ASYMPTOTIC INDEPENDENCE IN THE SPECTRUM OF THE GAUS- SIAN UNITARY ENSEMBLE

in PROBABILITY

ASYMPTOTIC INDEPENDENCE IN THE SPECTRUM OF THE GAUS- SIAN UNITARY ENSEMBLE

PASCAL BIANCHI

Télécom Paristech - 46 rue Barrault, 75634 Paris Cedex 13, France.

email: bianchi@telecom-paristech.fr MÉROUANE DEBBAH

Alcatel-Lucent Chair on flexible radio,

SUPELEC - Plateau de Moulon, 3 rue Joliot-Curie, 91192 Gif sur Yvette cedex, France.

email: merouane.debbah@supelec.fr JAMAL NAJIM

Télécom Paristech & CNRS- 46 rue Barrault, 75634 Paris Cedex 13, France.

email: najim@telecom-paristech.fr

SubmittedJanuary 6, 2010, accepted in final formSeptember 11, 2010 AMS 2000 Subject classification: Primary 15B52, Secondary 15A18, 60F05.

Keywords: Random matrix, eigenvalues, asymptotic independence, Gaussian unitary ensemble Abstract

Consider a nnmatrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets(∆i,n, 1≤i≤p)with positive distance from one another, even- tually included in any neighbourhood of the support of Wigner’s semi-circle law and properly rescaled (with respective lengthsn⁻¹in the bulk andn^−2/3around the edges), we prove that the related counting measuresNn(∆i,n),(1≤i≤p), whereNn(∆)represents the number of eigen- values within ∆, are asymptotically independent as the size ngoes to infinity, p being fixed.

As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the ratio of the extreme eigenvalues of a matrix from the GUE.

1 Introduction and main result

Denote byHnthe set ofnnrandom Hermitian matrices endowed with the probability measure P_n(dM):=Z_n⁻¹exp

§

−n

2Tr(M)²ª dM, whereZ_nis the normalization constant and where

dM=

n

Y

i=1

d M_ii Y

1≤i<j≤n

R”

d M_{i j}— Y

1≤i<j≤n

I” d M_{i j}—

376

for every M= (M_{i j})1≤i,j≤n inHn (R[z]being the real part of z ∈ C andI[z] its imaginary part). This set is known as the Gaussian Unitary Ensemble (GUE) and corresponds to the case where a nnHermitian matrix Mhas independent, complex, zero mean, Gaussian distributed entries with varianceE|M_{i j}|²= ¹_nabove the diagonal while the diagonal entries are independent real Gaussian with the same variance. Much is known about the spectrum ofM. Denote by λ⁽₁ⁿ⁾,λ⁽₂ⁿ⁾,· · ·,λ⁽_nⁿ⁾the eigenvalues ofM(all distinct with probability one), then:

• [1]The joint probability density function of the (unordered) eigenvalues(λ⁽₁ⁿ⁾,· · ·,λ⁽nⁿ⁾) is given by

p_n(x₁,· · ·,x_n) =C_ne⁻

nP x2

i 2

Y

j<k

|x_j−x_k|², whereC_nis the normalization constant.

• [19] The empirical distribution of the eigenvalues ¹

n

Pn i=1δ_λ⁽ⁿ⁾

i

(δx stands for the Dirac measure at point x) converges toward Wigner’s semi-circle law asn→ ∞, whose density is:

1

2π1_(−2,2)(x)p 4−x².

Fluctuations of linear statistics of the eigenvalues of large random matrices (and of the GUE in particular) have also been extensively addressed in the literature, see for instance [2,9]and the references therein; for a determinantal point of view, one can refer to[15].

• [3]The largest eigenvalueλ⁽_maxⁿ⁾ (resp. the smallest eigenvalueλ⁽_minⁿ⁾) almost surely con- verges to 2 (resp.−2), the right-end (resp. left-end) point of the support of the semi-circle law asn→ ∞.

• [16]The centered and rescaled quantityn²³€

λ⁽ⁿ⁾_max−2Š

converges in distribution toward Tracy-Widom distribution function F_{GU E}⁺ asn→ ∞, which can be defined in the following way

F_{GU E}⁺ (s) =exp

‚

− Z∞

s

(x−s)q²(x)d x

Œ , whereqsolves the Painlevé II differential equation

q⁰⁰(x) =xq(x) +2q³(x), q(x)∼Ai(x) as x→ ∞,

and Ai(x)denotes the Airy function. In particular,F_{GU E}⁺ is continuous. Similarly,n²³

λ⁽_minⁿ⁾ +2 _D

−→ F_{GU E}⁻ where

F_{GU E}⁻ (s) =1−F_{GU E}⁺ (−s). If∆is a Borel set inR, denote by

Nn(∆) =#n

λ⁽iⁿ⁾∈∆o ,

the number of eigenvalues ofMin∆. The following theorem is the main result of the article.

