CIRCULAR JACOBI ENSEMBLES AND DEFORMED VERBLUNSKY COEFFICIENTS

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1 CIRCULAR JACOBI ENSEMBLES AND DEFORMED VERBLUNSKY COEFFICIENTS P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT Abstract. Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: c (n) δ,β Y <l n Y n e iθ e iθ l β ( e iθ j ) δ ( e iθ j ) δ j= with Re δ > /2. If e is a cyclic vector for a unitary n n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsy coefficients (α 0,..., α n ). We introduce here a deformation (γ 0,..., γ n ) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ 0) r(γ n ) of elementary reflections parameterized by these coefficients. If γ 0,..., γ n are independent random variables with some remarable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsy coefficients also allow to prove that, in the regime δ = δ(n) with δ(n)/n βd/2, the spectral measure and the empirical spectral distribution wealy converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution. Contents. Introduction 2.. The circular Jacobi ensemble Orthogonal polynomials on the unit circle Organization of the paper Deformed Verblunsy coefficients and reflections Analytical properties Geometric interpretation 2 3. Deformed Verblunsy coefficients and independence 5 4. Matrix models for the circular Jacobi ensemble 9 5. Limiting spectral measure and large deviations Wea convergence of measures Mathematics Subject Classification. 5A52, 60F05, 60F5, 60F0. Key words and phrases. Random Matrices, circular Jacobi ensembles, Characteristic polynomial, Orthogonal polynomials on the unit circle, Verblunsy coefficients, spectral measure, Limit theorems.

2 2 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT 5.2. Large deviations for the ESD 27 References 32. Introduction.. The circular Jacobi ensemble. The theory of random unitary matrices was developed using the existence of a natural probability uniform measure on compact Lie groups, namely the Haar measure. The statistical properties of the eigenvalues as well as the characteristic polynomial of these random matrices have played a crucial role both in physics (see [27] for an historical account) and in analytic number theory to model L-functions (see [20] and [2] where Keating and Snaith predict moments of L-functions on the critical line using nowledge on the moments of the characteristic polynomial of random unitary matrices). The circular unitary ensemble (CUE) is U(n), the unitary group over C n, equipped with its Haar probability measure µ U(n). Weyl s integration formula allows one to average any (bounded measurable) function on U(n) which is conjugation-invariant fdµ U(n) = (e iθ,..., e iθn ) 2 f(diag(e iθ,..., e iθn )) dθ n! 2π... dθ n 2π, (.) where (e iθ,..., e iθn ) = j< n (eiθ e iθ j ) denotes the Vandermonde determinant. The circular orthogonal ensemble (COE) is the subset of U(n) consisting of symmetric matrices, i.e. U(n)/O(n) = {V V T ; V U(n)} equipped with the measure obtained by pushing forward µ U(n) by the mapping V V V T. The integration formula is similar to (.) but with (e iθ,..., e iθn ) 2 replaced by (e iθ,..., e iθn ) and with the normalizing constant changed accordingly. For the circular symplectic ensemble (CSE), which will not be recalled here, the integration formula uses (e iθ,..., e iθn ) 4. Dyson observed that the induced eigenvalue distributions correspond to the Gibbs distribution for the classical Coulomb gas on the circle at three different temperatures. More generally, n identically charged particles confined to move on the unit circle, each interacting with the others through the usual Coulomb potential log z i z j, give rise to the Gibbs measure at temperature /β (see the discussion and references in [24] and in [3] chap. 2): where c (n) 0,β E β n(f) = c (n) 0,β f(e iθ,..., e iθn ) (e iθ,..., e iθn ) β dθ... dθ n, (.2) is a normalizing constant chosen so that h (n) 0,β (θ,..., θ n ) = c (n) 0,β (eiθ,..., e iθn ) β (.3)

3 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 3 is a probability density on (0, 2π) n and where f is any symmetric function. The unitary, orthogonal and symplectic circular ensembles correspond to matrix models for the Coulomb gas at β =, 2, 4 respectively, but are there matrix models for general β > 0 for Dyson s circular eigenvalue statistics? Killip and Nenciu [24] provided matrix models for Dyson s circular ensemble, using the theory of orthogonal polynomials on the unit circle. In particular, they obtained a sparse matrix model which is five-diagonal, called CMV (after the names of the authors Cantero, Moral, Velásquez [9]). In this framewor, there is not a natural underlying measure such as the Haar measure; the matrix ensemble is characterized by the laws of its elements. There is an analogue of Dyson s circular ensembles on the real line: the probability density function of the eigenvalues (x,..., x n ) for such ensembles with temperature /β is proportional to (x,..., x n ) β n j= e x2 j /2 (.4) For β =, 2 or 4, this corresponds to the classical Gaussian ensembles. Dimitriu and Edelman [] gave a simple tridiagonal matrix model for (.4). Killip and Nenciu [24], gave an analogue matrix model for the Jacobi measure on [ 2, 2] n, which is up to a normalizing constant, (x,..., x n ) β n j= (2 x j ) a (2 + x j ) b dx... dx n, (.5) where a, b > 0, relying on the theory of orthogonal polynomials on the unit circle and its lins with orthogonal polynomials on the segment. When a and b are strictly positive integers, the Jacobi measure (.5) can be interpreted as the potential (x,..., x n+a+b ) β on [ 2, 2] n+a+b conditioned to have a elements located at 2 and b elements located at 2. Consequently, the Jacobi measure on the unit circle should be a two parameters extension of (.3), corresponding to conditioning to have specific given eigenvalues. Such an analogue was defined as the circular Jacobi ensemble in [3] and [5]. If δ R, we recover the cjue as in Witte and Forrester [37]. Definition.. Throughout this paper, we note h (n) δ,β the probability density function on (0, 2π) given by: h (n) δ,β (θ,..., θ n ) = c (n) δ,β (eiθ,..., e iθn ) β with δ C, Re(δ) > 2. n ( e iθ j ) δ ( e iθ j ) δ (.6) j= If δ β 2 N, this measure coincides with (.3) conditioned to have eigenvalues at. For β = 2, such measures were first considered by Hua [9] and Picrell [30], [3]. This case was also widely studied in [29] and [6] for its

