Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble

Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2019

Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble

Maples, Kenneth ; Najnudel, Joseph ; Nikeghbali, Ashkan

Abstract: It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble. In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.

DOI: https://doi.org/10.1214/18-aop1311

Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-177359

Journal Article Published Version

Originally published at:

Maples, Kenneth; Najnudel, Joseph; Nikeghbali, Ashkan (2019). Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble. The Annals of Probability, 47(4):2417-2458.

https://doi.org/10.1214/18-AOP1311 ©Institute of Mathematical Statistics, 2019

STRONG CONVERGENCE OF EIGENANGLES AND EIGENVECTORS FOR THE CIRCULAR UNITARY ENSEMBLE BYKENNETHMAPLES∗, JOSEPHNAJNUDEL†ANDASHKANNIKEGHBALI∗

Universität Zürich∗and University of Cincinnati† It is known that a unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and that these reflections are in-dependent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If we take an infinite sequence of such reflections, and consider their successive products, then we get an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble.

In this coupling, we show that the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed accord-ing to a sine-kernel point process, and we get some estimates of the rate of convergence. Moreover, we also prove that the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.

Notation. Ifv_∈Cn is a vector, then we writev_[m_] for the image ofv under the canonical projection mapCn_→Cmonto the firstmstandard basis vectors and we write(v)k for itskth coordinate.

We write O(n)for the orthogonal group of dimension n, that is, the group of invertible operators onRnwhich preserve the standard real inner product. We write

U (n)for the unitary group of dimensionnwhich preserves the standard complex inner product. For any vector spaceV, real or complex, we writeGL(V )for the group of invertible transformations. We always write 1 for the identity operator in every space.

We writeU₌U (1)for the unit circle inC, that is, those complex numbers with modulus 1.

Calligraphic characters denoteσ-algebras, that is,A_,B_,C_{, etc. If}A_andB_are σ-algebras on a common set, thenA_∨Bdenotes the smallestσ-algebra containing bothA_andB_.

We also write a _∨b, for a, b_≥ 0, to mean max(a, b) and a _∧b to mean min(a, b).

We employ asymptotic notation for inequalities where precise constants are not important. In particular, we write X ₌ O(Y ) to mean that there exists a

Received May 2017; revised September 2018. MSC2010 subject classifications.60B20, 60F15.

Key words and phrases.Random matrix, circular unitary ensemble, convergence of eigenvalues, convergence of eigenvectors, virtual isometries, complex reflections.

constant C >0 such that _|X_{| ≤}CY. All constants are assumed to be absolute unless indicated otherwise by an appropriate subscript; for example, we write

fn(x)=On(gn(x)) to mean that there are constants Cn, one for each value of n, such that_|fn(x)| ≤Cngn(x)for allx. We also use (modified) Vinogradov nota-tion, whereXY meansX₌O_{(Y ), for convenience.}

Iftis a real number, we write_⌊t_⌋its integer part.

IfH is a Hilbert space with scalar product_·,_·, and ifF _⊂H, thenF⊥_{= {}x_∈ H_; x, y ₌0_∀y_∈F_}. IfH is a complex Hilbert space, then we will always use the scalar product which is linear in the first variable and conjugate linear in the second, that is,ax, by ₌abx, y.

1. Introduction. It has been observed that for many models of random ma-trices, the eigenvalues have a limiting short-scale behavior when the dimension goes to infinity which depends on the global symmetries of the model, but not on its detailed features. For example, the Gaussian Orthogonal Ensemble (GOE), for which the matrices are real symmetric with independent Gaussian entries on and above the diagonal, corresponds to a limiting short-scale behavior for the eigen-values that is also obtained for several other models of random real symmetric matrices. Similarly, the limiting spectral behavior of a large class of random her-mitian and unitary ensembles, including the Gaussian Unitary Ensemble (GUE, with independent, complex Gaussians above the diagonal), and the Circular Uni-tary Ensemble (CUE, corresponding to the Haar measure on the uniUni-tary group of a given dimension), involves a remarkable random point process, called the deter-minantal sine-kernel process. It is a point process for which thek-point correlation function is given by ρk(x1, . . . , xk)=det _sin(π(x p−xq)) π(xp−xq) 1_≤p,q≤k .

From an observation of Montgomery in 1972, it has also been conjectured that the limiting short-scale behavior of the imaginary parts of the zeros of the Riemann zeta function is also described by a determinantal sine-kernel process. This similar behavior supports the conjecture of Hilbert and Pólya, who suggested that the non-trivial zeros of the Riemann zeta functions should be interpreted as the spectrum of an operator 1_{2 +}iH withH an unbounded Hermitian operator.

In 1999, Katz and Sarnak [6] gave a proof of the Montgomery conjecture in the function field case. It appears from this work (which relies, among other things, on existing work developed for the proof of Weil’s conjectures, most notably Deligne’s equidistribution theorems) that the classical compact groups (e.g., the unitary group, the orthogonal group, the symplectic group, etc.) endowed with the Haar measure play a central role in the corresponding spectral interpretation. The regime studied by Katz and Sarnak (fixed genus, and number of elements of the base field going to infinity) does not contain arithmetic in the limiting spectral interpretation (this is an effect of equidistribution theorems when the number of

elements of the base field goes to infinity). On the other hand, the problem is still open in theproblematic regimewhere the number of elements of the field is fixed and the genus goes to infinity. There it is not clear how random matrix statistics and arithmetic mix together in the limit (see [5] and [9] for more discussion on this particular aspect). A similar phenomenon occurs for the Riemann zeta func-tion. Inspired by the work of Katz and Sarnak, Keating and Snaith [7] suggest to use the characteristic polynomial of random unitary matrices to model the distri-bution of values of the Riemann zeta function on the critical line and propose a conjecture for the moments of the Riemann zeta function on the critical line where again random matrix statistics and arithmetic mix in a mysterious way.

Following this body of work, several questions have naturally emerged. In par-ticular, Katz and Sarnak asked whether it is possible to give a meaning to strong convergence (i.e., almost sure convergence) for the eigenvalues of random unitary matrices to the determinantal sine kernel point process. This problem was first solved by Borodin and Olshanski in [1] and then together with Paul Bourgade, the second and third authors of this paper proposed in [3] an alternative solution with the so called virtual isometries (see below for more details on the construction of virtual isometries). Our approach was probabilistic (we used coupling techniques ideas) and also quantitative: we were able to quantify the rate of convergence to the sine kernel point process. The goal of this paper is twofold:

• to improve on our estimates in the rate of convergence to the sine-kernel point process by refining several other estimates;

• to give a complete panorama of the spectral analysis of virtual isometries by establishing quantitative strong convergence for the eigenvectors as well. We believe that these new and/or refined estimates can be very useful in tack-ling other problems at the interface of random matrix theory and analytic number theory. For instance, in the companion paper [4], we provide a new approach to ratios of characteristic polynomials and solve a conjecture on the limit of ratios of characteristic polynomials, where we use the estimates of this paper. In another companion paper, we use the spectral analysis of this paper to construct a flow of operators, constructed from virtual isometries, and whose spectrum is the sine-kernel point process.

We now briefly recall the notion ofvirtual isometry, which has been introduced in [3], generalizing both the notion ofvirtual permutationstudied by Kerov, Ol-shanski and Vershik [8], and the previous notion of virtual unitary group intro-duced by Neretin [11]. A virtual isometry is a sequence of random unitary matrices

(un)n≥1constructed in the following way:

1. One considers a sequence(xn)x≥1 of independent random vectors,xnbeing uniform on the unit sphere ofCn.

2. Almost surely, for alln_≥1, xn is different from the last basis vectorenof

Cn, which implies that there exists a uniquern∈U (n)such thatrn(en)=xnand rn−Inhas rank one.

3. We define(un)n≥1by induction as follows:u1=x1and for alln≥2, un=rn un−1 0 0 1 .

It was proven in [2] that with this construction, un follows, for alln≥1, the Haar measure onU (n). From now on, we always assume that the sequence(un)n≥1

is defined with this coupling.