Theorem 1. LetMbe a nn matrix from the GUE with eigenvalues(λ⁽₁ⁿ⁾,· · ·,λ⁽ⁿ⁾n ). Let p≥2be a fixed integer and letµ= (µ1,· · ·,µp)∈R^p be such that−2=µ1< µ2<· · ·< µp=2. Denote by∆= (∆1,· · ·,∆p)a collection of p bounded Borel sets inRand consider∆n= (∆1,n,· · ·,∆p,n) defined by

(ed g e) ∆1,n := −2+ ∆1

n^2/3 , ∆p,n := 2+ ∆p

n^2/3 , (bul k) ∆i,n := µi+∆i

n , 2≤i≤p−1 . Let(`1,· · ·,`p)∈N^p, then

nlim→∞ P€

Nn(∆1,n) =`1,· · ·,Nn(∆p,n) =`p

Š−

p

Y

k=1

P€

Nn(∆k,n) =`k

Š

!

=0 .

Remark1. An important corollary of Theorem 1 is the asymptotic independence of the random variables n²³

λ⁽_minⁿ⁾ +2

andn²³€

λ⁽ⁿ⁾_max−2Š

, whereλ⁽_minⁿ⁾ andλ⁽ⁿ⁾_max are the smallest and largest eigenvalues ofM. This in turn enables us to describe the fluctuations of the ratio^λ(n)max

λ⁽ⁿ⁾min

.

Remark 2. For fluctuations of the eigenvalues within the bulk or near the spectrum edges at various scales (different from those studied here), one can refer to[6,7,8].

Proof of Theorem 1 is postponed to Section 3. In Section 2, we prove the asymptotic indepen- dence of the rescaled smallest and largest eigenvalues of M; we then describe the asymptotic fluctuations of the ratio ^λ(n)max

λ⁽ⁿ⁾_min. Remaining proofs are provided in Section 4.

Acknowlegment

This work was partially supported by “Agence Nationale de la Recherche” program, project SESAME ANR-07-MDCO-012-01 and by the “GDR ISIS” via the program “jeunes chercheurs”.

The authors are grateful to Walid Hachem for fruitful discussions and to Eric Amar for useful references related to functions of several complex variables.

2 Asymptotic independence of extreme eigenvalues

In this section, we prove that the random variablesn²³€

λ⁽_maxⁿ⁾ −2Š

andn²³

λ⁽ⁿ⁾_min+2

are asymp- totically independent as the size of matrixMgoes to infinity. We then apply this result to describe the fluctuations of ^λ(n)max

λ⁽ⁿ⁾_min. For a nice and short operator-theoretic proof of this result (subsequent to the present article, although previously published), one can also refer to[5]. In the sequel, we drop the superscript⁽ⁿ⁾to lighten the notations.

2.1 Asymptotic independence

Specifying p=2,µ1=−2,µ2=2 and getting rid of the boundedness condition over∆1and

∆2in Theorem 1 yields the following

Corollary 1. LetMbe a nn matrix from the GUE. Denote byλminandλmaxits smallest and largest eigenvalues, then the following holds true

P

n²³ λmin+2

<x, n²³ λmax−2

< y

−P

n²³ λmin+2<x P

n²³ λmax−2<y

−−→_n

→∞ 0 . Thus

n²³(λmin+2),n²³(λmax−2) _D

−−→n→∞ (λ₋,λ₊),

whereλ−andλ+are independent random variables with distribution functions F_{GU E}⁻ and F_{GU E}⁺ . Proof. Denote by(λ(i))the ordered eigenvalues ofMλmin=λ(1)≤λ(2)≤ · · · ≤λ(n)=λmax. Let (x,y)∈R²and takeα≥max(|x|,|y|). Let∆1= (−α,x)and∆2= (y,α)so that