4 4 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT connections with the theory of representations and in [8] for its analogies with the Ewens measures on permutation groups. One of our goals in this paper is to provide matrix models for the circular Jacobi ensemble, i.e. a distribution on U(n) such that the arguments of the eigenvalues (e iθ,..., e iθn ) are distributed as in (.6). One can guess that additional problems may appear because the distribution of the eigenvalues is not rotation invariant anymore. Nevertheless, some statistical information for the circular Jacobi ensemble can be obtained from Dyson s circular ensemble by a sampling (or a change of probability measure) with the help of the determinant. More precisely, let us first define the notion of sampling. Definition.2. Let (X, F, µ) be a probability space, and h : X R + a measurable and integrable function with E µ (h) > 0. Then a measure µ is said to be the h-sampling of µ if for all bounded measurable functions f E µ (f) = E µ(fh) E µ (h). If we consider a matrix model for h (n) 0,β, we can define a matrix model for h (n) δ,β by the means of a sampling, noticing that when the charges are actually the eigenvalues of a matrix U, then (.3) differs from (.6) by a factor which is a function of det(id U). We define det δ for a unitary matrix U as det δ (U) = det(id U) δ det(id U) δ, and we will use this det δ -sampling. Actually we loo for an effective construction of a random matrix, for instance starting from a reduced number of independent random variables with nown distributions. Notice that in the particular case β = 2, the density h 0,2 corresponds to eigenvalues of a matrix under the Haar measure on U(n) and the det δ -sampling of this measure is the Hua-Picrell measure studied in our previous paper [8]..2. Orthogonal polynomials on the unit circle. We now wish to outline the main ingredients which are needed from the theory of orthogonal polynomials on the unit circle to construct matrix models for the general Dyson s circular ensemble. The reader can refer to [35] and [36] for more results and references; in particular, all the results about orthogonal polynomials on the unit circle (named hereafter OPUC) can be found in these volumes. Let us explain why OPUC play a prominent role in these constructions. In all this paper, D denotes the open unit dis {z C : z < } and T the unit circle {z C : z = }. Let (H, u, e) be a triple where H is a Hilbert space, u a unitary operator and e a cyclic unit vector, i.e. {u j e} j= is total in H. We say that two triples (H, u, e) and (K, v, e ) are equivalent if and only if there exits an isometry : H K such that v = u and for Re(δ) > /2, due to an integrability constraint

5 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 5 e = e. The spectral theorem says that for each equivalence class, there exists a unique probability measure µ on T such that e, u e H = z dµ(z), = 0, ±,.... T Conversely, such a probability measure µ gives rise to a triple consisting of the Hilbert space L 2 (T, µ), the operator of multiplication by z, i.e. h (z zh(z)) and the vector, i.e. the constant function. When the space H is fixed, the probability measure µ associated with the triple (H, u, e) is called the spectral measure of the pair (u, e). Let us consider the finite n-dimensional case. Assume that u is unitary and e is cyclic. It is classical that u has n different eigenvalues (e iθ j, j =,..., n) with θ j [0, 2π). In any orthonormal basis whose first vector is e, say (e = e, e 2,..., e n ), u is represented by a matrix U and there is a unitary matrix Π diagonalizing U. It is then straightforward that the spectral measure is n µ = π j δ e iθ j (.7) j= where the weights are defined as π j = e, Πe j 2. Note that π j > 0 because a cyclic vector cannot be orthogonal to any eigenvector (and we also have n j= π j = because Π is unitary). The eigenvalues (e iθ j, j =,... n) and the vector (π,..., π n ) can then be used as coordinates for the probability measure µ. Keeping in mind our purpose, we see that the construction of a matrix model from a vector (e iθ j, j =,..., n) may be achieved in two steps: first give a vector of weights (π,..., π n ), then find a matricial representative of the equivalence class with a rather simple form. The ey tool for the second tas is the sequence of orthogonal polynomials associated with the measure µ. In L 2 (T, µ) equipped with the natural basis {, z, z 2,..., z n }, the Gram-Schmidt procedure provides the family of monic orthogonal polynomials Φ 0,..., Φ n. We can still define Φ n as the unique monic polynomial of degree n with Φ n L 2 (T,µ)= 0, namely Φ n (z) = n (z e iθ j ). (.8) j= The Φ s ( = 0,..., n) obey the Szegö recursion relation: where Φ j+ (z) = zφ j (z) ᾱ j Φ j(z) (.9) Φ j(z) = z j Φ j ( z ). (.0) The coefficients α j s (0 j n ) are called Verblunsy coefficients and satisfy the condition α 0,, α n 2 D and α n T.

6 6 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT When the measure µ has infinite support, one can define the family of orthogonal polynomials (Φ n ) n 0 associated with µ for all n. Then there are infinitely many Verblunsy coefficients (α n ) which all lie in D. Verblunsy s Theorem (see for example [35], [36]) states that there is a bijection between probability measures on the unit circle and sequences of Verblunsy coefficients. The matrix of the multiplication by z in L 2 (T, µ), in the basis of orthonormal polynomials, has received much attention. This unitary matrix, noted G(α 0,..., α n ), called GGT by B. Simon (see Chapter 4. in [35] for more details and for an historical account), is in the Hessenberg form: all entries below the subdiagonal are zero, whereas the entries above the subdiagonal are nonzero and the subdiagonal is nonnegative (see formulae (4..5) and (4..6) in [35] for an explicit expression for the entries in terms of the Verblunsy coefficients, or formula (.) in Lemma.3 below). For H a n n complex matrix, the subscript H ij stands for e i, H(e j ), where x, y = n = x y. Killip and Nenciu state that any unitary matrix in the Hessenberg form with nonnegative subdiagonal is the matrix of multiplication by z in L 2 (T, µ) for some measure µ. More precisely, from Killip-Nenciu [24], Lemma 3.2, we have Lemma.3. Let µ be a probability measure on T supported at n points. Then the matrix H of f(z) zf(z) in the basis of orthonormal polynomials of L 2 (T, µ), is in the Hessenberg form. More precisely α i α j j p=i ρ p if i < j + H i+,j+ = ρ j if i = j +, (.) 0 if i > j + with ρ j = α j 2 and α =, the α s being the Verblunsy coefficients associated to µ. Conversely, if α 0,..., α n are given in D n T, and if we define the matrix H by (.), then the spectral measure of the pair (H, e ) is the measure µ whose Verblunsy coefficients are precisely α 0,..., α n. Besides, there is a very useful decomposition of these matrices into product of bloc matrices, called the AGR decomposition by Simon ([33]), after the paper [3]. For 0 n 2, let ( ) Θ () α ρ (α) = Id Id ρ α n 2. and set Θ (n ) (α n ) = Id n (α n ), with α n =. Then the AGR decomposition states that ([33] Theorem 0.) G(α 0,..., α n ) = Θ (0) (α 0 )Θ () (α )... Θ (n ) (α n ). Now we state a crucial result of Killip and Nenciu which enabled them to obtain a matrix model in the Hessenberg form for Dyson s circular ensemble.

