Central limit theorem for the heat kernel measure on the unitary group
CCentral limit theorem for the heat kernel measure onthe unitary group
Thierry L´evy a , Myl`ene Ma¨ıda b, ∗ a D´epartement de Math´ematiques, Ecole Normale Sup´erieure, 45, rue d’Ulm, F-75230 ParisCedex 05 b Laboratoire de Math´ematiques, Facult´e des Sciences d’Orsay, Universit´e Paris-Sud,F-91405 Orsay Cedex
Abstract
We prove that for a finite collection of real-valued functions f , . . . , f n on thegroup of complex numbers of modulus 1 which are derivable with Lipschitzcontinuous derivative, the distribution of (tr f , . . . , tr f n ) under the properlyscaled heat kernel measure at a given time on the unitary group U ( N ) hasGaussian fluctuations as N tends to infinity, with a covariance for which wegive a formula and which is of order N − . In the limit where the time tends toinfinity, we prove that this covariance converges to that obtained by P. Diaconisand S. Evans in a previous work on uniformly distributed unitary matrices.Finally, we discuss some combinatorial aspects of our results. Keywords:
Central Limit Theorem, Random Matrices, Unitary Matrices,Heat Kernel, Free Probability
1. Introduction
In [8], P. Diaconis and S. Evans studied the fluctuations of the trace of func-tions of a unitary matrix picked uniformly at random. Let us recall briefly theirmain result. If U is a unitary matrix of size N ≥ f a real-valued functionon the set U of complex numbers of modulus 1, then the eigenvalues λ , . . . , λ N of U belong to U and tr f ( U ) = N (cid:80) Ni =1 f ( λ i ) , where tr is the normalized trace(so that tr( I N ) = 1) and the matrix f ( U ) is obtained from U and f by functionalcalculus. Using Weyl’s integration formula and the rotational invariance of theHaar measure, it is easy to see that if f : U → R is defined almost everywhere, ∗ Corresponding author. Postal address : Laboratoire de Math´ematiques, Facult´e des Sci-ences d’Orsay, Universit´e Paris-Sud, F-91405 Orsay Cedex. Tel: +33 1 69 15 77 95. Fax: +331 69 15 72 34
Email addresses: [email protected] (Thierry L´evy), [email protected] (Myl`ene Ma¨ıda )
Preprint submitted to Journal Functional Analysis October 29, 2018 a r X i v : . [ m a t h . P R ] S e p s integrable and has zero mean on U then tr f ( U ) is defined for almost every U and, seen as a random variable under the Haar measure, also has zero mean.The function f being fixed, tr f can be seen as a random variable on theunitary group U ( N ), endowed with the Haar measure, for all N ≥
1. Thus,the single function f gives rise to a sequence of random variables indexed bythe integer N, which is their main object of study. In order to understandthe behaviour of this sequence, a fundamental fact, which has been proved andused extensively in this context in [8], is the following: for all p, q ∈ Z , one has E [tr( U p )tr( U q )] = δ p,q N − min( | p | , N ). Using this, one can easily check that, if f is square-integrable on U , then the variance of tr f converges to 0 as N tendsto infinity. Moreover, if f belongs to the Sobolev space H ( U ) (see Definition9.1 below), then the series of the variances of tr f on U ( N ) converges, whichgives a strong law of large numbers.The main result of [8] is that the fluctuations of tr f under the Haar measureare asymptotically Gaussian. More precisely, they have proved that if f belongsto H ( U ) and has zero mean on U , then N tr f converges in distribution to acentered Gaussian random variable with variance equal to the square of the H -norm of f (see Theorem 9.2 below for a precise statement).In this paper, we consider the fluctuations of tr f when the unitary matrix ispicked not under the Haar measure, but rather under the heat kernel measure ata certain time. The heat kernel measure at time T is the distribution of U N ( T ),where ( U N ( t )) t ≥ is the Brownian motion on U ( N ) issued from the identitymatrix, that is, the Markov process whose generator is the Laplace-Beltramioperator associated to a certain Riemannian metric on U ( N ). The choice of aRiemannian metric that we make is explicited at the beginning of Section 2.Apart from being one of the most natural stochastic processes with values inthe unitary group, the Brownian motion arises for example in the context oftwo-dimensional U ( N ) Yang-Mills theory ([18, 12, 11]).Let f : U → R be a function, as above. Once a time T ≥ f isa random variable on U ( N ) for each N ≥