∆1,n =

‚

−2− α n²³

,−2+ x n²³

Œ

and ∆2,n =

‚ 2+ y

n²³ , 2+ α

n²³

Π. We have

¦N(∆1,n) =0©

= n

n²³(λmin+2)>xo

∪

¨

∃i∈ {1,· · ·,n}; λ(i)≤ −2− α n²³

, λ(i+1)≥ −2+ x n²³

« ,

=: n

n²³(λmin+2)>xo

∪ {Π(−α,x)}, (1)

with the convention that ifi=n, the condition simply becomesλmax≤ −2−αn⁻²³. Note that both sets in the right-hand side of the equation are disjoint. Similarly

¦N(∆2,n) =0©

= n

n²³(λmax−2)< yo

∪

¨

∃i∈ {1,· · ·,n}; λ₍i−1)≤2+ y n²³

, λ₍i)≥2+ α n²³

«

, (2)

=: n

n²³(λmax−2)< yo

∪ {Π(˜ y,α)}, (3)

with the convention that ifi=1, the condition simply becomesλmin≥2+αn⁻²³. Gathering the two previous equalities enables to write{N(∆1,n) =0,N(∆2,n) =0}as the following union of disjoint events

¦N(∆1,n) =0 , N(∆2,n) =0©

= n

Π(−α,x), n²³(λmax−2)<yo

∪¦

Π(−α,x), ˜Π(y,α)©

∪n

n²³(λmin+2)>x, ˜Π(y,α)o

∪n

n²³(λmin+2)>x, n²³(λmax−2)<yo . (4) Define

u_n := P n

n²³(λmin+2)>x, n²³(λmax−2)<yo

−Pn

n²³(λmin+2)>xo P

n

n²³(λmax−2)< yo ,

= P

¦N(∆1,n) =0 , N(∆2,n) =0©

−P¦

N(∆1,n) =0© P¦

N(∆2,n) =0©

+εn(α), (5)

where by equations (1), (3) and (4) εn(α):=−Pn

Π(−α,x), n²³(λmax−2)<yo

−P¦

Π(−α,x), ˜Π(y,α)©

−P n

n²³(λmin+2)>x , ˜Π(y,α)o +P

¦N(∆1,n) =0© P

¦Π(˜ y,α)© +P{Π(−α,x)}P

¦N(∆2,n) =0©

−P{Π(−α,x)}P

¦Π(˜ y,α)© . Using the triangular inequality, we obtain:

|εn(α)| ≤6 max€

P{Π(−α,x)},P¦

Π(˜ y,α)©Š

. As{Π(−α,x)} ⊂ {n²³(λmin+2)<−α}, we have

P{Π(−α,x)} ≤P{n²³(λmin+2)<−α} −−→_n

→∞ F_{GU E}⁻ (−α)−−→_α→∞ 0 . We can apply the same arguments to{Π(˜ y,α)} ⊂ {n²³(λmax−2)> α}. We thus obtain:

α→∞lim lim sup

n→∞ |εn(α)|=0 . (6)

The differenceP

¦N(∆1,n) =0 , N(∆2,n) =0©

−P¦

N(∆1,n) =0© P

¦N(∆2,n) =0©

in the right­- hand side of (5) converges to zero as n → ∞ by Theorem 1 for everyα large enough. We therefore obtain

lim sup

n→∞ |u_n|=lim sup

n→∞ |εn(α)|.

The lefthand side of the above equation is a constant w.r.t. α while the second term (whose behaviour for smallαis unknown) converges to zero asα→ ∞by (6). Thus, lim_n→∞u_n=0.

The mere definition ofu_ntogether with Tracy and Widom fluctuation results yields

n→∞limP n

n²³(λmin+2)>x, n²³(λmax−2)<yo

=€

1−F_{GU E}⁻ (x)Š

F_{GU E}⁺ (y). This completes the proof of Corollary 1.

2.2 Application: Fluctuations of the ratio of the extreme eigenvalues in the GUE

As a simple consequence of Corollary 1, we can easily describe the fluctuations of the ratio

λmax

λmin

. The counterpart of such a result to Gaussian Wishart matrices is of interest in digital communication (see[4]for an application in digital signal detection).

Corollary 2. LetMbe a nn matrix from the GUE. Denote byλminandλmaxits smallest and largest eigenvalues, then

n²³ λmax

λmin

+1 D

−−→_n

→∞ −1

2(λ−+λ+),

where −→^D denotes convergence in distribution,λ−and λ+ are independent random variable with respective distribution F_{GU E}⁻ and F_{GU E}⁺ .

Proof. The proof is a mere application of Slutsky’s lemma (see for instance[18, Lemma 2.8]).