7 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 7 The challenge consists in randomizing (α 0,..., α n ) in D n T in such a way that the angles (θ,..., θ n ) of the spectral measure n µ = π j δ e iθ j j= have the density h (δ) 0,β (see (.6)). To mae the statement precise, let us introduce three definitions (for the properties of the Beta and Dirichlet distributions, see [32]). Definition.4. For a, a 2 > 0, let Beta(a, a 2 ) be the distribution on [0, ] with density Γ(a + a 2 ) Γ(a )Γ(a 2 ) xa ( x) a 2 Its generalization is the following. Definition.5. For n 2 and a,..., a n > 0, let Dir(a,..., a n ) be the distribution on the simplex {(x,..., x n ) [0, ] n : n i= x i = } with density Γ(a a n ) n x a Γ(a ) Γ(a n ). If a =... = a n = a, it is called the Dirichlet distribution of order n 2 with parameter a > 0 and denoted by Dir n (a). (For a =, this is the uniform distribution). Definition.6. For s > let ν s be the probability measure on D with density s 2π ( z 2 ) (s 3)/2. It is the law of re iψ where r and ψ are independent, ψ is uniformly distributed on (0, 2π) and r 2 has the Beta(, (s )/2) distribution. We adopt the convention that ν is the uniform distribution on the unit circle. We denote by η (n) 0,β the distribution on Dn T given by = η (n) 0,β = n =0 ν β(n )+. (.2) Proposition.7 (Killip-Nenciu [24], Proposition 4.2). The following formulae express the same measure on the manifold of probability distributions on T supported at n points: 2 n (e iθ,..., e iθn ) β n! in the (θ, π) coordinates and n j= π β/2 j dθ... dθ n dπ... dπ n n 2 ( α 2 ) (β/2)(n ) d 2 α 0... d 2 dφ α n 2 2π =0 (.3)

8 8 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT in terms of the Verblunsy coefficients. Proposition.7 may be restated as follows: to pic at random a measure µ such that (α 0,..., α n ) is η (n) 0,β distributed is equivalent to pic the support (θ,..., θ n ) according to h (n) 0,β (see (.3)) and to pic the weights (π,..., π n ) independently according to Dir n (β/2). As a consequence, if one taes independent coefficients (α 0,..., α n ) such that α is ν β(n )+ distributed for 0 n, then the GGT matrix G(α 0,..., α n ) will be a matrix model for Dyson s circular ensemble with temperature /β (see also Proposition 2. in [3]). Actually in [24], Killip and Nenciu provide a matrix model which is much sparser (five-diagonal) as shall be explained in Section 4. Let us now define the laws on U(n) which we will consider in the sequel. Definition.8. We denote by CJ (n) 0,β the probability distribution supported by the set of n n GGT matrices of the form (.), corresponding to the law of G(α 0,..., α n ) defined above. We denote by CJ (n) δ,β distribution on U(n) which is the det δ -sampling of CJ (n) 0,β. the probability The standard GGT approach is not sufficient to produce matrix models for the circular Jacobi ensemble because, as we shall see in Section 3, under the measure CJ (n) δ,β, the Verblunsy coefficients are not independent anymore. To overcome this difficulty, we associate to a measure on the unit circle, or equivalently to its Verblunsy coefficients, a new sequence of coefficients (γ ) 0 n, which we shall call deformed Verblunsy coefficients. There is a simple bijection between the original sequence (α ) 0 n and the new one (γ ) 0 n. These coefficients satisfy among other nice properties that α = γ, and that they are independent under CJ (n) δ,β (and for δ = 0 the α s and the γ s have the same distribution). They have a geometric interpretation in terms of reflections: this leads to a decomposition of the GGT matrix G(α 0,... α n ) as a product of independent elementary reflections (constructed from the γ s). The explicit expression of the densities allows an asymptotic study (as n ) of the γ s, and consequently of the spectral measure, and finally of the empirical spectral distribution..3. Organization of the paper. In Section 2, after recalling basic facts about the reflections introduced in [8], we define the deformed Verblunsy coefficients (γ ) 0 n and give some of its basic properties. In particular we prove that the GGT matrix G(α 0,... α n ) can be decomposed into a product of elementary complex reflections (Theorem 2.8). In Section 3, we derive the law of the γ s under CJ (n) δ,β, (Thorem 3.2); in particular we show that they are independent and that the actual Verblunsy coefficients are dependent if δ 0. We then prove an analogue of the above Proposition.7 on the (θ, π) coordinates of µ (Theorem 3.3).

9 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 9 In Section 4, we propose our matrix model (Theorem 4.). It is a modification of the AGR factorization, where we transform the Θ s so that they become reflections : ( Ξ () α e (α) = Id iφ ) ρ ρ e iφ Id α n 2, with e iφ Of course the CMV representation [9], which is fivediagonal, is also available, but this time the α s are not independent. Using the following elementary fact proven in Section 2, = α α. n Φ n () = det(id U) = ( γ ), we are able to generalize our previous results in [7] and [8] about the decomposition of the characteristic polynomial evaluated at as a product of independent complex variables (Proposition 4.3). In Section 5, we study asymptotic properties of our model as n, when δ = βnd/2, with Re d 0. We first prove that the Verblunsy coefficients have deterministic limits in probability. This entails that the spectral measure converges wealy in probability to the same deterministic measure (denoted by µ d ) which is supported by an arc of the unit circle (Theorem 5.). Besides, we consider the empirical spectral distribution (ESD), where the Dirac masses have the same weight /n. Bounding the distances between both random measures, we prove that the ESD has the same limit (Theorem 5.4). Moreover, starting from the explicit joint distribution (.6), we prove also that the ESD satisfies the large deviation principle at scale (β/2)n 2 whose rate function reaches its minimum at µ d (Theorem 5.5). =0 2. Deformed Verblunsy coefficients and reflections In this section, we introduce the deformed Verblunsy coefficients and we establish some of their relevant properties, in particular a geometric interpretation in terms of reflections. One remarable property of the Verblunsy coefficients, as it appears in Proposition.7, is that they are independent under CJ (n) 0,β. As we shall see in Section 3, this does not hold anymore under CJ(n) δ,β. This motivated us to introduce a new set of coefficients, (γ 0,..., γ n 2, γ n ), called deformed Verblunsy coefficients, which are uniquely associated with a set of Verblunsy coefficients. In particular, γ D for 0 n 2, γ n T and the map (γ 0,..., γ n ) (α 0,..., α n ) is a bijection. Moreover, the characteristic polynomial at can be expressed simply in terms of (γ 0,..., γ n ). 2.. Analytical properties. Let µ be a probability measure on the unit circle supported at n points. Keeping the notations of the introduction, we let (Φ (z)) 0 n denote the monic orthogonal polynomials associated with