For each value ofn, letλ(n)₁ , . . . , λ(n)n be the eigenvalues ofun, ordered counter-clockwise, starting from 1: they are almost surely pairwise distinct and different from 1. If 1_≤k_≤n, we denote by θ_k(n) the argument ofλ(n)_k , taken in the inter-val (0,2π ):θ_k(n) is the kth strictly positive eigenangle of un. If we consider all the eigenangles ofun, taken not only in(0,2π )but in the whole real line, we get a (2π )-periodic set with n points in each period. If the eigenangles are indexed increasingly byZ, we obtain a sequence

· · ·< θ₋(n)₁ < θ₀(n)<0< θ₁(n)< θ₂(n)<_{· · ·},

for whichθ_k(n)_+n=θ_k(n)₊2π for allk_∈Z.

It is also convenient to extend the sequence of eigenvalues as an-periodic se-quence indexed byZ, in such a way that for allk_∈Z,

λ(n)_{k =}expiθ_k(n).

Note that in the notion of virtual isometry defined here, the vectors of the canon-ical basis ofCnplay a particular role. One could attempt to generalize the notion of virtual isometries by considering sequences of unitary operators onEn,n≥1, where(En)n≥1is a sequence of complex inner product spaces,Enbeing of dimen-sionn. However, this reduces to the particular caseEn=Cnby a change of basis and so we have chosen to use the standard basis for simplicity.

Next, for 1_≤k_≤n, let f_k(n) _∈Cn denote a unit length representative of the eigenspace forλ(n)_k . Then if we expandrn+1(en+1)in the basis of eigenvectors

rn+1(en+1)= n

j=1

μ(n)_j f_j(n)₊νnen+1

then the eigenvalues ofun+1are precisely the zeros of the rational equation n j=1 μ(n)_j 2 λ (n) j λ(n)_{j −}z+ |1₋νn|2 1₋z =1−νn

and the eigenvectors ofun+1are given by then+1 equations Ckfk(n+1)= n j=1 μ(n)_j λ(n)_{j −}λ(n_k+1) f_j(n)₊ νn−1 1₋λ(n_k +1) en+1

for 1_≤k_≤n₊1; here,Ck∈R+ is a constant so thatf_k(n+1)has unit length. As above, we will also extend, when needed, the notationf_k(n),μ(n)_j , in such a way that the sequences(f_k(n))k∈Zand(μ(n)_j )j∈Zaren-periodic.

In [3], it is shown that if(un)n≥1follows this distribution, then for allk, thekth

positive (resp., negative) eigenangle θ_k(n) (resp. θ₁(n)_−k) ofun, multiplied by n/2π (i.e., the inverse of the average spacing between eigenangles for any matrix in

U (n)), converges almost surely to a random variableyk(resp.,y1−k). The random set(yk)k∈Zis a determinantal sine-kernel process, and for eachk, the convergence

holds with a rate dominated by some negative power ofn. In the present paper, we improve our estimate of this rate, and more importantly, we prove that almost sure convergence not only holds for the eigenangles ofun, but also for the components of the corresponding eigenvectors. More precisely, we show that, for allk, ℓ_≥1, the ℓth component of the eigenvector of un associated to the kth positive (resp., negative) eigenangle converges almost surely to a nonzero limit whenn goes to infinity, if the norm of the eigenvector is taken equal to√nand if the phases are suitably chosen. Note that taking a norm equal to√nis natural in this setting: with this normalization, the expectation of the squared modulus of each coordinate of a given eigenvector ofunis equal to 1, so we can expect a convergence to a nontrivial limit. If the norm of the eigenvectors is taken equal to 1 instead of√n, then the coordinates converge to zero whenngoes to infinity.

The precise statement of our main results are given as follows.

THEOREM1.1. With the notation above,the following estimate holds almost surely: n 2πθ (n) k =yk+O 1₊k2n−13+ε_,

for all n_≥1, _|k_{| ≤}n1/4 and ε >0, where the implied constant may depend on

(um)m≥1andε,but not onnandk.

THEOREM 1.2. Let (un)n≥1 be a virtual isometry,following the Haar

mea-sure.Fork_∈Zandn_≥1,letv_k(n) be a unit eigenvector corresponding to thekth smallest nonnegative eigenangle ofunfork≥1,and the(1−k)th largest strictly negative eigenangle ofun fork≤0. Then for allk∈Z,there almost surely exist some complex numbers (ψ_k(n))n≥1 of modulus 1, and a sequence (tk,ℓ)ℓ≥1, such

that for allℓ_≥1,

√

nψ_k(n)v(n)_k , eℓ _n−→

→∞tk,ℓ.

Almost surely, for allk_∈Z, the sequence (tk,ℓ)ℓ≥1 depends, up to a

multiplica-tive factor of modulus one, only on the virtual rotation (un)n≥1. Moreover, if (ψk)k∈Z is a sequence of i.i.d., uniform variables onU,independent of(tk,ℓ)ℓ≥1,

then (ψktk,ℓ)k∈Z,ℓ≥1 is an i.i.d. family of standard complex Gaussian variables

REMARK1.3. The vectorsv_k(n)are equal tof_k(n), up to a multiplicative factor of modulus 1. The independent phasesψkintroduced in the last part of the theorem are needed in order to get i.i.d. complex Gaussian variables. This is not the case, for example, if we normalize(tk,ℓ)ℓ≥1in such a way thattk,1∈R+.

In Section3, we make the following key observation: the sequence of eigenval-uesλ(n)_k , with 1_≤k_≤nandn_≥1, is independent of the argument of the coeffi-cientsμ(n)_j , with 1_≤j_≤nandn_≥1. Therefore, we can consider the sequence of eigenvalues of the virtual isometry and prove that it converges almost surely, and then,conditioning on the eigenvalues of every matrix in the virtual isometry, con-sider the sequence of eigenvectors and show that they also converge in a suitable sense.

The first part of this plan is carried out in Section 4, where Theorem 1.1 is proven.

Next, in Section 5 we condition on the eigenangles of the entire sequence of matrices. We show that for each fixedk there is a renormalization factorD(n)_k so that for eachℓ_≥1 the sequenceD_k(n)f_k(n), eℓis a martingale which converges in L2and almost surely to a limiting valuegk,ℓ. In Section6, Theorem1.2is deduced from this convergence.

2. Spectral analysis of the virtual isometries. It is classical that if un is a random unitary matrix following the Haar measure onU (n), then the distribution of the eigenangles ofun, multiplied byn/2π, converges in law to a determinantal sine-kernel process. In fact, this result can be found in the literature under the form of the convergence of the correlation functions against a suitably chosen family of test functions. However, we were not able to find a statement with its proof on the fact that the convergence takes place in the sense of weak convergence of point processes, using Laplace functionals. So for completeness, we give such a statement below and postpone its proof until AppendixB.

PROPOSITION 2.1. Let En denote the set of eigenvalues taken in (−π, π] and multiplied byn/2π of a random unitary matrix of size nfollowing the Haar measure.Let us also define fory₌y′,

K(∞)y, y′₌sin[π(y

′_−y)]

π(y′_−y)

and

K(∞)(y, y)₌1.

Then there exists a point process E_∞ such that for all r _≥1, and for all Borel measurable and bounded functions F with compact support from Rr to R, we

have E x1=···=xr∈E∞ F (x1, . . . , xr) = RrF (y1, . . . , yr)ρ (_∞) r (y1, . . . , yr) dy1· · ·dyr, where ρ_r(∞)(y1, . . . , yr)=detK(∞)(yj, yk)_1≤j,k≤r.

Moreover, the point processEn converges to E∞ in the following sense: for all

Borel measurable bounded functionsf with compact support fromRtoR,

x∈En f (x) _−→ n→∞ x∈E_∞ f (x),

where the convergence above holds in law.

In [3], it was proven that the eigenangles of a virtual isometry, taken according to Haar measure and renormalizing the eigenangles by the dimensionn, converge almost surely to a point process with this determinantal distribution. Precisely, we have the following.

PROPOSITION 2.2. Let(un)n≥1 be a random virtual isometry following the

Haar measure.The eigenvalues ofunare almost surely distinct and different from 1,and then,as explained before,it is possible to order the eigenangles as an in-creasing sequence indexed byZ:

· · ·< θ₋(n)₂ < θ₋(n)₁ < θ₀(n)<0< θ₁(n)< θ₂(n)<_{· · ·}.