1, the unitary group being endowedwith the heat kernel measure at time T . With our choice of Riemannian metric,it is known since the work of P. Biane [3] that if f is continuous, then tr f converges almost surely towards the integral of f against a probability measure ν T on U , which is characterized by the formula (4) below. By this almost sureconvergence, we mean that the expectations of these variables and the series oftheir variances converge. For all T >
0, the measure ν T is absolutely continuouswith respect to the uniform measure on U , with a density which unfortunatelycannot be expressed in terms of usual functions. Its support is the full circleonly for T ≥
4. For T ∈ (0 , T , and for the width of which a simple explicit formula exists. In fact, as N tends to infinity, not only the distribution of the eigenvalues of U N ( T ) butthe Brownian motion itself as a stochastic process converges in a certain sensetowards a limiting object called the free multiplicative Brownian motion , whichis defined in the language of free probability. The measure ν T is the non-2ommutative distribution of this free process at time T and can be consideredas a multiplicative analogue of the Wigner semi-circle law.The main result of this paper is that for any function f : U → R with Lip-schitz continuous derivative, the fluctuations of N tr f are asymptotically Gaus-sian with variance σ T ( f, f ), where σ T is the quadratic form defined in Definition2.4. This definition of σ T ( f, f ) involves three free multiplicative Brownian mo-tions which are mutually free and the functional calculus associated to f (cid:48) . Itmakes sense for functions of class C , or at best for absolutely continuous func-tions. An alternative definition of σ T ( f, f ) is given by Definition 9.10 in terms ofthe Fourier coefficients of f and the solution of an infinite triangular differentialsystem (see Lemma 9.7). We prove that, when T is large enough, this seconddefinition makes sense for functions in the Sobolev space H ( U ), which are noteven necessarily continuous.Moreover, we prove that, as T tends to infinity, σ T ( f, f ) converges towardsthe square of the H -norm of f . This convergence is consistent, at a heuristiclevel, with the result of P. Diaconis and S. Evans, since the Haar measure is theinvariant measure of the Brownian motion, and its limiting distribution as timetends to infinity.For small values of T , the analysis seems much harder to perform. We haveno expression of the covariance other than Definition 2.4 and it seems plausible,considering the limiting support of the distribution of the eigenvalues of U N ( T )and some puzzling numerical simulations (see Figure 1 in Section 9), that thelargest space of functions f for which N tr f has Gaussian fluctuations mightdepend on T , say for T ≤
4. Unfortunately, we have no precise conjecture tooffer in this respect.The understanding of global fluctuations of random matrices has been widelydeveloped in the literature using various techniques. By combinatorial meth-ods applied to the computation of moments, Ya. Sinai and A. Soshnikov [29]derived a central limit theorem (CLT) for moments of Wigner matrices grow-ing as o ( N / ) . An important breakthrough is the work of K. Johansson [17]where he got, using techniques of orthogonal polynomials on the explicit jointdensity of eigenvalues, a CLT for Hermitian or real symmetric matrices whoseentries have joint density e N tr V ( M ) , for a large class of potentials V . Recently,M. Shcherbina [28] has been able to lower, in the symmetric case, the regularityof those functions for which the CLT holds. The study of Stieltjes transformfor this purpose, initiated by L. Pastur and others [25, 26], has recently givensome striking results, among which one can cite the works of G. W. Andersonand O. Zeitouni [1] or W. Hachem, P. Loubaton and J. Najim [15]. RecentlyS. Chatterjee [6] proposed “a soft approach” based on second order Poincar´einequalities.The technique of proof that we have chosen is rather of the flavour of the oneintroduced in [5]. Therein, T. Cabanal-Duvillard proposed an approach based onmatricial stochastic calculus to get a CLT for Hermitian and Wishart Brownianmotions but also for several Gaussian Wigner matrices. In this direction we can3lso mention a CLT for band matrices obtained by