10 0 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT µ and (α ) 0 n its corresponding set of Verblunsy coefficients through Szegö s recursion formula (.9). The functions b (z) = Φ (z) Φ, n (2.) (z) are nown as the inverse Schur iterates ([36] p.476, after Khrushchev [23] p.273). They are analytic in a neighborhood of D and meromorphic in C. Each b is a finite Blashe product n ( ) z zj b (z) = z j z j= where z,..., z are the zeros of Φ. Let us now explain the term inverse Schur iterate. The Schur function is a fundamental object in the study of the orthogonal polynomials on the unit circle. Let us briefly recall its definition (see [35] or [34] for more details and proofs): if µ is a probability measure on the unit circle (supported at finitely many points or not), its Schur function f : D D is defined as: f(z) = z F (z) F (z) + where F (z) = e iθ + z e iθ z dµ(eiθ ). (2.2) It is a bijection between the set of probability measures on the unit circle and analytic functions mapping D to D. The Schur algorithm (which is described in [35] or [34] p.438) allows to parametrize the Schur function f by a sequence of so-called Schur parameters, which are actually the Verblunsy coefficients associated to µ (Geronimus theorem). In particular, there are finitely many Verblunsy coefficients (or equivalently the measure µ is supported at n points) if and only if f is a finite Blasche product. The name inverse Schur iterate ([22]) for b comes from the result (.2) of the latter paper where b is identified as the Schur function corresponding to the reversed sequence ( ᾱ,..., ᾱ 0, ) (see also [36] Prop ). Let us define our sequence of functions, which shall lead us the deformed coefficients. Definition 2.. If µ is supported at n points and with the notation above, define γ (z) for 0 n, as: γ (z) = z Φ +(z) Φ (z). (2.3) From the Szegö s recursion formula (.9) and notation (2.), this is equivalent to γ (z) = ᾱ b (z), (2.4) so that γ is meromorphic, with poles in D and zeros lying outside D.

11 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS The next proposition shows how the functions γ (z) can be defined recursively with the help of the coefficients α. As a consequence, we shall see that the γ (z) are very closely related to a fundamental object in the theory of random matrices: the characteristic polynomial. Proposition 2.2. For any z C, γ 0 (z) = ᾱ 0 and the following decomposition for Φ (z) holds: Φ (z) = (z γ j (z)), =,..., n. (2.5) j=0 The γ (z) s may be also defined by means of the α s through the recursion : γ (z) = ᾱ j=0 z γ j (z) z γ j (z), (2.6) γ (z) = γ ( z ). (2.7) Proof. The first claim is an immediate consequence of (2.3). Φ (z) = j=0 (z γ j(z)), we obtain and hence (we use (2.4)) Φ (z) = ( z γ j (z)), j=0 γ (z) = ᾱ j=0 z γ j (z) z γ j (z). Now, using Note that when z =, γ (z) = α. Combined with the above proposition, this leads us to introduce the following set of coefficients. Definition 2.3. Define the coefficients (γ ) 0 n by γ := γ (), = 0,..., n. (2.8) We shall refer to the γ s as the deformed Verblunsy coefficients. Proposition 2.4. The following properties hold for the deformed Verblunsy coefficients: a) For all 0 n, γ = α, and in particular γ n T; b) γ 0 = ᾱ 0 and γ = ᾱ e iϕ, e iϕ = j=0 γ j γ j, ( =,..., n ). (2.9) The last term is special. Since α n =, we set α n = e iψ n, so that γ n = e i( ψ n +ϕ n 2 ) := e iθ n. (2.0)

12 2 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT c) Let µ be the spectral measure associated to (U, e ), U U(n). Then Φ n (z) is the characteristic polynomial of U, in particular, n Φ n () = det(id U) = ( γ ). (2.) Proof. All the results are direct consequences of the definition 2.3 and the formulae in Proposition 2.2 evaluated at. Remar. In [26], Killip and Stoiciu have already considered variables which are the complex conjugate of our deformed Verblunsy coefficients as auxiliary variables in the study of the Prüfer phase (Lemma 2. in [26]). Nevertheless, the way we define them as well as the use we mae of them are different. Remar. The formula (2.9) shows that the γ s can be obtained from the α s recursively. Hence starting from a spectral measure associated to a unitary matrix, one can associate with it the Verblunsy coefficients and then the deformed Verblunsy coefficients. Conversely, one can translate any property of the deformed ones into properties for the spectral measure associated with it by inverting the transformations (2.9). Remar. The distribution of the characteristic polynomial of random unitary matrices evaluated at, through its Mellin-Fourier transform, plays a ey role in the theory of random matrices, especially through its lins with analytic number theory (see [28] for an account). In [7] it is proven that it can be decomposed in law into a product of independent random variables when woring on the unitary and orthogonal groups endowed with the Haar measure; since we will prove in Section 3 that the γ s are independent under CJ (n) δ,β, then we can conclude that this latter result holds for any circular Jacobi ensemble Geometric interpretation. We give a connection between the coefficients (γ ) 0 n and reflections defined just below. This allows us to obtain a new decomposition of the GGT matrix associated with a measure µ supported at n points on the unit circle as a product of n elementary reflections. Many distinct definitions of reflections on the unitary group exist, the most well-nown may be the Householder reflections. The transformations which will be relevant to us are the following ones. Definition 2.5. An element r in U(n) will be referred to as a reflection if r Id has ran 0 or. If v C n, we denote by v the linear form w v, w. The reflections can also be described in the following way. If e and m e are unit vectors of C n, there is a unique reflection r such that r(e) = m, and r = Id (m e) (m e). (2.2) m, e =0