Moreover,almost surely,for allm_∈Z,there existsymsuch that n

2πθ (n)

m =ym+On−ε,

when n goes to infinity, ε >0 being some universal constant, and the process

(ym)m∈Zis a determinantal sine-kernel process.

REMARK 2.3. In this proposition, the periodic extension of the sequence

(θ_k(n))1_≤k≤n is needed to defineymfor nonpositive values ofm. We also note that ymdepends only on the behavior ofθm(n)for a fixed value ofm. Informally, the lim-iting sine-kernel process(ym)m∈Zdepends only on the behavior of the eigenvalues

ofunwhich are close to 1.

We want to understand the behavior of the eigenvectors of un as n goes to infinity. Our method will give a formula for the spectrum and eigenvectors ofun+1

We assume throughout that for eachn_≥1, theneigenvalues ofunare distinct; this holds almost surely for virtual isometries constructed according to the Haar measure.

We recall that the eigenvalues ofun,λ(n)₁ , λ(n)₂ , . . . , λ(n)n , are ordered in such a way thatλ(n)_{k =}eiθk(n)_{, and}

0< θ₁(n)<_{· · ·}< θ_n(n)<2π.

Moreover, the eigenangles enjoy a property of periodicity: for all k_∈Z,θ_k(n)_+n= θ_k(n)₊2π.

As all the eigenvalues are distinct, each eigenvalue corresponds to a one-dimensional eigenspace. We can therefore write f₁(n), . . . , fn(n) for the family of unit length eigenvectors ofun, which are well defined up to a complex phase: the notationf_k(n)is then extendedn-periodically to allk_∈Z.

Letxn=un(en)and letrndenote the unique reflection onCnmappingentoxn. Therefore, we haveun+1=rn+1◦(un⊕1). It is natural to decomposexn+1 into

the basis given byι(f₁(n)), . . . , ι(fn(n)), en+1, whereι:Cn→Cn+1is the inclusion

which maps(x1, . . . , xn)to(x1, . . . , xn,0). Identifyingfk(n) andι(f (n) k ), we then have xn+1= n k=1 μ(n)_k f_k(n)₊νnen+1

for some μ(n)_k (1_≤k_≤n) and νn such that |μ(n)₁ | 2

+ · · · + |μ(n)n | 2

+ |νn|2=1. Again, it can be convenient to considerμ(n)_k for allk_∈Z, by an-periodic extension of the sequence. The following result gives the spectral decomposition ofun+1in

function of the decomposition ofunandxn+1.

THEOREM 2.4 (Spectral decomposition). On the event that the coefficients

μ(n)₁ , . . . , μ(n)n are all different from zero and that theneigenvalues of unare all distinct(which holds almost surely under the uniform measure onU∞),the eigen-values ofun+1 are the unique roots of the rational equation

n j=1 μ(n)_j 2 λ (n) j λ(n)_{j −}z+ |1₋νn|2 1₋z =1−νn

on the unit circle.Furthermore,they interlace between1and the eigenvalues ofun

0< θ₁(n+1)< θ₁(n)< θ₂(n+1)<_{· · ·}< θ_n(n)< θ_n(n₊+₁1)<2π,

rela-tion h(n_k +1)12_f(n+1) k = n j=1 μ(n)_j λ(n)_{j −}λ(n_k +1)f (n) j + νn−1 1₋λ(n_k+1)en+1,

whereh(n_k+1)is real and strictly positive.

PROOF. Let f be an eigenvector of un+1 with corresponding eigenvalue z.

Then we have f ₌ n j=1 ajfj(n)+ben+1,

wherea1, . . . , an,bare (as yet unknown) complex numbers, not all zero. Our goal is to write these coefficients in terms ofxn+1 and the eigenvalues ofun.

We have zf ₌un+1f =un+1 _n j=1 ajfj(n)+ben+1 = n j=1 ajun₊1f_j(n)+bun₊1en₊1 = n j=1 ajλ(n)_j rn+1fj(n)+bxn+1.

We recall that for allt_∈Cn+1,rn+1(t )is given by

rn+1(t )=t+ t, xn+1−en+1 en+1, xn+1−en+1 (xn+1−en+1) so that zf ₌ n j=1 ajλ(n)_j f_j(n)₊ f (n) j , xn+1−en+1 en+1, xn+1−en+1 (xn+1−en+1) +bxn+1. Now we decompose xn+1= n k=1 μ(n)_k f_k(n)₊νnen+1

and zf ₌ n j=1 ajλ(n)_j f_j(n)₊ μ (n) j νn−1 (xn+1−en+1) +bxn+1 = n j=1 ajλ(n)j f (n) j + _n ℓ=1 aℓλ(n)ℓ μ(n)_ℓ νn−1 (xn+1−en+1)+bxn+1.

Becausef₁(n), . . . , fn(n), en+1 is a basis forCn+1, we deduce the system ofn+1

equations zaj =ajλ(n)j +μ (n) j n ℓ=1 aℓλ(n)ℓ μ(n)_ℓ νn−1+ bμ(n)_j forj ₌1, . . . , nand zb₌b₊(νn−1) n ℓ₌1 aℓλ(n)_ℓ μ(n)_ℓ νn−1+ b(νn−1).

Forz /_{∈ {}λ(n)₁ , . . . , λ(n)n ,1}, let us consider the linear transformQ:Cn+1→Cn+1 whose matrix representation in the basisf₁(n), . . . , fn(n), en+1is

Q₌I₊wvt, where w₌ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ μ(n)₁ λ(n)_{1 −}z .. . μ(n)n λ(n)n −z νn−1 1₋z ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ and vt ₌ λ(n)₁ μ (n) 1 νn−1 , . . . , λ(n)_n μ (n) n νn−1 ,1 .

Then, the above system can be written

Qf ₌0.

Clearly, rankQ_{∈ {}n, n₊1_}. If it has full rank thenf ₌0, but we assume a priori thatzis an eigenvalue forun+1 and so has a nontrivial eigenspace. Thus we must

have rankQ₌nand

The right-hand side can only vanish iff is proportional tow, sof ₌αwfor some complex constantα_∈C_{\ {}0_}andvtw_{= −}1. In particular,

n j=1 λ(n)_{j |}μ(n)_{j |}2 λ(n)_{j −}z + |νn−1|2 1₋z =1−νn, as required.

Conversely, ifz /_{∈ {}λ(n)₁ , . . . , λ(n)n ,1}satisfies this equation, then Qw₌w₊wvtw₌w₊w(₋1)₌0,

which implies thatwis an eigenvector ofun+1for the eigenvaluez.

Let us now show that the eigenvalues z /_{∈ {}λ(n)₁ , . . . , λ(n)n ,1} of un+1 strictly

interlace between 1 and the eigenvalues ofun: sinceun+1has at mostn+1

eigen-values, this implies thatλ(n)₁ , . . . , λ(n)n ,1 are not eigenvalues ofun+1.

Define the rational function_:S1_→C_{∪ {∞}}by

(z)₌ n j=1 λ(n)_{j |}μ(n)_{j |}2 λ(n)_{j −}z + |νn−1|2 1₋z −(1−νn).

Note that vanishes precisely on the eigenvalues of un+1 which are different

from λ(n)₁ , . . . , λ(n)n ,1. Recalling that |μ(n)₁ | 2

+ · · · + |μ(n)n | 2

+ |νn|2=1, we can rearrange the expression ofto the equivalent form

(z)₌1 2 _n j=1 μ(n)_j 2λ (n) j +z λ(n)_{j −}z+ |1−νn| 21+z 1₋z−νn+νn .

Hence,takes values only iniR_{∪ {∞}}, since for allz₌z′_∈S1,(z₊z′)/(z₋z′)

is purely imaginary (the triangle (₋z′, z, z′)has a right angle atz). Note that for

z_{∈ {}λ(n)₁ , . . . , λ(n)n ,1}, a unique term of the sum defining is infinite, since by assumption, μ(n)₁ , . . . , μ(n)n ,1−νn are nonzero and λ(n)₁ , . . . , λ(n)n ,1 are distinct: (z)_{= ∞}.