A. Guionnet [13].Some tools of free probability will play a key role in our analysis. The notionof second order freeness was developed in a series of papers [24, 23, 7] in orderto give a general framework to CLT’s for large random matrices. In particular,the second paper [23] of the series deals with unitary matrices and the resultstherein might be relevant to the problem under consideration (see Section 8 formore details).Let us mention the work of F. Benaych-Georges [2], which is closely relatedto ours. He also considers unitary matrices taken under the heat kernel measure,and he obtains a CLT for functions of the entries of these matrices, whereas weare rather considering functions of their empirical measure.The paper is organized as follows : Section 2 is devoted to defining the Brow-nian motion on the unitary group, recalling from [3] its asymptotics, definingthe proper covariance functional and stating our main result (Theorem 2.6).In Section 3, we present the structure of the proof of our main theorem byintroducing a family of martingales (see Equation (6)) that will be the mainobject of study. The proof will in fact boil down to proving the convergenceof the bracket of these martingales (Section 5) and to controlling the varianceof this bracket (Section 6), relying on some technical results on the functionalcalculus on U ( N ) gathered in Section 4. In Section 7, we extend our result toother Brownian motions on the unitary group and to the Brownian motion onthe special unitary group. In Section 8, we deal with the fluctuations of uni-tary Brownian motions stopped at different times. Section 9 is devoted to thestudy of the covariance for large time, in connexion with the CLT for Haar uni-taries [8]. Finally, in Section 10, we discuss a combinatorial approach to someof our previous results and we obtain, via representation theoretic arguments,an explicit formula (Theorem 10.2) for mixed moments of the heat kernel on SU ( N ).
2. The Brownian motion on the unitary group
Let N ≥ U ( N ) the group of unitary N × N matrices and by u ( N ) its Lie algebra, which is the space of anti-Hermitian N × N matrices. We denote by I N the identity matrix. We will use systematicallythe following convention for traces: we denote the usual trace by Tr and thenormalized trace by tr, so that Tr( I N ) = N and tr( I N ) = 1.Let us endow u ( N ) with the real scalar product (cid:104) X, Y (cid:105) u ( N ) = N Tr( X ∗ Y ) = − N Tr( XY ). We denote by (cid:107) · (cid:107) u ( N ) the corresponding norm.The scalar product (cid:104) · , · (cid:105) u ( N ) determines a Brownian motion with values in u ( N ), namely the unique continuous Gaussian process ( K N ( t )) t ≥ with valuesin u ( N ) such that ∀ s, t ≥ , ∀ A, B ∈ u ( N ) , E [ (cid:104) A, K N ( s ) (cid:105) u ( N ) (cid:104) B, K N ( t ) (cid:105) u ( N ) ] = min( s, t ) (cid:104) A, B (cid:105) u ( N ) . B kl , C kl , D k ) k,l ≥ be independent standard real Brownianmotions. Then K N ( t ) has the same distribution as the anti-Hermitian matrixwhose upper-diagonal coefficients are the √ N ( B kl ( t ) + iC kl ( t )) and whose di-agonal coefficients are the i √ N D k ( t ).The linear stochastic differential equation dU N ( t ) = U N ( t ) dK N ( t ) − U N ( t ) dt (1)admits a strong solution which is a process with values in M N ( C ). This processsatisfies the identity d ( U N U ∗ N )( t ) = 0, as one can check by using Itˆo’s formula.Hence, this equation defines a Markov process on the unitary group U ( N ),which we call the unitary Brownian motion. The generator of this Markovprocess can be described as follows. Let ( X , . . . , X N ) be an orthonormal basisof u ( N ). Each element X of u ( N ) can be identified with the left-invariant first-order differential operator L X on U ( N ) by setting, for all differentiable function F : U ( N ) → R and all U ∈ U ( N ),( L X F )( U ) = ddt | t =0 F ( U e tX ) . (2)The generator of the unitary Brownian motion is the second-order differentialoperator 12 ∆ = 12 N (cid:88) k =1 L X k . This operator does not depend on the choice of the orthonormal basis of u ( N ).We denote the associated semi-group by ( P t ) t ≥ . From now on, we will alwaysconsider the Brownian motion issued from the identity matrix, so that U N (0) = I N .The stochastic differential equation satisfied by U N can be translated intoan Itˆo formula, as follows. Proposition 2.1.