13 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 3 Let F := span{e, m} be the 2-dimensional vector space which is spanned by the vectors e and m. It is clear that the reflection given by formula (2.2) leaves F invariant. Now set γ = e, m, ρ = γ 2, e iϕ = γ γ, (2.3) and let g F be the unit vector orthogonal to e obtained by the Gram- Schmidt procedure. Then in the basis (e, g) of F, the matrix of the restriction of r is ( ) γ ρe iϕ Ξ(γ) := ρ γe iϕ. (2.4) Conversely, for γ D, such a matrix represents the unique reflection in C 2 provided with its canonical basis, mapping e onto γe + γ 2 e 2. The eigenvalues of r are and e iϕ. Let u be a unitary operator in C n and e a unit cyclic vector for u. We define n reflections r,..., r n recursively as follows. Let (ε,..., ε n ) be the orthonormal basis obtained from the Gram-Schmidt procedure applied to (e, ue,..., u n e). Let r be the reflection, mapping e = ε onto u e = uε. More generally, for 2 let r be the reflection mapping ε onto r r 2... r uε. We will identify these reflections and establish the decomposition of u. Following the basics recalled about GGT matrices in the introduction, we note that the matrix of u in the basis (ε,..., ε n ) is the GGT matrix associated to the measure µ, i.e. the matrix G(α 0,, α n 2, α n ), where (α 0,, α n 2, α n ) are the Verblunsy coefficients associated with the measure µ. We will use formula (4..6) of [35] or formula (.) of (our) Lemma.3 for the identification of scalar products. Proposition 2.6. () For every n, the reflection r leaves invariant the n 2-dimensional space Span{ε,..., ε, ε +2,..., ε n }. The reflection r n leaves invariant Span{ε,..., ε n }. (2) The following decomposition holds : u = r r n. (2.5) Proof. () In view of Section 2., it is enough to prove that for j / {, +}, the vectors ε j and r ε are orthogonal. For =, ε j, r ε = ε j, uε = 0 as soon as j 3 from (.). Assume that for every l, the reflection r l leaves invariant Span{ε,..., ε l, ε l+2,..., ε n }. For every j =,..., n, we have ε j, r ε = ε j, r r 2... r uε = r r ε j, uε. (2.6) For j + 2, by assumption, the reflections r,..., r leave invariant ε j, so that the above expression reduces to ε j, uε which is 0 again by (.). For j =, we have r r ε = uε by definition of r, so that (2.6) gives ε, r ε = uε, uε, which is 0 since u is unitary.

14 4 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT For j <, by assumption, the reflections r j+,..., r leave invariant ε j, so that the right hand side of (2.6) reduces to r r j ε j, uε. By definition of r j, it is uε j, uε which is 0. (2) For fixed, it is clear from () that r r n ε = r r ε which is uε by definition of r. Proposition 2.7. For =,..., n, the matrix of the restriction of r to the basis (ε, ε + ) is Ξ(γ ) as defined in (2.4). In particular ε, r ε = γ. (2.7) The restriction of r n to Cε n is the multiplication by γ n. Proof. Note that for every n ε +, r ε = r r ε +, uε = ε +, uε = ρ. (2.8) Since r is a reflection acting on the subspace Span{ε, ε + }, identities (2.8) and (2.7) entail that the matrix representing r in the basis (ε,..., ε n ) is precisely Ξ(γ ) (see (2.4)). It is then enough to prove (2.7). For = it is immediate that: ε, r ε = ε, uε = ᾱ 0 = γ 0. Let us proceed by induction. For j set q j := ε j, r j ε j. Assume that q j = γ j for j. We have q + = ε +, r + ε + = r... r ε +, uε +. Equation (2.2) implies r ε + = ε + and since r j ε l = ε l for l j + 2, we get: r... r ε + = ε + γ ( r ε ε ) r ε, ε +, (2.9) γ ( r... r ε r... r ε ) uε, ε +. Now, it is nown that uε, ε + = ε +, uε = ρ. If we set v = ε, v j = r... r j ε j, a j = ρ j, w j+ = ε j+ a j uε j γ j we get the recursion v j+ = a j v j + w j+, (j ), (2.20) which we solve in : ( + ( v + = a j )ε + a j )w l. (2.2) j= l=2 Taing the scalar product with uε + yields j=l ( + ( q + = ā j ) ε, uε + + ā j ) w l, uε +. j= l=2 j=l

15 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 5 But w l, uε + = ε l, uε + ā l uε l, uε +, and since l +, we have w l, uε + = ε l, uε + = ᾱ α l 2 ρ m, which yields (with α = ) q + ᾱ = = = + ( ) ā j l= + j=l ρ 2 m l= m=l j=0 α l 2 ρ m m=l l 3 ( γ j ) + [ s=0 ( γ s) ( γ s ) = ( γ s ), s=0 and eventually q + = γ. l= γ l 2 m=l s=0 ( γ s) ( γ l 2) ρ 2 m m=l 2 j=0 l 3 ( γ j ) ρ 2 m m=l j=0 Now, we can summarize the above results in the following theorem. l 2 ] ( γ j ) Theorem 2.8. Let u U(n) and e a cyclic vector for u. Let µ be the spectral measure of the pair (u, e), and (α 0,..., α n ) its Verblunsy coefficients. Let (ε,..., ε n ) be the orthonormal basis obtained from the Gram-Schmidt procedure applied to (e, ue,..., u n e). Then, u can be decomposed as a product of n reflections (r ) n : u = r... r n (2.22) where r is the reflection mapping ε onto uε and by induction for each 2 n, r maps ε onto r r 2... r uε. This decomposition can also be restated in terms of the GGT matrix : G(α 0,, α n ) = Ξ (0) (γ 0 )Ξ () (γ )... Ξ (n ) (γ n ), (2.23) where for 0 n 2, the matrix Ξ () is given by Ξ ( ) (γ ) = Id Ξ(γ ) Id n, (2.24) with Ξ(γ) defined in (2.4). For = n, Ξ (n ) (γ n ) = Id n (γ n ). (2.25) 3. Deformed Verblunsy coefficients and independence We now use the point of view of sampling (or change of probability measure) to compute the distribution of the deformed Verblunsy coefficients

16 6 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT under CJ (n) δ,β. Let us first remar that, if the α s are independent and with rotational invariant distribution, then from (2.9) (α 0,..., α n ) law = (γ 0,..., γ n ). (3.) This is the case under CJ (n) 0,β. We first prove that when δ 0 the Verblunsy coefficients are not independent anymore by studying the simple case n = 2, β = 2, and then we compute the distribution of (γ 0,..., γ n ) under CJ (n) δ,β. We then show that under this distribution, the weights of the measure associated to the Verblunsy coefficients (α 0,..., α n ) are independent from the points at which the measure is supported and follow a Dirichlet distribution. Let δ C such that Re δ > /2. The formula λ (δ) (ζ) = Γ( + δ)γ( + δ) ( ζ) δ ( ζ) δ, ζ T (3.2) Γ( + δ + δ) defines a probability density with respect to the Haar measure on T, which is discontinuous at when Im δ 0 (see [8]). When β = 2 and δ 0, the Verblunsy coefficients are dependent. Indeed, let M U(2) with Verblunsy coefficients α 0 and α. Then det(id M) = [ ᾱ 0 ᾱ ( α 0 )], with α 0 < and α =. Under CJ (2) 0,2, the variables α 0 and α are independent and uniformly distributed on D and T respectively (see [24] or Proposition.7). The CJ (2) δ,2 is a det δ sampling of CJ (n) 0,2 (see the Introduction for the definition and notation for the det δ - sampling). So, the joint density of (α, ϕ) on D T is proportional to where f(α 0, α ) = [ ᾱ 0 ᾱ ( α 0 )] δ[ α 0 α ( ᾱ 0 )] δ = ( ᾱ 0 ) δ( α 0 ) δ [ γᾱ ] δ[ γα ] δ. γ = α ᾱ. It is then clear that the conditional density of α given α 0 is α λ (δ) (γᾱ ), α T, and this last quantity does not depend on α 0 (i.e. on γ) if and only if δ = 0. Otherwise the Verblunsy coefficients α 0 and α are dependent. The next theorem illustrates our interest in the deformed Verblunsy coefficients: under CJ (n) δ,β, they are independent. For the proof of this theorem, we shall need the following lemma which will also be useful when we study limit theorems:

17 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 7 Lemma 3.. Let s, t, l C such that: Re(s + l + ) > 0, Re(t + l + ) > 0. Then, the following identity holds: ( z 2 ) l ( z) s ( z) t d 2 πγ(l)γ(l + + s + t) z = Γ(l + + s)γ(l + + t). (3.3) D Proof. A Taylor expansion yields ( z) s ( z) t = m,n 0 ρ m+n ( s) n( t) m e i(m n)θ, n!m! with z = ρe iθ and 0 ρ <. We obtain by integrating z D( 2 ) l ( z) s ( z) t d 2 z = 2π ( s) n ( t) n n!n! n 0 = π n 0 ( s) n ( t) n n! (l )! (n + l)! = π l 2 F ( s, t; l + ; ), 0 ( ρ 2 ) l ρ 2n+ dρ where 2 F is the classical hypergeometric function (see [4]) and an application of Gauss formula (see [4]) shows that the last expression is exactly the right hand side of (3.3). Theorem 3.2. Let δ C with Re δ > /2 and β > 0. Set β = β/2. Under CJ (n) δ,β, the distribution of (γ 0,..., γ n ), denoted hereafter η (n) δ,β, is the following: () the variables γ 0,..., γ n 2, γ n are independent ; (2) for = 0,..., n 2 the density of γ with respect to the Lebesgue measure d 2 z on C is where c,n (δ) ( z 2) β (n ) ( z) δ( z) δ D (z), c,n (δ) = Γ( β (n ) + + δ ) Γ ( β (n ) + + δ ) πγ ( β (n ) ) Γ ( β (n ) + + δ + δ ) ; (3.4) (3) the density of γ n with respect to the Haar measure on T is λ (δ) (given by (3.2)). Proof. The distribution of the α s in the β-circular unitary ensemble is η (n) 0,β. More precisely, as seen in Definition.5 they are independent and if α = r e iψ (0 n 2), α n = e iψ n, then r and ψ are independent, ψ is uniformly distributed and r 2 has the Beta(, β (n )) distribution. Moreover α n is uniformly distributed on T. From (2.), the sampling factor is n 2 det(id U) δdet(id Ū)δ = ( γ n ) δ( γ n ) δ ( γ ) δ( γ ) δ. =0

18 8 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT so that, under CJ (n) δ,β, the density of (r 0,..., r n 2, ψ 0,..., ψ n ) is proportional to λ (δ)( ) n 2 γ n ( r 2 (n ) )β r ( γ ) δ( γ ) δ (0,) (r ), =0 with γ = r e iθ (0 n 2), γ n = e iθ n. Thans to the relations (2.9) and (2.0), the Jacobian matrix of the mapping (r 0,..., r n 2, ψ 0,..., ψ n ) (r 0,..., r n 2, θ 0,..., θ n ) is lower triangular with diagonal elements ±, so that, under CJ (n) δ,β, the density of (r 0,..., r n 2, θ 0,..., θ n ), is proportional to n 2 λ (δ) (γ n ) ( r 2 (n ) )β r ( γ ) δ( γ ) δ (0,) (r ), (3.5) =0 which proves the required independence and the expression of the distributions, up to the determination of c,n (δ). This quantity is obtained by taing l = β (n ), s = δ, t = δ in (3.3), which gives (3.4) and completes the proof of the Theorem. Starting with a set of deformed Verblunsy coefficients, with distribution η (n) δ,β, we obtain the coefficients (α 0,..., α n ) by inverting formula (2.). These are the coordinates of some probability measure µ supported at n points on the unit circle: n µ = π δ e iθ, = with π > 0 and n = π =. The next theorem gives the distribution induced on the vector (π,..., π n, θ,..., θ n ) by (γ 0,..., γ n ). Theorem 3.3. The following formulae express the same measure on the manifold of probability distribution on T supported at n points: n K (n) n δ,β (eiθ,..., e iθn ) β ( e iθ ) δ ( e iθ ) δ π β dθ... dθ n dπ... dπ n = in the (θ, π) coordinates and K (n) n 2 δ,β =0 = n ( γ 2 ) β (n ) ( γ ) δ( γ ) δ d 2 γ 0... d 2 γ n 2 dφ =0 in terms of the deformed Verblunsy coefficients, with γ n = e iφ. Here, is a constant: K (n) δ,β K (n) δ,β n 2 Γ( + δ)γ( + δ) = 2 n πγ( + δ + δ) c,n (δ), =0

19 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 9 with c,n (δ) given in Theorem 3.2. Consequently, if (γ 0,..., γ n ) is η (n) δ,β distributed, then (π,..., π n ) and (θ,..., θ n ) are independent; the vector of weights (π,..., π n ) follows the Dir n (β ) distribution and the vector (θ,..., θ n ) has the density h (n) δ,β. Proof. In the course of this proof, we shall adopt the following point of view. Starting with a measure supported at n points on the unit circle, we associate with it its Verblunsy coefficients (α 0,..., α n ) and then the corresponding GGT matrix which we note G for simplicity. Then e is a cyclic vector for G and µ is the spectral measure of (G, e ). Conversely, starting with the set of deformed Verblunsy coefficients with η δ,β distribution, we construct the coefficients (α 0,..., α n ) with the transformations (2.9), then the GGT matrix associated with it and finally µ the spectral measure associated with this matrix and e. We use the following well-nown identity (see [35] or [24] Lemma 4.): (e iθ,..., e iθn ) 2 n = Since γ = α, we can also write Moreover, from (2.), (e iθ,..., e iθn ) 2 det(id G) = n = n 2 π = ( α 2 ) n. (3.6) =0 n 2 π = ( γ 2 ) n. (3.7) =0 n n ( e iθ ) = ( γ ). (3.8) = In our setting, π i is modulus squared of the first component of the i-th eigenvector of the matrix G. Now, define =0 q 2 = π, for =,..., n. It is nown (see for example Forrester [3], Chapter 2 and [4] Theorem 2) that the Jacobian of the map (α 0,..., α n ) (θ,... θ n, q,..., q n ) is given by n 2 =0 ( α 2 ) q n n = q. Moreover, the map (γ 0,..., γ n ) (α 0,..., α n ) is invertible and its Jacobian is, as already seen. The result now follows from simple integral manipulations combined with the identities (3.7) and (3.8). 4. Matrix models for the circular Jacobi ensemble The results of the previous sections can now be used to propose some simple matrix models for the circular Jacobi ensemble. There are mainly two ways to generate matrix models for a given spectral measure encoded by its Verblunsy coefficients.