Next, we considert_→(eit)in a short interval(θ_j(n)₋δ, θ_j(n)₊δ). Then, for

t₌θ_j(n)₊uin this interval, λ(n)_{j +}λ(n)_j eiu λ(n)_{j −}λ(n)_j eiu = 1₊eiu 1₋eiu=2iu −1 +O₍₁₎

while the other terms in(eit)are uniformly bounded asδ_→0; likewise for the interval(₋δ, δ). In particular, _→i_∞asu_→0 from the right and_{→ −}i_∞

asu_→0 from the left. We therefore conclude, as is continuous, that on each interval of the partition

of the unit circle into n₊1 parts, t_→(eit) must assume every value on the line iR, and in particular must have at least one root. But we know that has onlyn₊1 roots on the circle so there must be exactly one root in each part of the partition, which proves the interlacing property.

It remains to check the expression of the eigenvectors(f_k(n+1))1_≤k≤n+1given in

the theorem, but this expression is immediately deduced from the expression of the vectorwinvolved in the operatorQdefined above, and the fact thatf_k(n+1) ₌1.

3. A filtration adapted to the virtual isometry. In our proof of convergence of eigenvalues and eigenvectors, we will use some martingale arguments. That is why in this section, we introduce a filtration related to our random virtual isometry. Forn_≥1, we define theσ-algebraA_n=σ{λj(m)_|1_≤m_≤n,1_≤j _≤m_}and its limitA₌∞_n=1A_n.

LEMMA 3.1. For all n_≥1,theσ-algebra A_{nis equal},up to completion,to theσ-algebra generated byu1the variables_|μ(m)_{j |}andνmfor1≤m≤n−1and 1_≤j _≤m.

PROOF. By Theorem2.4, the eigenvalues ofun+1are almost surely the roots

of the equation n j=1 μ(n)_j 2 λ (n) j λ(n)_{j −}z+ |1₋νn|2 1₋z =1−νn.

This equation depends only on_|μ(n)_{j |},λ_j(n)forj ₌1, . . . , n, andνn. By induction, we deduce that λ(n_j +1) is a measurable function ofu1, _{|μ_j(m)_|}1≤j≤m,1≤m≤n and {νm}1≤m≤n.

Conversely, the above equation withzspecialized to λ(n₁+1), . . . , λ(n_n++₁1)can be written in the form

Rv₌w.

Here,wis a column vector of 1s,vis the column vector with entries

vt₌ |μ(n)_{1 |}2 1₋νn , . . . ,|μ (n) n | 2 1₋νn ,1₋νn

andRis an(n₊1)_×(n₊1)matrix with entries

Rk,j=

λ(n)_j λ(n)_{j −}λ(n_k +1)

for 1_≤j_≤nand

Rk,n+1=

1 1₋λ(n_k +1).

If we write R₌ RS, where S is the diagonal matrix with entries Sjj = (μ(n)_j λ(n)_j )−1 (1_≤ j _≤ n) and Sn+1,n+1 =νn −1, then we see that the rows of Rare the representations of the eigenvectors f₁(n+1), . . . , f_n(n₊+₁1) in the basis

f₁(n), . . . , fn(n), en+1 up to constants and, therefore, orthogonal. We conclude that

R, and thus R, are invertible, so Rv₌w has a unique solution, which can be written in terms of the eigenvalues λ(m)_j for 1_≤j _≤mand m_≤n₊1. Thus we conclude that _|μ(n)_{j |}andνn are measurable functions ofλ(m)j for 1≤j≤mand m_≤n₊1 as was to be shown.

For 1_≤j _≤n, we define the phaseφ_j(n)byμ_j(n)₌φ(n)_{j |}μ(n)_{j |}, and theσ-algebras

B_n=A_∨σ_{φ_j(m)_|1_≤m_≤n₋1,1_≤j_≤m_}andB₌∞_n

=1Bn.

LEMMA 3.2. Theσ-algebraB_{n is equal,up to completion,to theσ-algebra}

generated byA_{and the eigenvectorsfj}(m)_{for1≤j≤mand1≤m≤n.}

PROOF. Again by Theorem2.4, we can write the eigenvectors ofun+1as

func-tions ofμ(n)_{j =}φ_j(n)_|μ(n)_{j |}(1_≤j _≤n),νn,λ(m)_j (1≤j ≤m,m∈ {n, n+1}), and the eigenvectors of un. Clearly, |μ(n)_j |, νn, and λ(m)_j for m∈ {n, n+1} are A -measurable andφ_j(n)isB_{n+1-measurable.}

Conversely, we write h(n_k +1)1/2f_k(n+1), f_j(n) ₌ φ (n) j |μ (n) j | λ(n)_{j −}λ(n_k +1)

to find eachφ_j(n)as a function of the other variables.

4. Convergence of the eigenangles. In order to prove the convergence of the normalized eigenangles ofun whenngoes to infinity, we need the following lemma.

LEMMA4.1. Letε >0.Then,almost surely under the Haar measure onU∞, forn_≥1and0< k_≤n1/4,we have

θ_k(n+1)_|μ(n)_{k |}2 θ_k(n)₋θ_k(n+1)=1+

and forn_≥1and₋n1/4_≤k_≤0, θ_k(n+1)_|μ(n)_{k |}2 θ_k(n)₋θ_k(n+1) = θ_k(n+1)_|μ(n)_k+n|2 θ_k(n)₋θ_k(n+1) =1+ O1_{+ |}k_|n−13+ε_.

REMARK4.2. The implied constant in theO(_·)notation depends on(um)m≥1

andε: in particular, it is a random variable. However, for given(um)m≥1 andε, it

does not depend onkandn.

PROOF. By symmetry of the situation, we can assumek >0. Moreover, let us fix ε_∈(0,0.01). We will suppose that the eventE_:=E0_∩E1_∩E2_∩E3 holds, where E0₌θ₀(1)₌0_{∩ {∀}n_≥1, νn=0} ∩∀n≥1,1≤k≤n, μ(n)_k =0, E1_=∃n0_≥1,_∀n_≥n0,_|νn| ≤n− 1 2+ε_, E2_=∃n0_≥1,_∀n_≥n0,1_≤k_≤n,μ(n)_{k ≤}n−12+ε_, E3_=∃n0_≥1,_∀n_≥n0, k_≥1, n−53−ε_≤θ(n) k₊1−θ (n) k ≤n− 1₊ε_.

It is possible to do this assumption, since by the result proven in AppendixAof the present paper, the eventEoccurs almost surely. As we will see now, thisa priori information on the distribution of the eigenvalues of the random virtual isome-try implies strong quantitative bounds on the change in eigenvalues of successive unitary matrices.

Recall from Theorem2.4that n j=1 λ(n)_{j |}μ(n)_{j |}2 λ(n)_{j −}λ(n_k +1)+ |1₋νn|2 1₋λ(n_k +1)=1−νn.

By using then-periodictiy ofλ(n)_j ,μ(n)_j ,f_j(n)with respect toj, we can write

(1) j∈J λ(n)_{j |}μ(n)_{j |}2 λ(n)_{j −}λ(n_k+1) + |1₋νn|2 1₋λ(n_k+1) =1−νn,

whereJ is the random set ofnconsecutive integers, such thatθ_k(n+1)₋π < θ_j(n)_≤ θ_k(n+1)₊π. Iterating the lower bound on the distance between adjacent eigenval-ues, given by the definition of the eventE3, we get, forj _∈J_{\ {}k₋1, k_},

θ_j(n)₋θ_k(n+1)_|k₋j_|n−53−ε_, and then λ(n)_{j −}λ(n_k +1)_|k₋j_|n−53−ε_, since_|θ_j(n)₋θ_k(n+1)_{| ≤}π.

Likewise, we have by E3, 1₋λ(n_k +1)₌O_(kn−1+ε₎, and by E2, _|μ(n)_{j |}2 ₌ O_(n−1+2ε_{), which gives, forj}

∈J_{\ {}k₋1, k_}, λ_j(n)(1₋λ_k(n+1))_|μ(n)_{j |}2 λ(n)_{j −}λ(n_k+1) k |k₋j_|n −13+4ε_.