Let F : R × U ( N ) → R be a function of class C . Then forall t ≥ , F ( t, U N ( t )) = F (0 , I N ) + N (cid:88) k =1 (cid:90) t ( L X k F )( s, U N ( s )) d (cid:104) X k , K N (cid:105) u ( N ) ( s )+ (cid:90) t (cid:18)
12 ∆ F + ∂ t F (cid:19) ( s, U N ( s )) ds, (3) and the processes {(cid:104) X k , K N (cid:105) u ( N ) : k ∈ { , . . . , N }} are independent standardreal Brownian motions. This result is classical in the framework of stochastic analysis on manifolds(see for example [16]), but since our whole analysis relies on this formula andfor the convenience of the reader, we offer a sketch of proof in this particularsetting. 5 roof.
For all a, b ∈ { , . . . , N } , let ε ab : M N ( C ) → C denote the coordinatemapping which to a matrix M associates the entry M ab . Let also ∂ ab denotethe partial derivation with respect to the ab -entry. The definition of L X givenby (2) makes sense for any matrix X . One can check the following identities: ∀ X ∈ M N ( C ) , L X = N (cid:88) a,b,c =1 ε ac X cb ∂ ab and L X − L X = N (cid:88) a,b,c,a (cid:48) ,b (cid:48) ,c (cid:48) =1 ε ac X cb ε a (cid:48) c (cid:48) X c (cid:48) b (cid:48) ∂ ab ∂ a (cid:48) b (cid:48) , ∆ = L C + N (cid:88) k =1 N (cid:88) a,b,c,a (cid:48) ,b (cid:48) ,c (cid:48) =1 ε ac ( X k ) cb ε a (cid:48) c (cid:48) ( X k ) c (cid:48) b (cid:48) ∂ ab ∂ a (cid:48) b (cid:48) , where C = (cid:80) N i =1 X i . Moreover, C = − I N , regardless of the choice of theorthonormal basis ( X , . . . , X N ).Any smooth function F : R × U ( N ) is the restriction of a smooth functiondefined on R × M N ( C ). Applying the usual Itˆo formula to this extended functionand using the identities above leads immediately to (3). We are interested in the large N behaviour of the stochastic process U N issued from I N . P. Biane has described in [3] the limiting distribution of thisprocess seen as a collection of elements of the non-commutative probability space( L ∞ ⊗ M N ( C ) , E ⊗ tr). We start by describing the limiting object. As a generalreference on non-commutative probability and freeness, we recommend [30]. Definition 2.2.
Let ( A , τ ) be a (non-commutative) ∗ -probability space. A col-lection of unitaries ( u t ) t ≥ in A is called a free multiplicative Brownian motion if the following properties hold.1. For all ≤ t ≤ . . . ≤ t n , the elements u t , u t u ∗ t , . . . , u t n u ∗ t n − are free.2. For all ≤ s ≤ t , the element u t u ∗ s has the same distribution as u t − s .3. For all t ≥ , the distribution of u t is the probability measure ν t on U = { z ∈ C : | z | = 1 } characterized by the identity (cid:90) U − zz +1 e tz e t ξ dν t ( ξ ) = 1 + z, (4) valid for z in a neighbourhood of . The following result was proved by P. Biane. The second assertion followsfrom the first by a general result of D. Voiculescu.6 heorem 2.3.