Write

n²³ λmax

λmin

+1

= 1 λmin

h

n²³(λmax−2) +n²³(λmin+2)i

. (7)

Now, λmin

−1

goes almost surely to -2 asn→ ∞, andn²³(λmax−2) +n²³(λmin+2)converges in distribution to the convolution ofF_{GU E}⁻ andF_{GU E}⁺ by Corollary 1. Thus, Slutsky’s lemma yields the convergence (in distribution) of the right-hand side of (7) to−¹₂(λ⁻+λ⁺)withλ⁻andλ⁺ independent and distributed according toF_{GU E}⁻ andF_{GU E}⁺ . Proof of Corollary 2 is completed.

3 Proof of Theorem 1

3.1 Useful results

3.1.1 Kernels

Let{H_k(x)}k≥0be the classical Hermite polynomialsH_k(x):=e^x2€

−_{d x}^d Šk

e^−x2and consider the functionψ⁽_kⁿ⁾(x)defined for 0≤k≤n−1 by:

ψ⁽_kⁿ⁾(x):=n 2

‹¹₄ e^−nx

2 4

(2^kk!p π)¹²H_k

‚Çn 2x

Œ . Denote byK_n(x,y)the following kernel onR²

K_n(x,y) :=

n−1

X

k=0

ψ⁽_kⁿ⁾(x)ψ⁽_kⁿ⁾(y), (8)

= ψ⁽nⁿ⁾(x)ψ⁽ⁿ⁾_n−1(y)−ψ⁽nⁿ⁾(y)ψ⁽ⁿ⁾_n−1(x)

x−y . (9)

Equation (9) is obtained from (8) by the Christoffel-Darboux formula. We recall the two well- known asymptotic results

Proposition 1. a) Bulk of the spectrum.Letµ∈(−2, 2).

∀(x,y)∈R², lim

n→∞

1 nK_n

µ+x n,µ+ y

n

‹=sinπρ(µ)(x−y)

π(x−y) , (10) whereρ(µ) =p

4−µ²

2π . Furthermore, the convergence(10)is uniform on every compact set of R².

b) Edge of the spectrum.

∀(x,y)∈R², lim

n→∞

1 n^2/3K_n

 2+ x

n^2/3, 2+ y n^2/3

‹=Ai(x)Ai⁰(y)−Ai(y)Ai⁰(x)

x−y , (11)

where Ai(x)is the Airy function. Furthermore, the convergence (11) is uniform on every compact set ofR².

We will need as well the following result on the asymptotic behavior of functionsψ⁽_kⁿ⁾. Proposition 2. Letµ∈(−2, 2), k∈ {0, 1}and denote by K a compact set ofR.

a) Bulk of the spectrum.There exists a constant C such that sup

x∈K

ψ⁽_n−kⁿ⁾ 

µ+ x n

‹

≤C. (12) b) Edge of the spectrum.There exists a constant C such that

sup

x∈K

ψ⁽_n−kⁿ⁾ 

2 x n^2/3

‹

≤n^1/6C. (13) The proof of these results can be found in[11, Chapter 7], see also[1, Chapter 3].

3.1.2 Determinantal representations, Fredholm determinants

There are determinantal representations using kernel K_n(x,y)for the joint density p_n of the eigenvalues(λ⁽iⁿ⁾; 1≤i≤n), and for its marginals (see for instance[10, Chapter 6]):

p_n(x₁,· · ·,x_n) = 1 n!det¦

K_n(x_i,x_j)©

1≤i,j≤n , (14)

Z

R^n−m

p_n(x₁,· · ·,x_n)d x_m+1· · ·d x_n = (n−m)! n! det¦

K_n(x_i,x_j)©

1≤i,j≤m (m≤n). (15) Definition 1. Consider a linear operator S defined for any bounded integrable function f :R→R by

S f :x7→

Z

R

S(x,y)f(y)d y,

where S(x,y)is a bounded integrable Kernel on R² →R with compact support. The Fredholm determinant D(z)associated with operator S is defined as follows

∀z∈C, D(z) := det(1−zS) = 1+ X∞ k=1

(−z)^k k!

Z

R^k

det¦

S(x_i,x_j)©

1≤i,j≤kd x₁· · ·d x_k. (16) It is in particular an entire function and its logarithmic derivative has a simple expression [17, Section 2.5]given by

D⁰(z) D(z) =−

X∞ k=0

T(k+1)z^k, (17)

where

T(k) = Z

R^k

S(x₁,x₂)S(x₂,x₃)· · ·S(x_k,x₁)d x₁· · ·d x_k. (18) For details related to Fredholm determinants, see for instance[14,17].