Summing forj inJ_{\ {}k₋1, k_}, which is included in the interval_[k₋1₋n, k₊n_], gives j∈J\{k−1,k} λ_j(n)(1₋λ_k(n+1))_|μ(n)_{j |}2 λ(n)_{j −}λ(n_k +1) = O_kn−13+4ε_logn=O_kn− 1 3+5ε_.

Now, subtracting this equation from the product of (1) by 1₋λ(n_k +1), and bounding

νn=O(n−

1

2+ε_{)(by the propertyE1) gives us the resulting equation}

λ_k(n)(1₋λ_k(n+1))_|μ(n)_{k |}2 λ(n)_{k −}λ(n_k+1) 1k∈J +

λ_k(n)₋₁(1₋λ_k(n+1))_|μ(n)_k−1|2 λ_k(n)₋₁₋λ_k(n+1) 1k−1∈J = −1₊O_kn−13+5ε_.

Next, we estimate the first two terms in terms of the eigenangles. We find 1₋λ(n_k+1)_{= −}iθ_k(n+1)₊Oθ_k(n+1)2 and λ(n)_{j −}λ(n_k +1)₌iθ_j(n)₋θ_k(n+1)λ(n)_{j +}O_θ(n) j −θ (n+1) k 2

forj ₌k₋1,k. Collecting terms and using the trivial bounds give

θ_k(n+1)_|μ(n)_{k |}2 θ_k(n)₋θ_k(n+1) 1₊O_kn−1+ε_1k∈J +θ (n+1) k |μ (n) k−1| 2 θ_k(n)₋₁₋θ_k(n+1) 1₊O_kn−1+ε_{1k−1∈J =1+}O_kn−13+5ε_. (2)

From Theorem2.4, the eigenvalues ofunandun+1interlace, so fornsufficiently

large the real part of the first term is positive and the real part of the second term is negative. The real part of the right-hand side tends to 1 asngrows withkfixed, so the first term has real part bounded below fornsufficiently large. In particular,

θ_k(n+1)_|μ(n)_{k |}2 θ_k(n)₋θ_k(n+1)1.

Using the a priori bounds forθ_k(n+1)and_|μ(n)_{k |}2, we find θ_k(n)₋θ_k(n+1)kn−2+3ε. Hence, θ_k(n+1)₋θ_k(n)₋₁₌θ_k(n)₋θ_k(n)₋₁₋θ_k(n)₋θ_k(n+1) n−53−ε₋O_kn−2+3ε_n− 5 3−ε_,

sincekn−2+3ε/n−53−ε₌O_(n1/4−2+0.03+5/3+0.01_{)=o(1). We deduce that the}

sec-ond term of (2) is dominated bykn−1/3+4ε, and then

θ_k(n+1)_|μ(n)_{k |}2 θ_k(n)₋θ_k(n+1) =1+

O_kn−13+5ε_.

Changing the value ofε appropriately gives the desired result.

This lemma is enough for us to estimate the change in θ_k(n) asn grows, and in particular to find a limit for the renormalized angle. We can now complete the proof of Theorem1.1.

PROOF OFTHEOREM1.1. The proof proceeds exactly as in [3]. It is sufficient to prove the result forε equal to the inverse of an integer: hence, it is enough to show the estimate for fixedε. By symmetry, one can takek >0. We rearrange the equation in Lemma4.1to find

μ(n)_k 2_{= θ}(n) k θ_k(n+1) −1 1₊O_kn−13+ε_.

Because almost surely,_|μ(n)_{k |}2₌O_(n−1+2ε₎, we get

μ(n)_k 2₌ θ (n) k θ_k(n+1) −1+ Okn−43+3ε_.

Using the asymptotic log(1₋δ)_{= −}δ₊O_(δ2_)forδ

=o(1), we conclude, ifε is small enough, log θ (n) k θ_k(n+1) = μ(n)_k 2₊O_kn−43+3ε_.

Define the random variableL(n)_{k =}logθ_k(n)_+jn₌−1_1|μ(j )_{k |}2; we have just shown

L(n_k +1)₋L(n)_{k =}O_(kn−43+3ε_{)so forkfixed,L}(n)

k converges to a limitL (∞) k almost

surely asn_{→ ∞}, with_|L(n)_{k −}L_k(∞)_{| =}O(kn−13+3ε). Now, expL(n)_{k =}θ_k(n)exp n−1 j=k μ(j )_k 2 =nθ_k(n)exp −logn₊ n−1 j=1 1 j + n−1 j=1 μ(j )_k 2₋1 j .

Recall ₋logn₊n_j−₌1₁1_{j =}γ ₊O_(n−1_{) whereγ is the Euler–Mascheroni}

con-stant. Next, we define

M_k(n)_:= n−1 j=1 μ(j )_k 2₋1 j

and observe that each term of the sum is an independent mean-zero random vari-able. Therefore, for k fixed, (M_k(n))n≥k is a martingale. We claim that M_k(n) is bounded inL2; in fact, Eμ(n)_k 2₋1 n 2 =O_n−2_, so that EM_k(∞)₋M_k(n)2₌ j≥n Eμ(j )_k 2₋1 j 2 =O_n−1_,

where M_k(∞) is the claimed limit of M_k(n) (this limit exists since M_k(n) is a sum of centered and independent random variables with summable variances). To see this, we observe that_|μ(n)_{k |}2 is a Beta random variable of parameters 1 andn₋1, whose variance can be explicitly computed, and is easily checked to be dominated by n−2. Now, by the triangle inequality and Doob’s maximal inequality, for q

positive integer,k_≤2q, Esup n≥2q M_k(∞)₋M_k(n)2EM_k(∞)₋M_k(2q)2₊Esup n≥2q M_k(n)₋M_k(2q)2 EM_k(∞)₋M_k(2q)2 =O₂−q_. Hence, E sup 2q_≤n≤2q+1 sup k_≤n1/4 M_k(∞)₋M_k(n)2_≤E sup k≤2(q+1)/4 sup 2q_≤n≤2q+1 M_k(∞)₋M_k(n)2 ≤ k≤2(q+1)/4 E sup 2q_≤n≤2q+1 M_k(∞)₋M_k(n)2 2(q+1)/42−q ₌O₂−3q/4

and Esup n≥2q_ksup ≤n1/4 M_k(∞)₋M_k(n)2_≤ r≥q E sup 2r_≤n≤2r+1 sup k≤n1/4 M_k(∞)₋M_k(n)2 r≥q 2−3r/4₌O₂−3q/4_.

By Markov’s inequality, we get Psup n≥2q sup k≤n1/4 M_k(∞)₋M_k(n)_≥2−q/3_≤22q/3Esup n≥2q sup k≤n1/4 M_k(∞)₋M_k(n)2 =O2−q/12,

which, by the Borel–Cantelli lemma, shows that almost surely for someq0_≥1, all

q_≥q0,n_≥2q andk_≤n1/4 satisfy_|M_k(∞)₋M_k(n)_{| ≤}2−q/3. Hence,

M_k(∞)₋M_k(n)₌O_n−13

almost surely. Collecting these estimates and applying them to the equation expL(n)_{k =}nθ_k(n)expγ ₊O_n−1_+M(n) k gives us expL(_k∞)₊O_kn−13+3ε_=nθ(n) k exp γ ₊M_k(∞)₊O_n−13 Rearranging, nθ_k(n)₌expL(_k∞)₋M_k(∞)₋γ1₊Okn−13+3ε_=:2πyk(1+Okn−13+3ε_.

Now, by [3], (yk)k∈Z is a determinantal sine-kernel process, so we have almost

surely the estimateyk=O(1+ |k|), which proves Theorem1.1.

5. Weak convergence and renormalization of the eigenvectors. We are now ready to show that the eigenfunctionsf_k(n)ofunconverge in a suitable sense. We assume that for a given value ofε, the event E from Section 4holds, which happens almost surely. Recall from Theorem 2.4that we can choose representa-tivesf_k(n)for the eigenvectors of eachunin such a way that

h(n_k +1)12_f(n+1) k = n j=1 μ(n)_j λ(n)_{j −}λ(n_k +1)f (n) j + νn−1 1₋λ(n_k+1)en+1, where h(n_k+1)₌ n j=1 |μ(n)_{j |}2 |λ(n)_{j −}λ(n_k+1)_|2 + |νn−1| 2 |1₋λ(n_k +1)_|2 .