The collection ( U N ( t )) t ≥ of non-commutative random vari-ables converges in distribution, as N tends to + ∞ , towards a free multiplicativeBrownian motion.Moreover, if U (1) N , U (2) N , . . . , U ( n ) N are n independent sequences of unitary Brow-nian motions, then the family (( U (1) N ( t )) t ≥ , ( U (2) N ( t )) t ≥ , . . . , ( U ( n ) N ( t )) t ≥ ) con-verges in non-commutative distribution, as N tends to infinity, towards (( u (1) t ) t ≥ , ( u (2) t ) t ≥ , . . . , ( u ( n ) t ) t ≥ ) where u (1) , . . . , u ( n ) are n free multiplicativeBrownian motions which are mutually free.2.3. Statement of the Central Limit Theorem Recall that U denotes the group of complex numbers of modulus 1. Let f : U → R be a function. Then, by the functional calculus, f induces a function,still denoted by f , from U ( N ) to M N ( C ). Moreover, for all unitary matrix U ,the matrix f ( U ) is Hermitian.We endow U with the usual length distance, that is, the distance such that d ( e iα , e iβ ) = | α − β | for all α, β ∈ R such that | α − β | ≤ π . Accordingly, wedefine the Lipschitz norm of a function f : U → R as follows: (cid:107) f (cid:107) Lip = sup z,w ∈ U ,z (cid:54) = w | f ( z ) − f ( w ) | d ( z, w ) . Note that if f is Lipschitz continuous and z, w belong to U , then the followinginequalities hold: | f ( z ) − f ( w ) | ≤ (cid:107) f (cid:107) Lip d ( z, w ) ≤ π (cid:107) f (cid:107) Lip | z − w | .By the derivative of a differentiable function f : U → R , we mean thefunction f (cid:48) : U → R defined by ∀ z ∈ U , f (cid:48) ( z ) = lim h → f ( ze ih ) − f ( z ) h . We denote by L ( U ) the space of integrable functions on U , with respect to theLebesgue measure. We denote by C ( U ) the space of continuously differentiablefunctions and by C , ( U ) the subspace of C ( U ) consisting of those functionswhose derivative is Lipschitz continuous. We define a family of bilinear formson C ( U ) as follows. Definition 2.4.
Let ( A , τ ) be a C ∗ -probability space which carries three freemultiplicative Brownian motions u, v, w which are mutually free. Let T ≥ bea real number. Let f, g : U → R be two functions of C ( U ) . For all s ∈ [0 , T ] ,we set σ T,s ( f, g ) = τ ( f (cid:48) ( u s v T − s ) g (cid:48) ( u s w T − s )) . Then, we define σ T ( f, g ) = (cid:90) T σ T,s ( f, g ) ds = (cid:90) T τ ( f (cid:48) ( u s v T − s ) g (cid:48) ( u s w T − s )) ds. Lemma 2.5.
For all T ≥ , σ T is a symmetric non-negative bilinear form on C ( U ) . roof. The symmetry of σ T comes from the fact that the triples ( u, v, w ) and( u, w, v ) have the same distribution. In order to prove the non-negativity, let usrealize ( u, v, w ) on the free product of three non-commutative probability spaces.So, let ( A u , τ u ), ( A v , τ v ) and ( A w , τ w ) be three non-commutative probabilityspaces which carry respectively u , v and w . We consider their free product, sowe define A = A u ∗ A v ∗ A w and τ = τ u ∗ τ v ∗ τ w . We also use the notation τ u , τ v , τ w for the partial traces on A . Then σ T ( f, f ) = (cid:90) T τ u ( τ v ( f (cid:48) ( u s v T − s )) τ w ( f (cid:48) ( u s w T − s ))) ds = (cid:90) T τ u ( τ v ( f (cid:48) ( u s v T − s )) ) ds ≥ , the positivity coming from the fact that f (cid:48) ( u s v T − s ) is self-adjoint.We will use the notation σ T ( f ) = σ T ( f, f ). Let us state our main result. Theorem 2.6.