The following kernel will be of constant use in the sequel S_n(x,y;λ,∆):=

p

X

i=1

λi1_∆i(x)K_n(x,y), (19) where λ= (λ1,· · ·,λp)∈R^p orλ ∈C^p, depending on the need, and ∆= (∆1,· · ·,∆p)is a collection ofpbounded Borel sets inR.

Remark3. The kernelK_n(x,y)is unbounded and one cannot consider its Fredholm determinant without caution. The kernelS_n(x,y)is bounded inxsince the kernel is zero if xis outside the compact closure of the set∪^p_i=1∆i, but a priori unbounded in y. In all the forthcoming compu- tations, one may replaceS_nwith the bounded kernel ˜S_n(x,y) =Pp

i,`=1λi1<sub>∆i</sub>(x)1<sub>∆`</sub>(y)K_n(x,y) and get exactly the same results. For notational convenience, we keep on working withS_n. Proposition 3. Let p≥1be a fixed integer,`= (`1,· · ·,`p)∈N^p and denote∆= (∆1,· · ·,∆p), where every∆iis a bounded Borel set. Assume that the∆i’s are pairwise disjoint. Then the following identity holds true

P¦

N(∆1) =`1,· · ·,N(∆p) =`p

©

= 1

`1!· · ·`p!

− ∂

∂ λ1

`1

· · ·

‚

− ∂

∂ λp

Œ`p

det 1−S_n(λ,∆) _λ

1=···=λp=1

, (20) where S_n(λ,∆)is the operator associated to the kernel defined in(19).

Proof of Proposition 3 is postponed to Section 4.1.

3.1.3 Useful estimates for kernelS_n(x,y;λ,∆)and its iterations

Considerµ,∆and∆n as in Theorem 1. Assume moreover thatnis large enough so that the Borel sets(∆i,n; 1≤i≤p)are pairwise disjoint. Fori∈ {1,· · ·,p}, defineκias

κi=

¨ 1 if −2< µi<2

2

3 ifµi=2 . (21)

Otherwise stated,κ1=κp=²₃andκi=1 for 1<i<p.

Letλ∈C^p. With a slight abuse of notation, denote byS_n(x,y;λ)the kernel

S_n(x,y;λ):=S_n(x,y;λ,∆n). (22) For 1≤m,`≤pandΛ⊂C^p, define

Mm`,n(Λ) := sup

λ∈Λ sup

(x,y)∈∆m,n∆`,n

S_n(x,y;λ)

, (23) whereS_n(x,y;λ)is given by (22).

Proposition 4. Let Λ ⊂ C^p be a compact set. There exist two constants R := R(Λ) > 0 and C:=C(Λ)>0, independent from n, such that for n large enough,

¨ Mii,n(Λ) ≤ R⁻¹n^κi , 1≤i≤p Mi j,n(Λ) ≤ C n¹⁻

κi+κj

2 , 1≤i,j≤p, i6=j . (24)

Proposition 4 is proved in Section 4.2.

Consider the iterated kernel|S_n|^(k)(x,y;λ)defined by

¨ |S_n|⁽¹⁾(x,y;λ) =|S_n(x,y;λ)|

|S_n|^(k)(x,y;λ) =R

R^k−1|S_n(x,u;λ)||S_n|^(k−1)(u,y;λ)du k≥2 , (25)

whereS_n(x,y;λ)is given by (22). The next estimates will be stated withλ∈C^p fixed. Note that|S_n|^(k)is nonnegative and write

Z

R^k−1

|S_n(x,u₁;λ)S_n(u₁,u₂;λ)· · ·S_n(u_k−1,y;λ)|du₁· · ·du_k−1. As previously, define for 1≤m,`≤p

M_m`,n^(k) (λ):= sup

(x,y)∈∆m,n∆`,n

|S_n|^(k)(x,y;λ). The following estimates hold true

Proposition 5. Consider the compact set Λ = {λ} and the associated constants R = R(λ)and C=C(λ)as given by Prop. 4. Letβ >0be such thatβ >R⁻¹and considerε∈(0,¹

3). There exists an integer N₀:=N₀(β,ε)such that for every n≥N₀and for every k≥1,

( M_mm,n^(k) (λ) ≤ β^kn^κm, 1≤m≤p M_m^(k)_`,n(λ) ≤ Cβ^k−1n

1+ε−^κm₂^+κ`

, 1≤m,`≤p, m6=` . (26) Proposition 5 is proved in Section 4.3.