Forn₌1, we adopt the conventionf₁(1)_{= −}e1. We deduce, forn_≥ℓ, f_k(n+1), eℓ =h(n_k +1)− 1 2 n j=1 μ(n)_j λ(n)_{j −}λ(n_k +1) f_j(n), eℓ.

The invariance by conjugation of the Haar measures implies that each eigenvector

f_k(n+1), multiplied by an independent random phase of modulus 1, is a uniform vector on the complex sphere Sn+1. Hence, the scalar product f_k(n+1), eℓ con-verges to zero in probability. In order to get a limit which is different from zero, we need to consider a suitable normalization. We introduce the following eigen-vectors, forn_≥k:

g_k(n)_:=D_k(n)f_k(n),

whereD(n)_{k ∈}Cis the random variable

D_k(n)₌ n−1 s=k h(s_k+1)12λ (s) k −λ (s+1) k μ(s)_k .

We claim that for each renormalized eigenvectorg_k(n), the scalar productg(n)_k , eℓ converges.

THEOREM5.1. For eachk_≥1andℓ_≥1,there exists an increasing sequence

(Hj)j≥1 of events in A, with probability tending to 1, such that for all j ≥1, (1Hjg

(n)

k , eℓ)n≥k∨ℓ is a martingale with respect to the filtration(Bn)n≥k∨ℓ,and the conditional expectation of1Hj|g

(n)

k , eℓ|2,givenA,is almost surely bounded whennvaries.

REMARK5.2. We introduce the eventsHj in order to avoid to define condi-tional expectation of variables which are not necessarily integrable or nonnegative.

PROOF. From the equation above,

g_k(n+1), eℓ =Dk(n+1) h(n_k+1)−12 n j=1 μ(n)_j λ(n)_{j −}λ(n_k+1) f_j(n), eℓ =D_k(n)λ (n) k −λ (n+1) k μ(n)_k n j=1 μ(n)_j λ(n)_{j −}λ(n_k +1) f_j(n), eℓ =g(n)_k , eℓ +D_k(n) λ(n)_{k −}λ(n_k +1) μ(n)_k 1_≤j≤n j=k μ(n)_j λ(n)_{j −}λ(n_k +1) f_j(n), eℓ. (3)

Now, recall from Lemma 3.2 that the sequence of eigenvectors (f_k(n))n≥k is adapted to the filtration ofσ-algebras(B_n)n≥k. If we decompose

μ(n)_{j =}φ_j(n)μ(n)_j

then for n fixed, _{φ_j(n)_}1≤j≤n is a family of i.i.d. random phases uniformly dis-tributed on the unit circle, independent of the σ-algebra B_n. Indeed, B_n is the

σ-algebra generated by A and_{φ_j(m)_}1≤m≤n−1,1≤j≤m, and then, by Lemma3.1, it is the σ-algebra generated by u1, (νm)m≥1, (|μ(m)_j |)m≥1,1≤j≤m, and (φ(m)_j )1_≤m≤n−1,1≤j≤m. All these variables are independent of(φ_j(n))1≤j≤n, which are i.i.d. uniform on the unit circle, since the vectors((μ(m)_j )1_≤j≤m, νm)are inde-pendent, uniform on the unit complex sphere ofCm+1.

Now, sinceh(n_k+1)is real, we have

(n)_{k :=} D (n) k |D(n)_{k |}= n−1 s=k φ(s)_k −1 λ (s) k −λ (s+1) k |λ(s)_{k −}λ(s_k+1)_|

for 1_≤k_≤n, and we can write

g_k(n+1)₋g(n)_k , eℓ =(n)_k D_k(n)λ (n) k −λ (n+1) k |μ(n)_{k |} 1≤j≤n j=k φ(n)_j φ(n)_k |μ(n)_{j |} λ(n)_{j −}λ(n_k+1) f_j(n), eℓ.

We would like to compute the conditional expectation of the difference

Hjg_k(n+1)−g_k(n), eℓ, given the σ-algebra Bn, for suitable events Hj. We first verify the measurability of each quantity on the right; in particular:

1. _|D_k(n)_|, λ (n) k −λ (n+1) k |μ(n)_k | , |μ(n)_j | λ(n)_j −λ(n_k+1) areA-measurable. 2. (n)_k andf_j(n), eℓareBn-measurable.

3. _{φ_j(n)(φ(n)_k )−1_}1_≤j≤n j=k

are i.i.d. and independent ofB_n.

We also have g(n_k +1)₋g_k(n), eℓ ≤ D_k(n)|λ (n) k −λ (n+1) k | |μ(n)_{k |} 1≤j≤n j=k |μ(n)_{j |} |λ(n)_{j −}λ(n_k +1)_|

which gives an A_{-measurable bound for the scalar product gk}(n+1)_−g(n)_{k , eℓ,}

which is almost surely finite. Let us denote this bound byXn, and let us define the eventHj as the intersection of the events {Xn≤Rn,j} forn≥k∨ℓ,Rn,j being increasing in j and chosen in such a way that P_[Xn> Rn,j] ≤j−12−n, which implies that the probability ofHj tends to 1 whenj goes to infinity.

By Fubini’s theorem, measurability and the fact that the quantity inside the ex-pectation is uniformly bounded byRn,j,

E1Hj g(n_k +1)₋g_k(n), eℓ|Bn=1Hj (n) k D(n)_k λ (n) k −λ (n+1) k |μ(n)_{k |} × 1≤j≤n j=k E _φ(n) j φ_k(n) B_n |μ (n) j | λ(n)_{j −}λ(n_k+1) f_j(n), eℓ. However, by independence, E _φ(n) j φ_k(n) B_{n =}E _φ(n) j φ_k(n) =0, so(1Hjg (n) k , eℓ)n≥k∨ℓis a(Bn)n≥k∨ℓ-martingale.

To check its conditional boundedness inL2, givenA_{, it is sufficient to show}

E1Hj g_k(k∨ℓ), eℓ 2 |A₊ n≥k∨ℓ E1Hj g_k(n+1)₋g_k(n), eℓ 2 |A<_∞

almost surely. In fact, we will prove this inequality without the eventHj (here, the conditional expectations are well defined since the variables are nonnegative). The first term is smaller than or equal tog(k_k∨ℓ)2_{= |}D_k(k∨ℓ)_|2, which isA_-measurable

and almost surely finite. Hence, it is sufficient to bound the sum. Expanding it gives (4) Eg(n_k +1)₋g_k(n), eℓ 2 |A₌D_k(n)2|λ (n) k −λ (n+1) k | 2 |μ(n)_{k |}2 S, where S₌ 1≤i,j≤n i,j=k |μ(n)_{i |} λ(n)_{i −}λ(n_k +1) |μ(n)_{j |} λ(n)_{j −}λ(n_k+1)E φ(n)_i φ_j(n)f_i(n), eℓ f_j(n), eℓ|A.

In order to show this equality even if some of the variables involved inS are not integrable, we observe that the equality is true if we multiply all the variables by the indicator of theA_{-measurable event}

D(n)_k 2|λ (n) k −λ (n+1) k | 2 |μ(n)_{k |}2 1≤i,j≤n i,j=k |μ(n)_{i |} |λ(n)_{i −}λ(n_k+1)_| |μ(n)_{j |} |λ(n)_{j −}λ(n_k+1)_|≤R,

for anyR >0. Then we can letR_{→ ∞}. Now

Eφ_i(n)φ_j(n)f_i(n), eℓ f_j(n), eℓ|A=EEφ_i(n)φ_j(n)|Bnf_i(n), eℓ f_j(n), eℓ|A =δi,jE f_i(n), eℓ fj(n), eℓ|A ,

whereδi,j is the Kronecker delta. Thus (5) S₌ 1≤j≤n j=k |μ(n)_{j |}2 |λ(n)_{j −}λ(n_k+1)_|2 Ef_j(n), eℓ 2 |A_.