Let T ≥ be a real number. Let n ≥ be an integer. Let f , . . . , f n : U → R be n functions of C , ( U ) . Let us define a n × n real non-negative symmetric matrix by setting Σ T ( f , . . . , f n ) = ( σ T ( f i , f j )) i,j ∈{ ,...,n } .Then, as N tends to infinity, the following convergence of random vectors in R n holds in distribution: N (tr f i ( U N ( T )) − E [tr f i ( U N ( T ))]) i ∈{ ,...,n } ( d ) −→ N →∞ N (0 , Σ T ( f , . . . , f n )) . (5)
3. Structure of the proof
For T = 0, the result is straightforward. Let us choose once for all a real T >
0. In order to study the left-hand side of (5), we write each componentof this random vector as the difference between the final and the initial valueof a martingale. To do this, let ( F N,t ) t ≥ denote the filtration generated bythe unitary Brownian motion U N . To each function f of L ( U ) we associate areal-valued martingale ( M fN ( t )) t ∈ [0 ,T ] by setting M fN ( t ) = E [tr f ( U N ( T )) |F N,t ] . (6)The left-hand side of (5) is simply N (cid:16) M f i N ( T ) − M f i N (0) (cid:17) i ∈{ ,...,n } and weare going to study the quadratic variations and covariations of the martingales M f i N . In order to state the main technical results, let us introduce some notation.Recall that the gradient of a differentiable function F : U ( N ) → C is thevector field on U ( N ) defined by ∇ F = (cid:80) N k =1 ( L X k F ) X k , where ( X , . . . , X N )is an orthonormal basis of u ( N ). To each pair of functions f, g ∈ L ( U ) weassociate a function E f,gN on [0 , T ) × U ( N ) by setting E f,gN ( s, U ) = N (cid:104)∇ ( P T − s (tr f ))( U ) , ∇ ( P T − s (tr g ))( U ) (cid:105) u ( N ) . f is integrable on U implies that tr f is an integrable function on U ( N ). Hence, for all s ∈ [0 , T ), P T − s (tr f ) is a function of class C ∞ on U ( N )and E f,gN is well defined. Proposition 3.1.
Consider f, g ∈ L ( U ) . With the notation introduced above,the following properties hold. For all t ∈ [0 , T ] , the quadratic covariation of the martingales N M fN and N M gN is given by (cid:104) N M fN , N M gN (cid:105) t = (cid:90) t E f,gN ( s, U N ( s )) ds. Assume that f and g are Lipschitz continuous. Then for all s ∈ [0 , T ) and all U ∈ U ( N ) , | E f,gN ( s, U ) | ≤ ( (cid:107) f (cid:107) Lip + (cid:107) g (cid:107) Lip ) . Moreover, if f and g belong to C ( U ) , then the following convergence holds: E [ E f,gN ( s, U N ( s ))] −→ N →∞ σ T,s ( f, g ) . Assume that f and g belong to C , ( U ) . Then the following estimate holds: sup s ∈ [0 ,T ) Var( E f,gN ( s, U N ( s ))) = O ( N − ) . Let us show that these results imply Theorem 2.6.
Proof of Theorem 2.6.
