꢂ
ꢇ
ꢇ
2
1
k
1
AV
LS
−
dρ (x; N, k, α)(φ) −
e
−xφ(x) dx
dα = O
+ O
.
(37)
N
N
k
"
0
Prof of Corollary 4.8. The corollary follows directly from Proposition 4.7 and the es-
timate for the convergence of the integrated, averaged level-spacings measure in (30).
ꢁꢂ
Remark. We should point out that one can also quite easily determine the weak limit of
the level-spacings measures before “unfolding the zeroes” (i.e. rescaling to unit mean
level-spacing). Indeed, by carrying out exactly the same analysis as above, one can show
that
σ(N,k)
k−1
ꢀ
ꢀ
1
1
w − lim
N→∞ σ(N, k)
δ(x − (θl(,kj)+1(α) − θ(k)(α))) = δ0(x), (38)
l,j
k − 1
l=1
j=1
both in the mean, and pointwise for an asymptotically full measure of α ∈ "N . Indeed,
the computation of the weak-limit in (38) turns out to be a simpler problem that the
corresponding one after “unfolding”, since the error terms are much easier to control.

Asymptotic Statistics of Zeroes for the Lamé Ensemble
493
Acknowledgements. We thank Dmitry Jakobson, Maung Min-Oo and Steve Zelditch for many helpful discus-
sions regarding the paper. We would also like to thank the referee for his insightful comments.
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Communicated by P. Sarnak