3.2 End of proof

Considerµ,∆and∆nas in Theorem 1. Assume moreover thatnis large enough so that the Borel sets(∆i,n; 1≤i≤ p)are pairwise disjoint. As previously, denoteS_n(x,y;λ) =S_n(x,y;λ,∆n); denote also S_i,n(x,y;λi) = S_n(x,y;λi,∆i,n) = λi1_∆i(x)K_n(x,y), for 1 ≤ i ≤ p. Note that S_n(x,y;λ) =S_i,n(x,y;λi)ifx∈∆i,n.

For everyz∈Candλ∈C^p, we use the following notations

D_n(z,λ):=det(1−zS_n(λ,∆n)) and D_n,i(z,λi):=det(1−zS_n(λi,∆i,n)). (27) The following controls will be of constant use in the sequel.

Proposition 6. 1. Letλ∈C^pbe fixed. The sequences of functions

z7→D_n(z,λ) and z7→D_i,n(z,λi), 1≤i≤p are uniformly bounded in n on every compact subset ofC.

2. Let z=1. The sequences of functions

λ7→D_n(1,λ) and λ7→D_1,n(1,λi), 1≤i≤p are uniformly bounded in n on every compact subset ofC^p.

3. Letλ∈C^p be fixed. For everyδ >0, there exists r>0such that sup

n

sup

z∈B(0,r)|D_n(z,λ)−1| < δ, sup

n

sup

z∈B(0,r)|D_i,n(z,λi)−1| < δ, 1≤i≤p, where B(0,r) ={z∈C, |z|<r}.

The proof of Proposition 6 is provided in Section 4.4.

We introduce the following functions

d_n:(z,λ) 7→ det 1−zS_n(λ,∆n)

−

p

Y

i=1

det€

1−zS_n(λi,∆i,n)Š

, (28)

f_n:(z,λ) 7→ D⁰_n(z,λ) D_n(z,λ)−

p

X

i=1

D⁰_i,n(z,λi)

D_i,n(z,λi), (29)

where⁰ denotes the derivative with respect toz ∈C. We first prove that f_n goes to zero as n→ ∞.

3.2.1 Asymptotic study off_nin a neighbourhood ofz=0

In this section, we mainly consider the dependence of f_ninz∈Cwhileλ∈C^pis kept fixed. We therefore drop the dependence inλfor readability. Equality (17) yields

D⁰_n(z) D_n(z)=−

X∞ k=0

T_n(k+1)z^k and D⁰_i,n(z) D_i,n(z)=−

X∞ k=0

T_i,n(k+1)z^k (1≤i≤p), (30) where ⁰ denotes the derivative with respect to z ∈ C and T_n(k) and T_i,n(k) are as in (18), respectively defined by

T_n(k) := Z

R^k

S_n(x₁,x₂)S_n(x₂,x₃)· · ·S_n(x_k,x₁)d x₁· · ·d x_k, (31) T_i,n(k) :=

Z

R^k

S_i,n(x₁,x₂)S_i,n(x₂,x₃)· · ·S_i,n(x_k,x₁)d x₁· · ·d x_k. (32)

Recall thatD_nandD_i,nare entire functions (ofz∈C). The functionsD⁰_n

D_n andD⁰_i,n

D_i,nadmit a power series expansion around zero given by (30). Therefore, the same holds true for f_n(z). Moreover Lemma 1. Define R as in Proposition 4. For n large enough, f_n(z)defined by (29) is holomorphic on B(0,R):={z∈C, |z|<R}, and converges uniformly to zero as n→ ∞on each compact subset of B(0,R).

Proof. Denote byξ⁽_iⁿ⁾(x):=λi1_∆i,n(x)and recall thatT_n(k)is defined by (31). Using the identity

k

Y

m=1 p

X

i=1

a_im= X

σ∈{1,···,p}^k k

Y

m=1

a_σ(m)m, (33)

wherea_imare complex numbers,T_n(k)writes (k≥2) T_n(k) =

Z

R^k k

Y

m=1 p

X

i=1

ξ⁽ⁿ⁾i (x_m)

!

K_n(x₁,x₂)· · ·K_n(x_k,x₁)d x₁· · ·d x_k,

= X

σ∈{1,···,p}^k

j_n,k(σ), (34)

where we define j_n,k(σ):=

Z

R^k k

Y

m=1

ξ⁽ⁿ⁾_σ(m)(x_m)

!