For all N ≥
1, define a R n -valued martingale Q N =( Q N , . . . , Q nN ) by setting Q N ( t ) = N (cid:16) M f j N ( t ) − M f j N (0) (cid:17) j ∈{ ,...,n } . It is a mar-tingale indexed by [0 , T ], issued from 0 and with the same bracket as N (cid:16) M f j N (cid:17) j ∈{ ,...,n } . For all ξ = ( ξ , . . . , ξ n ) ∈ R n and all t ∈ [0 , T ], set R N ( t ) = exp i n (cid:88) j =1 ξ j Q jN ( t ) + 12 n (cid:88) j,k =1 ξ j ξ k (cid:90) t σ T,s ( f j , f k ) ds . Itˆo’s formula yields E [ R N ( t )] = 1 + 12 n (cid:88) j,k =1 ξ j ξ k E (cid:90) t R N ( s ) (cid:16) σ T,s ( f j , f k ) − E f j ,f k N ( s, U N ( s )) (cid:17) ds. Thus, | E [ R N ( t ) − | ≤ n (cid:107) ξ (cid:107) e nT (cid:107) ξ (cid:107) max j =1 ...n (cid:107) f (cid:48) j (cid:107) ∞ max j,k =1 ...n E (cid:90) t (cid:12)(cid:12)(cid:12) σ T,s ( f j , f k ) − E f j ,f k N ( s, U N ( s )) (cid:12)(cid:12)(cid:12) ds. j and k , the last integral is smaller than (cid:90) t (cid:12)(cid:12)(cid:12) σ T,s ( f j , f k ) − E (cid:104) E f j ,f k N ( s, U N )( s ) (cid:105)(cid:12)(cid:12)(cid:12) ds + E (cid:90) t (cid:12)(cid:12)(cid:12) E f j ,f k N ( s, U N )( s ) − E (cid:104) E f j ,f k N ( s, U N )( s ) (cid:105)(cid:12)(cid:12)(cid:12) ds. By the second part of Proposition 3.1, and by the dominated convergence the-orem, the first integral tends to 0 as N tends to infinity. The square of thesecond integral is smaller than t (cid:82) t Var( E f j ,f k N ( s, U N ( s ))) ds , which, thanks tothe third part of Proposition 3.1 and by dominated convergence again, tendsalso to 0. Finally, we have proved that ∀ ξ ∈ R n , lim N →∞ E (cid:104) e i (cid:80) nj =1 ξ j Q jN ( t ) (cid:105) = exp − n (cid:88) j,k =1 ξ j ξ k (cid:90) t σ T,s ( f j , f k ) ds , which, for t = T , yields the expected result.In Section 4, we collect some technical results that we use in Sections 5 and6 to prove Proposition 3.1.
4. Regularity of the functional calculus
In this section, we relate the regularity of a function f : U → R to theregularity of the functional calculus mapping f : U ( N ) → M N ( C ) and thefunction tr f : U ( N ) → R . We start with a result which, logically speaking, isnot necessary for our exposition, but which is the simplest instance of a crucialphenomenon. The group U ( N ) becomes a metric space when it is endowed with the Rie-mannian distance, denoted by d , associated to the Riemannian metric inducedby the scalar product (cid:104)· , ·(cid:105) u ( N ) on u ( N ). We denote by (cid:107) F (cid:107) Lip the correspondingLipschitz norm of a function F : U ( N ) → R , that is, (cid:107) F (cid:107) Lip = sup (cid:26) | F ( U ) − F ( V ) | d ( U, V ) :
U, V ∈ U ( N ) , U (cid:54) = V (cid:27) . As a reference for the notions of Riemannian geometry that we use, werecommend [9].
Proposition 4.1.
Let f : U → R be a Lipschitz continuous function. Then tr f : U ( N ) → R is also Lipschitz continuous and (cid:107) tr f (cid:107) Lip = 1 N (cid:107) f (cid:107) Lip . Lemma 4.2.
Let U and V be two elements of U ( N ) . Then there exists A, B ∈ U ( N ) such that AU A − and BV B − are diagonal and d ( AU A − , BV B − ) ≤ d ( U, V ) .Proof. Let O be the conjugacy class of V . It is a compact submanifold of U ( N ).Let V (cid:48) be a point of O which minimizes the distance to U . Let γ : [0 , → U ( N )be a minimizing geodesic path from V (cid:48) to U parametrized at constant speed.It is thus of the form γ ( t ) = V (cid:48) e tZ for some Z ∈ u ( N ). Since V (cid:48) minimizesthe distance to U , the vector ˙ γ (0) is orthogonal to the tangent space T V (cid:48) O .This space T V (cid:48) O , identified with a subspace of u ( N ) by a left translation, is therange of the linear mapping Ad( V (cid:48)− ) − Id. Hence, Z belongs to the kernel ofthe adjoint linear mapping, that is, to the kernel of Ad( V (cid:48) ) − Id. In other words, V (cid:48) ZV (cid:48)− = Z . It follows that Z and V (cid:48) can be simultaneously diagonalized, inan orthonormal basis, and the same is true for V (cid:48) and V (cid:48) e Z = U . Finally, V (cid:48) and U are conjugated by a same unitary matrix to two diagonal unitary matrices.The result follows easily from the fact that translation are isometries on U ( N ). Proof of Proposition 4.1.