20 20 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT The AGR decomposition : if U = Θ (0) (α 0 )Θ () (α )... Θ (n ) (α n ) (the Θ s are defined in the introduction), the Verblunsy coefficients for the spectral measure associated to (U, e ) are precisely (α 0,..., α n ) (see [3] or [33] Section 0). Therefore, taing independent α s with law η (n) 0,β, the density of the eigenvalues of U = Θ (0) (α 0 )Θ () (α )... Θ (n ) (α n ) is proportional to (e iθ,..., e iθn ) β. The matrix U obtained above is the GGT matrix associated with the α s. It is in the Hessenberg form. The CMV form : set { L = Θ (0) (α 0 )Θ (2) (α 2 )... M = Θ () (α )Θ (3) (α 3 )... Cantero, Moral, and Velazquez [9] proved that the Verblunsy coefficients associated to (LM, e ) are precisely (α 0,..., α n ). Therefore, taing as previously independent α s with distribution η (n) 0,β, the density of the eigenvalues of the spectral law of LM is proportional to (e iθ,..., e iθn ) β ([24]). This matrix model is very sparse: it is pentadiagonal. We now propose a matrix model for the circular Jacobi ensemble: it is reminiscent of the AGR factorization with the noticeable difference that it is based on the deformed Verblunsy coefficients and actual reflections as defined in Section 2. Theorem 4.. If (γ 0,..., γ n ) is η (n) δ,β of (2.24) and (2.25), Ξ (0) (γ 0 )Ξ () (γ )... Ξ (n ) (γ n ) distributed, then with the notation is a matrix model for the circular Jacobi ensemble, i.e. the density of the eigenvalues is h (n) δ,β (see (.6)). Proof. We now from Theorem 2.8 that G(α 0,, α n ) = Ξ (0) (γ 0 )Ξ () (γ )... Ξ (n ) (γ n ). (4.) We also proved in Theorem 3.3 that the set of deformed Verblunsy coefficients with probability distribution η (n) δ,β induces a distribution on the eigenvalues of the GGT matrix G(α 0,, α n ) which has exactly the density h δ,β. This completes the proof of the Theorem. Remar. We now say a few extra words on the CMV form obtained by Killip and Nenciu in [24]. Cantero, Moral and Velazquez [9] introduced the basis χ 0,..., χ n obtained by orthogonalizing the sequence, z, z,.... They prove that in this basis the matrix is pentadiagonal. We name this matrix C(α 0,, α n ). It turns out that there exists a unitary P such that: P G(α 0,, α n )P = C(α 0,, α n ), P ϕ 0 = χ 0.

21 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 2 The two pairs (G(α 0,, α n ), ϕ 0 ) and (C(α 0,, α n ), χ 0 ) are equivalent, they admit the α s as Verblunsy coefficients, and have the same spectral measure. We conclude that if we start with the γ s distributed as η (n) δ,β, and build the α s by inverting the transformation (2.9), then C(α 0,, α n ) will be a matrix model for the circular Jacobi ensemble. But we do not now how to construct the CMV matrix from the γ s directly. We saw at the beginning of this section that Cantero et al. introduced the matrices L and M, as direct product of small blocs Θ () (α ) and obtained C as C = LM. It would be interesting to have an analogue construction based on the independent γ s. Theorem 2.8 which is a deterministic result, has also the following consequence: Proposition 4.2. Let (α 0,..., α n 2, α n ) D n T be independent random variables with rotationally invariant distribution. Then Θ (0) (α 0 )Θ () (α )... Θ (n ) (α n ) law = Ξ (0) (α 0 )Ξ () (α )... Ξ (n ) (α n ). Proof. We give two proofs of this result. The first one is a consequence of Theorem 2.8 from which we now that Θ (0) (α 0 )Θ () (α )... Θ (n ) (α n ) = Ξ (0) (γ 0 )Ξ () (γ )... Ξ (n ) (γ n ), and the remar at the beginning of Section 3. For the second proof, we proceed by induction on n. For n = the result is obvious. Suppose the result holds at ran n : thans to the recurrence hypothesis, Ξ (0) (α 0 )Ξ () (α )... Ξ (n ) (α n ) law = Ξ (0) (α 0 )Θ () (α )... Θ (n ) (α n ). Let e iφ 0 = α 0 α 0. An elementary calculation gives Ξ (0) (α 0 )Θ () (α )... Θ (n 2) (α n 2 )Θ (n ) (α n ) = Θ (0) (α 0 )Θ () (e iφ 0 α )... Θ (n 2) (e iφ 0 α n )Θ (n ) (e iφ 0 α n ). As the α s are independent with law invariant by rotation, (α 0, e iφ 0 α,..., e iφ 0 α n 2, e iφ 0 α n ) law = (α 0, α,..., α n 2, α n ), which completes the proof. Now that we have a matrix model for the circular Jacobi ensemble, we can study the characteristic polynomial for such matrices; the ey formula will be (2.). Proposition 4.3. Let U be a unitary matrix of size n and let Z n = det(id U) be its characteristic polynomial evaluated at. Then, in the circular