K_n(x₁,x₂)· · ·K_n(x_k,x₁)d x₁· · ·d x_k. (35)

We split the sum in the right-hand side of (34) into two subsums. The first is obtained by gathering the terms withk-tuplesσ= (i,i,· · ·,i)for 1≤i≤pand writes

p

X

i=1

Z

R^k k

Y

m=1

λi1_∆i,n(x_m)

!

K_n(x₁,x₂)· · ·K_n(x_k,x₁)d x₁· · ·d x_k=

p

X

i=1

T_i,n(k),

where T_i,n(k)is defined by (32). The remaining sum consists of those terms for which there exists at least one couple(m,`)∈ {1,· · ·,k}<sup>2</sup>such thatσ(m)6=σ(`). Let

S =¦

σ∈ {1,· · ·,p}^k : ∃(m,`)∈ {1,· · ·,k}<sup>2</sup>,σ(m)6=σ(`)© . We obtainT_n(k) =Pp

i=1T_i,n(k) +s_n(k)where s_n(k):= X

σ∈S

j_n,k(σ)

for every k≥2. For each q∈ {1, . . . ,k−1}, denote by πq the following permutation for any k-tuple(a₁, . . . ,a_k)

πq(a₁, . . . ,a_k) = (a_q,a_q+1, . . . ,a_k,a₁, . . . ,a_q−1).

In other words, πq operates a circular shift ofq−1 elements to the left. Clearly, any k-tuple σ∈ S can be written asσ=πq(m,`, ˜σ)for someq∈ {1, . . . ,k−1},(m,`)∈ {1, . . . ,p}²such thatm6=`, and ˜σ∈ {1, . . . ,p}^k−2. This simply expresses the fact that ifσ∈ S, there exists two consecutive elements that differ at some point. Thus

|s_n(k)| ≤ Xk−1 q=1

X

(m,`)∈{1···p}<sup>2</sup> m6=`

X

σ∈{1···˜ p}^k−2

|j_n,k(πq(m,`, ˜σ))|.

From (35), function j_n,k is invariant up to any circular shift πq, so that j_n,k(σ)coincides with j_n,k(πq(m,`, ˜σ))for anyσ=πq(m,`, ˜σ)as above. Therefore,|s_n(k)|writes

|s_n(k)| ≤ Xk−1 q=1

X

(m,`)∈{1···p}<sup>2</sup> m6=`

X

σ∈{1···p}˜ ^k−2

|j_n,k(πq(m,`, ˜σ))|,

≤ k X

(m,`)∈{1···p}<sup>2</sup> m6=`

X

σ∈{1···˜ p}^k−2

Z

R^k

|ξ⁽mⁿ⁾(x₁)ξ⁽ⁿ⁾_` (x₂)ξ⁽ⁿ⁾_σ(1)˜ (x₃)· · ·ξ⁽ⁿ⁾_σ(k−2)˜ (x_k)|

|K_n(x₁,x₂)· · ·K_n(x_k,x₁)|d x₁· · ·d x_k.

The latter writes

|s_n(k)| ≤ k X

1≤m,`≤p m6=`

Z

∆m,n∆`,n

K_n(x₁,x₂)ξ⁽mⁿ⁾(x₁)ξ⁽_`ⁿ⁾(x₂)





 Z

R^k−2

X

σ∈{1···p}˜ ^k−2

ξ⁽ⁿ⁾_σ(1)˜ (x₃)· · ·ξ⁽ⁿ⁾_σ(k−2)˜ (x_k)

K_n(x₂,x₃)· · ·K_n(x_k,x₁)

d x₃· · ·d x_k





d x₁d x₂,

= k X

1≤m,`≤p m6=`

Z

∆m,n∆`,n

K_n(x₁,x₂)

p

X

i=1

ξ⁽iⁿ⁾(x₁)

p

X

i=1

ξ⁽iⁿ⁾(x₂)





 Z

R^k−2

X

σ∈{1···˜ p}^k−2

ξ⁽_σ(1)˜ⁿ⁾ (x₃)· · ·ξ⁽_σ(˜ⁿ⁾_k−2)(x_k)

K_n(x₂,x₃)· · ·K_n(x_k,x₁)

d x₃· · ·d x_k





d x₁d x₂.

It remains to notice that

p

X

i=1

ξ⁽ⁿ⁾i (x₂) Z

R^k−2

X

σ∈{1···˜ p}^k−2 k

Y

m=3

ξ⁽ⁿ⁾_σ(m−2)˜ (x_m)

K_n(x₂,x₃)· · ·K_n(x_k,x₁)

d x₃· · ·d x_k

(a)=

p

X

i=1

ξ⁽iⁿ⁾(x₂) Z

R^k−2 k

Y

m=3 p

X

i=1

ξ⁽iⁿ⁾(x_m)

!