In order to effectively bound this sum, we need a posteriori information from the convergence of the eigenvalues in Section4. We define the eventFkto be

Fk:=

∃n0_≥1,_∀n_≥n0,θ_k(n)₊₁₋θ_k(n)_∧θ_k(n)₋θ_k(n)_−1≥n−1−ε.

LEMMA5.3. FkisA-measurable and holds with probability one.

PROOF. Fkdepends only on the eigenangles (hence eigenvalues) and is there-fore clearlyA_{-measurable. By Theorem1.1applied tok−1,k,k+1, we see that}

each of the associated eigenangles satisfies

nθ_j(n)₌2πyj+On− 1 4_. In particular, θ_k(n)₋θ_k(n)₋₁₌2π n−1(yk−yk−1)+O n−54_.

Since_{yj}j_∈Z is a sine-kernel point processyk−yk−11 almost surely and

sim-ilar fork₊1 andk.

Now we assume that Fk holds and estimate the quantities involved in

E_[|g(n_k +1)₋g_k(n), eℓ| 2

|A_{]. We will begin by estimating D}(n)

k . By Lemma 4.1, almost surely, θ_k(n+1)₋θ_k(n)₌θ_k(n+1)μ(n)_k 21₊On−13+ε_, hence (6) λ_k(n+1)₋λ(n)_{k =}θ_k(n+1)μ(n)_k 21₊O_n−13+ε_. Similarly, |νn−1|2 |1₋λ(n_k+1)_|2 =θ_k(n+1)−21₊On−12+ε_.

Now, as in the proof of convergence of the eigenangles, let us consider the setJ

of indicesj such thatθ_k(n+1)₋π < θ_j(n)_≤θ_k(n+1)₊π. From the eventFk, we see that forj_∈J_{\ {}k_}, andnlarge enough,

θ_j(n)₋θ_k(n+1)_≥θ_k(n)₋θ_j(n)₋θ_k(n)₋θ_k(n+1)

≥n−1−ε₋O_θk(n+1)_μ(n)_k 2₁₊O_n−13+ε

and then

(7) λ(n)_{j −}λ_k(n+1)n−1−ε,

ifε is taken small enough. We can get a stronger estimate for_|j ₋k_|sufficiently large. In fact, sinceE3holds, we have

(8) λ(n)_{j −}λ(n_k+1)_|j₋k_|n−53−ε_.

These two lower bounds let us estimate

1≤j≤n j=k |μ(n)_{j |}2 |λ(n)_{j −}λ(n_k +1)_|2 n−1+ε j∈J j=k λ(n)_{j −}λ(n_k+1)−2 n−1+εn2+εn23 _+n 10 3+ε_n− 2 3 n53+ε_,

where we split the sum into the interval with_|j₋k_{| ≤}n23 _{and its complement.}

Now we can estimateh(s_k+1). In fact,

h(s_k+1)|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 =1₊|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 1_≤j≤n j=k |μ(s)_{j |}2 |λ(s)_{j −}λ(s_k+1)_|2 + |νs−1| 2 |1₋λ(s_k+1)_|2 =1₊|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 Os53+ε_+θ(s+1) k ₋2 1₊Os−12+ε_.

Since almost surely,θ_k(s+1)₌O(1/s),s53+ε=O((θ(s+1)

k )−2s− 1 3+ε), and then h(s_k+1)|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 = 1₊(θ (s+1) k )−2|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 1₊O_s−13+ε_. Now, λ(s)_{k −}λ(s_k+1)2₌θ_k(s)₋θ_k(s+1)21₊O_|θ_k(s)₋θ_k(s+1)_| =θ_k(s)₋θ_k(s+1)21₊Os−1.

Using Lemma4.1, one obtains h(s_k+1)|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 =1₊(θ (s+1) k )−2|θ (s) k −θ (s+1) k | 2 |μ(s)_{k |}2 1₊Os−13+ε =1₊μ(s)_k 21₊O_s−13+ε_.

Applying the bound on_|μ(s)_{k |}2given byE2 and changing the value ofε gives

h(s_k+1)|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 =1₊μ(s)_k 2₊Os−43+ε_.

Thus from the expression

D(n)_k 2₌ n−1 s=k h(s_k+1)|λ (s) k −λ (s+1) k | 2 |μ(s)_{k |}2 , we deduce D_k(n)2₌ n−1 s=k 1₊μ(s)_k 2₊Os−43+ε =exp _n−1 s=k 1 s exp _n−1 s=k μ(s)_k 2₋1 s exp n−1 s=k Os−43+ε_. As before, exp n−1 s=k 1 s =k −1_nexpγ1 +On−1,

whereγ is the Euler–Mascheroni constant, exp n−1 s=k μ(s)_k 2₋1 s

is equal to exp(M_k(n−1)₋M_k(k))from Section4, and the last term converges to a limitN_∞with errorO_(n−13+ε_).

Thus, we have D_k(n)2₌k−1nexpγ ₊M_k(∞)₋M_k(k)₊N_∞1₊O_n−13+ε =:Dkn 1₊O_n−13+ε_, (9)

We are now ready to estimateE_[|g(n_k +1)₋g_k(n), eℓ| 2

|A_]. In fact we have, using (6), (9), the estimateθ_k(n+1)₌O_{(1/n)and the bound on|μ}(n)_{k |}2_{given byE2,}

Eg_k(n+1)₋g(n)_k , eℓ 2 |A₌D(n)_k 2|λ (n) k −λ (n+1) k | 2 |μ(n)_{k |}2 S nμ(n)_k 2θ_k(n+1)2S n−2+εS

and using (7) and (8),

S₌ 1≤j≤n j₌k |μ(n)_{j |}2 |λ(n)_{j −}λ(n_k +1)_|2 Ef_j(n), eℓ 2 |A n−1+ε 1≤j≤n j=k λ(n)_{j −}λ(n_k +1)−2Ef_j(n), eℓ 2 |A n−1+ε j∈J,0<|j−k|<n23 n2+εEf_j(n), eℓ 2 |A +n−1+ε j∈J,|j−k|≥n23 |j₋k_|−2n103+εE_f(n) j , eℓ 2 |A_.

Therefore, it is now sufficient to prove that for someε >0, and almost surely,

n≥k n−3+ε j∈J,0<|j−k|<n23 n2+εEf_j(n), eℓ 2 |A + j∈J,|j−k|≥n23 |j₋k_|−2n103+ε_Ef(n) j , eℓ 2 |A <_∞.

It is then sufficient to prove that the expectation of the left-hand side is finite. Since

J_{⊂ [}k₋n₋1, k₊n_], one deduces that it is enough to have

n≥k n−3+ε 0<|j−k|<n23 n2+εEf_j(n), eℓ 2 + j∈J,n23_≤|j₋k_|≤n₊1 |j₋k_|−2n103+ε_Ef(n) j , eℓ 2 <_∞.

Now, sincef_j(n)is, up to a phase of modulus 1, uniform on the sphere ofCn, one has

Ef_j(n), eℓ 2

=1/n,

and then one needs only to check

n≥k n−3+ε 0<|j−k|<n23 n1+ε₊ j∈J,n23_≤|j−k|≤n+1 |j₋k_|−2n73+ε <_∞, which is easy.

The result we have proven has the following consequence.

COROLLARY 5.4. Almost surely,for allk_∈Zandℓ_≥1,the scalar product g(n)_k , e_ℓconverges to a limitgk,ℓ whenngoes to infinity.

PROOF. We can fixkandℓ. Then, forj_≥1,R >0 we consider the sequence of variables

1Hj,Yj≤R

g(n)_k , eℓ _n≥k∨ℓ,

whereYj denotes the supremum innof the conditional expectation of1Hj|g (n) k , eℓ|2 givenA. Since Yj isA-measurable, this sequence is a martingale in the fil-tration(B_n)n≥k∨ℓ, and this martingale is bounded inL2by construction. Hence, it a.s. converges, andg_k(n), eℓconverges with probability at least 1−P[Hj, Yj ≤R]. Since we have shown that Yj is a.s. finite, we letR→ ∞ and deduce a conver-gence with probability at least 1₋P_[Hj], and then we letj→ ∞.