22 22 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT Jacobi ensemble, Z n can be written as a product of n independent complex random variables: n Z n = ( γ ), =0 where the laws of the γ s are given in Theorem 3.2. Consequently for any s, t C, with Re(t) > 2, the Mellin-Fourier transform of Z n is : n =0 E[ Z n t e is arg Zn ] = Γ(β + + δ)γ(β + + δ)γ(β + + δ + δ + t) Γ(β + + δ + δ)γ(β + + δ + t s 2 )Γ(β + + δ + t+s 2 ). (4.2) Proof. The first part is an easy consequence of (2.) and Theorem 3.2. To prove the second part, we note that if X = ( γ ), then X t e is arg X = ( γ ) a ( γ ) b, where a = (t + s)/2 and b = (t s)/2. Consequently, by independence of the γ s, we obtain: E[ Z n t e is arg Zn ] = n =0 E[( γ ) a ( γ ) b ], and formula (4.2) then easily follows from Lemma Limiting spectral measure and large deviations In (.7) we defined the spectral measure which is a central tool for the study of our circular ensembles. Let us re-write this measure on T as n µ (n) sp := = π (n) δ, (5.) e iθ(n) where we put a superscript (n) to stress on the dependency on n. Besides, in classical Random Matrix Theory, many authors are mainly interested in the empirical spectral distribution (ESD) defined by µ (n) esd = n n = δe iθ(n) We are concerned with their asymptotics under CJ (n) δ,β δ = δ(n) = β nd. (5.2) when n with where Re d 0 (and as usual β = β/2). In this framewor the variables (θ (n),..., θ(n) n ) and (π (n),..., π(n) n ) are distributed as in Theorem 3.3. It is well-nown that in the CUE (i.e. for d = 0 and β = ), the sequence (µ (n) esd ) converges wealy in probability to the uniform measure on T and that it satisfies the Large Deviation Principle at scale n 2 ([7] chap. 5). In this section we prove that in the cjue, both sequences (µ (n) sp ) and (µ (n) esd ) converge wealy in probability to the same limit measure supported by an

23 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 23 arc of the unit circle, and that (µ (n) esd ) satisfies the Large Deviation Principle at scale n 2. In the next two subsections, we first recall some definitions (which may be sipped by probabilist readers) and then state the main results which are Theorems 5., 5.4 and Wea convergence of measures. Let M (T) be the space of probability measures on T. The wea topology on M (T) is defined from the duality with the space of continuous functions on T; that is, ν n wealy converges to ν means fdν n fdν for every continuous function f. Since T is compact, it is equivalent to the convergence of moments. If we define the distribution function of ν M (T) as F ν (t) = ν({e iθ ; θ [0, t)}), then ν n ν wealy if and only if F νn (t) F ν (t) for all t [0, 2π) at which F ν (t) is continuous. The Lévy distance 2 d L is compatible with this topology and maes M (T) a compact metric space. It should be noticed that for all pair of elements of M (T) d L (µ, ν) sup F µ (t) F ν (t). (5.3) t In the following we consider random probability measures (ν n ) and the wea convergence in probability of (ν n ) to a deterministic ν M (T). That means that for every n we have a probability space (Ω n, F n, Q n ), a M (T)-valued random variable ν n, and that d(ν n, ν) 0 in probability, where d is any distance compatible with the wea topology (e.g. the Lévy distance d L ). In other words lim n ν n = ν (in probability) lim n d(ν n, ν) = 0 (in probability) ε > 0 lim n Q n (d(ν n, ν) > ε) = 0. (5.4) 5... The spectral measure. The following theorem states the convergence of the sequence (µ (n) sp ) to an explicit limit, which we identify now, with the help of some more notation. We assume that Re d 0. Let α d = d + d and let θ d [0, π) and ξ d [ θ d, θ d ] be such that sin θ d 2 = d + d, e iξ d = + d + d. (5.5) 2 whose precise definition is not needed here (see [0] Theorem D8)

24 24 P. BOURGADE, A. NIKEGHBALI, AND A. ROUAULT If Re d 0, d 0, let w d be defined by sin 2 ( (θ ξ d )/2 ) sin 2 (θ d /2) w d (θ) = if θ (θ d + ξ d, 2π θ d + ξ d ) + α d sin(θ/2) 0 otherwise. (5.6) In this case let dµ d (ζ) := w d (θ) dθ (ζ = e iθ ), (5.7) 2π and for d = 0 let dµ 0 (ζ) := dθ (ζ = e iθ ), 2π the Haar measure on T. Theorem 5.. Assume Re d 0. As n, lim µ (n) sp = µ d (in probability). To prove this convergence we use the parameterization of measures by their modified Verblunsy coefficients and the following two lemmas whose proofs are postponed until the end of the section. Lemma 5.2. For every fixed 0, as n, and consequently lim γ (n) = d + d (in probability), (5.8) lim α (n) = α d e i(+)ξ d (in probability). (5.9) Lemma ( 5.3. The sequence of Verblunsy coefficients of µ d is precisely αd e i(+)ξ d) 0. Proof of Theorem 5.. For ν M (T), let m (ν) = e iθ dν(θ) be its -th moment and let α (ν) be its -th Verblunsy - in short V- coefficient. Moments are related to V-coefficients in a continuous way : for every j, m j (ν) is a continuous function of (α 0,..., α (j N) ) where N is the cardinal of the support of ν. We now for Lemma 5.2 that for every fixed, lim α j (µ (n) sp ) = α j (µ d ) for j, in probability. It entails the convergence of m (µ (n) sp ) to m (µ d ), in probability, hence the wea convergence of the measures. Remar. The above method can easily be adapted to show the trivial asymptotics of two different scaling regimes : if δ(n) = o(n), the limit is the uniform measure on T, if δ(n) is real and n = o(δ(n)), the limit is the Dirac measure at.

25 CIRCULAR JACOBI ENSEMBLES AND DEFORMED COEFFICIENTS 25 Proof of Lemma 5.2. For γ (n) we use the Mellin transform ( E ( γ (n) )s) = Γ( β (n ) + δ + δ + s + ) Γ ( β (n ) + δ + ) Γ ( β (n ) + δ + δ + ) Γ ( β (n ) + δ + s + ), (this comes immediately from (3.3)). Since for fixed z C Γ(n + z) lim n Γ(n)n z =, we get, for fixed s (and ) which implies that ( lim E ( γ (n) n )s) = lim n ( γ(n) ) = + d + d + d ( ) s + d + d, + d in probability, which is equivalent to (5.8). The statement (5.9) is a direct consequence of (5.8) and (2.9). Remar. The convergences in probability in the above lemma actually hold in L p, for all p > 0 because all variables are bounded by. Proof of Lemma 5.3. If d is real, the Verblunsy coefficients are all equal. In fact, measures satisfying this property are nown to have an absolute continuous part supported by an arc of the unit circle, and a possible additional Dirac mass ([35] p.87). More precisely, let α D and let ν be the measure on T such that α (ν) = α for every 0. If θ(α) and ξ(α) are defined by θ(α) = 2 arcsin α, e iξ(α) = + α + α, then dν(ζ) = w(θ) dθ (ζ = e iθ ), 2π where sin 2 (θ/2 ) sin 2 (θ(α)/2) w(θ) = if θ (θ(α), 2π θ(α) + α sin((θ + ξ(α))/2), (5.0) 0 otherwise as soon as α (5.) (otherwise, there is an additional Dirac mass). The orthogonal polynomials with respect to this measure are nown as the Geronimus polynomials.