K_n(x₂,x₃)· · ·K_n(x_k,x₁)

d x₃· · ·d x_k,

= Z

R^k−2

|S_n(x₂,x₃)S_n(x₃,x₄)· · ·S_n(x_k,x₁)|d x₃· · ·d x_k,

(b)= |S_n|^(k−1)(x₂,x₁),

where(a)follows from (33), and(b)from the mere definition of the iterated kernel (25). Thus, fork≥2, the following inequality holds true

|s_n(k)| ≤k X

1≤m,`≤p m6=`

Z

∆m,n∆`,n

|S_n(x₁,x₂)||S_n|^(k−1)(x₂,x₁)d x₁d x₂. (36)

Fork=1, lets_n(1) =0 so that equationT_n(k) =P

iT_i,n(k) +s_n(k)holds for everyk≥1.

According to (29), f_n(z)writes:

f_n(z) =− X∞ k=1

s_n(k+1)z^k.

Let us now prove thatf_n(z)is well-defined on the desired neighbourhood of zero and converges

uniformly to zero asn→ ∞. Letβ >R⁻¹, then Propositions 4 and 5 yield

|s_n(k)| ≤ k X

1≤m,`≤p m6=`

Z

∆m,n∆`,n

|S_n(x,y)||S_n|^(k−1)(y,x)d x d y ,

≤ k X

1≤m,`≤p m6=`

Mm`,nM<sub>`</sub>^(k−1)_m,n |∆m,n||∆`,n|,

≤ kβ^k−2 X

1≤m,`≤p m6=`

C²n

1−^κm+₂^κ`

n

1+ε−^κm+₂^κ`

n<sup>−(κm+κ`)</sup>|∆m∆<sub>`</sub>|,

≤ kβ^k−2 X

1≤m,`≤p m6=`

C²|∆m∆<sub>`</sub>| n<sup>2(κm+κ`−1)−ε</sup> ,

(a)

≤ kβ^k−2

1≤maxm≤p|∆m|

2p(p−1)C² n^23−ε

,

where(a)follows from the fact thatκm+κ`−1≥¹₃. Clearly, the power seriesP∞

k=1(k+1)β^k−1z^k converges for|z|< β⁻¹. Asβ⁻¹is arbitrarily lower thanR, this implies thatf_n(z)is holomorphic in B(0,R). Moreover, for each compact subset K included in the open diskB(0,β⁻¹)and for eachz∈K,

|f_n(z)| ≤ X∞ k=1

(k+1)β^k−1(sup

z∈K|z|)^k

!

1≤m≤pmax |∆m|

2p(p−1)C² n^23−ε

.

The right-hand side of the above inequality converges to zero as n → ∞. Thus, the uniform convergence of f_n(z)to zero onKis proved; in particular, asβ⁻¹<R, f_n(z)converges uniformly to zero onB(0,R). Lemma 1 is proved.

3.2.2 Convergence ofd_nto zero asn→ ∞

In this section,λ∈C^pis fixed. We therefore drop the dependence inλin the notations. Consider functionF_ndefined by

F_n(z):=log D_n(z) Qp

i=1D_i,n(z), (37)

where log corresponds to the principal branch of the logarithm and D_nandD_i,nare defined in (30). As D_n(0) =D_i,n(0) =1, there exists a neighbourhood of zero whereF_nis holomorphic.

Moreover, using Proposition 6-3), one can prove that there exists a neighbourhood of zero, say B(0,ρ), where (F_n(z)) is a uniformly compactly bounded family, hence a normal family (see for instance [13]). Assume that this neighbourhood is included in B(0,R), where Ris defined in Proposition 4 and notice that in this neighbourhood,F_n⁰(z) = f_n(z)as defined in (29).

Consider a compactly converging subsequence F_φ(n)→F_φ in B(0,ρ)(by compactly, we mean that the convergence is uniform over any compact set K ⊂B(0,ρ)), then one has in particular F_φ(⁰ _n)(z)→F_φ⁰ butF_φ(⁰ _n)(z) = f_φ(n)(z)→0. Therefore,F_φis a constant overB(0,ρ), in particular, F_φ(z) =F_φ(0) =0. We have proved that every converging subsequence ofF_nconverges to zero