For eachk_∈Z, the infinite sequencegk:=(gk,ℓ)ℓ≥1∈C∞ can be considered

as the weak limit of the eigenvectorg_k(n) ofun, when ngoes to infinity. We will study the behavior of a suitable renormalization ofgk in the next section.

6. The law of the coefficients of the limiting eigenvector. In the previous section, we have proven the almost sure convergence of each coordinate of the eigenvectors of un, after normalization by a factorDk(n). Using the estimate (9), we see that almost surely,_|D_k(n)_|/√ntends to a nonzero random limit whenngoes to infinity. It is then more elegant to formulate the result of convergence by taking eigenvectors of norm exactly√n. Moreover, such a normalization provides the dis-tribution of the limiting coordinates of the eigenvectors, as stated in Theorem1.2 proven just below.

PROOF OF THEOREM1.2. For fixedk_∈Z, let us first show the existence of the sequence (tk,ℓ)ℓ≥1. By symmetry, one can assume k≥1: in this case, for all n_≥k, there existsτ_k(n) of modulus 1 such thatf_k(n)₌τ_k(n)v_k(n). Hence,

D_k(n)_k(n)τ_k(n)v(n)_k , eℓ

almost surely converges when n goes to infinity. Now, by the estimate (9), |D(n)_{k |}/√n converges almost surely to a strictly positive constant. One deduces the existence of(tk,ℓ)ℓ≥1, by taking

ψ_k(n)_:=(n)_k τ_k(n).

Let us now check the uniqueness, by supposing that two sequences(tk,ℓ)ℓ≥1 and (t_k,ℓ′ )ℓ≥1 can be constructed from the same virtual rotation(un)n≥1. In this case,

there exist, for alln_≥1, two unit eigenvectorsw_k(n)andw_k(n)′corresponding to the same eigenvalue, and such that for allℓ_≥1,

(10) √nw_k(n), eℓ −→ n→∞tk,ℓ and √ nw_k(n)′, eℓ −→ n→∞t ′ k,ℓ.

Since the eigenvalues are almost surely simple, for alln_≥1, there existsχ_k(n)_∈U such thatw_k(n)′₌χ_k(n)w(n)_k , which implies

(11) χ_k(n)√nw(n)_k , eℓ −→ n→∞t

′

k,ℓ.

By comparing (10) and (11), one deduces thatt_k,ℓ′ ₌χktk,ℓ, whereχk∈Udenotes the limit of any converging subsequence of(χ_k(n))n≥1.

Moreover, let us choose the random vectors(w(n)_k )k∈Z,n≥1and the random

vari-ables (tk,ℓ)k∈Z,ℓ≥1 as measurable functions of (un)n≥1, in such a way that (10)

is satisfied almost surely. Let(ψk)k∈Z be i.i.d. random variables, independent of

(un)n≥1. For all n≥1, the invariance by conjugation of the Haar measure on U (n)implies that the family of eigenvectors(ψkw(n)k )1≤k≤nofun forms a Haar-distributed unitary matrix inU (n). One deduces that ifLis a finite set of strictly positive integers, and ifKis a finite set of integers, then

√

nψkwk(n), eℓ _{k∈K,ℓ∈L}

converges in law to a family of i.i.d. standard complex Gaussian variables. Since this family of variables also converges almost surely, one deduces that the limiting variables(ψktk,ℓ)k∈K,ℓ∈Lare i.i.d. standard complex and Gaussian. Since the finite setsKandLcan be taken arbitrarily, we are done.

COROLLARY 6.1. The limiting coordinatesgk,ℓ introduced in Corollary5.4 are almost surely different from zero.

PROOF. We know that for a suitable normalization oftk,ℓ,

gk,ℓ= lim n→∞ D_k(n)(n)_k f_k(n), eℓ, tk,ℓ= lim n→∞ √ n(n)_k f_k(n), eℓ, and by (9), ! Dk=_nlim →∞ D_k(n)/√n.

Combining these limits gives

(12) gk,ℓ=

!

Dktk,ℓ,

which is almost surely nonzero, sincetk,ℓis a standard complex Gaussian variable up to multiplication by an independent uniform random variable on the unit circle. The eigenspaces ofun, generated by the vectorsf_k(n), can also be considered as elements of the projective spacePn−1(C). Moreover, one can define the infinite-dimensional projective spaceP∞(C), as the space of nonzero infinite sequences of complex numbers, quotiented by scalar multiplication. The convergence of renor-malized eigenvectors proven above can be viewed as a convergence of the corre-sponding points on the projective spaces.

There exists a uniform measure on all these projective spaces, obtained by tak-ing the equivalence class of a sequence of i.i.d. standard complex Gaussian vari-ables. Forn_≥1 finite, the uniform measure onPn(C)can also be obtained from a uniform point on the sphere inCn+1. Form < n_∈N_{∪ {∞}}, there exists a nat-ural projection n,m from Pn(C) to Pm(C), obtained by taking only the m+1 first coordinates of the sequences, and this projection is well defined when these coordinates are not all vanishing: in particular, almost surely under the uniform measure onPn(C). Note that the image of this measure byn,m is the uniform measure onPm(C). Moreover, we can define the notion of weak convergence on the projective spaces as follows. Let(xn)n≥1 be a sequence such thatxn∈Pn(C) for alln_≥1 and letx_∞∈P∞(C). We say that(xn)n≥1weakly converges tox∞if

and only if the following holds: for allm_≥1 such that them₊1 first coordinates ofx_∞are not all vanishing, the projectionn,m(xn)∈Pm(C)is well defined forn large enough and tends to_∞,m(x∞)whenngoes to infinity. From Theorem1.2,

we can easily deduce the following.

THEOREM6.2. Let(un)n≥1be a virtual rotation,following the Haar measure.

kth smallest nonnegative eigenangle of un for k≥1, and the (1−k)th largest strictly negative eigenangle ofun fork≤0 (this eigenspace is almost surely one-dimensional). Then there almost surely exists some random points (x_k(∞))k∈Z in P∞(C) such that for all k_∈Z, x_k(n) weakly converges to x_k(∞) when n goes to infinity. The points(x_k(∞))k∈Z are represented by the sequences(tk,ℓ)ℓ≥1, k∈Z

given above:they are independent and uniform onP∞(C).

APPENDIX A: A PRIORI ESTIMATES FOR UNITARY MATRICES Let us fixε >0. The goal of this section is the proof that the eventE_:=E0_∩ E1_∩E2_∩E3holds almost surely under the Haar measure on the space of virtual isometries, for E0₌θ₀(1)₌0_{∩ {∀}n_≥1, νn=0} ∩∀n≥1,1≤k≤n, μ(n)k =0 , E1_=∃n0_≥1,_∀n_≥n0,_|νn| ≤n− 1 2+ε_, E2_=∃n0_≥1,_∀n_≥n0,1_≤k_≤n,μ(n)_{k ≤}n−12+ε_, E3_=∃n0_≥1,_∀n_≥n0, k_≥1, n−53−ε_≤θ(n) k+1−θ (n) k ≤n− 1+ε_.

REMARK A.1. In [3], an analogous event was defined: in the present paper, we choose the exponents more carefully to sharpen our results.

We begin by showing that for any fixed basis ofCn, the coefficients of a uniform random vector on the unit sphere are almost surelyO(n−12+ε)for anyε >0.

LEMMA A.2. Supposev1, . . . , vn∈Cnis an orthonormal basis andx∈Cn, x ₌1 is chosen uniformly from the unit sphere. Then if we writex ₌x1v1₊ · · · +xnvn,we have bound

P_|xj|2> δ=Oexp(−δn/2), for allδ >0andj₌1, . . . , n.

REMARKA.3. We will prove this statement for deterministic vectorsv1, . . . , vn∈Cn. However, by conditioning, one deduces that the result remains true if the vectorsv1, . . . , vnare random, as soon as they are independent ofx.

PROOF. The random variable_|xj|2 is Beta distributed with parameters 1 and n₋1, and then the probability that it is larger thanδis

(n₋1)

1

δ∧1(