Let f : U → R be Lipschitz continuous. Consider U and V in U ( N ). Thanks to Lemma 4.2, let us choose U (cid:48) and V (cid:48) which areboth diagonal, conjugated respectively to U and V , and such that d ( U (cid:48) , V (cid:48) ) ≤ d ( U, V ). Let us write U (cid:48) = diag( e iα , . . . , e iα N ) and V (cid:48) = diag( e iβ , . . . , e iβ N ) insuch a way that | β j − α j | ≤ π for all j ∈ { , . . . , N } . Let us compute d ( U (cid:48) , V (cid:48) ).It is equal to d ( I N , U (cid:48)− V (cid:48) ), hence to d ( I N , e i diag( β − α ,...,β N − α N ) ) = (cid:107) i diag( β − α , . . . , β N − α N ) (cid:107) u ( N ) = (cid:118)(cid:117)(cid:117)(cid:116) N N (cid:88) j =1 ( β j − α j ) . It follows that d ( U, V ) ≥ (cid:80) Nj =1 | β j − α j | . On the other hand, | tr f ( V ) − tr f ( U ) | ≤ N N (cid:88) j =1 | f ( e iβ j ) − f ( e iα j ) | ≤ N (cid:107) f (cid:107) Lip N (cid:88) j =1 | β j − α j |≤ N (cid:107) f (cid:107) Lip d ( U, V ) . This proves the inequality (cid:107) tr f (cid:107) Lip ≤ N (cid:107) f (cid:107) Lip . By choosing α, β such that | f ( e iβ ) − f ( e iα ) | is close to (cid:107) f (cid:107) Lip | β − α | and by considering U = e iα I N , V = e iβ I N , one verifies that the opposite inequality holds.11et us make a short heuristic comment on this result. The scalar productwhich we have chosen on u ( N ) corresponds to a metric structure on U ( N ) whichgives this group the diameter d ( I N , − I N ) = (cid:107) i diag( π, . . . , π ) (cid:107) u ( N ) = N π , of theorder of N . The function f : U → R being fixed, the variations of the functiontr f : U ( N ) → R are of the same order of magnitude as those of f but occur ona space N times as large. This makes the equality that we have juste provedplausible.In the same order of ideas, note that the distance to the origin at time T of alinear Brownian motion in a Euclidean space of large dimension d is, by the lawof large numbers, of the order of √ dT . Assuming that the Brownian motion onthe unitary group behaves in a comparable way, and considering the fact thatthe dimension of U ( N ) is N , this indicates that the Brownian motion U N ( T )might be at a distance of order N √ T of I N , thus a fraction of the diameter of U ( N ) which does not depend on N . This gives an intuitive justification for thechoice of the normalization. We are now going to prove that the functional calculus induced by f isdifferentiable when f is differentiable, and to compute its differential. For this,we introduce some notation. Let f : U → C be a differentiable function. Let usdefine a function D f : U × U → C by setting ∀ z, w ∈ U , D f ( z, w ) = (cid:26) f ( z ) − f ( w ) z − w if z (cid:54) = w, − iz f (cid:48) ( z ) if z = w. The function D f is symmetric and, if f is C ( U ), it is continuous and boundedby π (cid:107) f (cid:48) (cid:107) ∞ . Note that D f takes its values in C even if f is real-valued.If the function f is only Lipschitz continuous, then it is differentiable withbounded differential outside a negligible subset of U , and the definition of D f still makes sense outside the corresponding negligible subset of the diagonal of U × U . Moreover, outside this subset, the inequality | D f ( z, w ) | ≤ π (cid:107) f (cid:48) (cid:107) ∞ holds.If U is a unitary matrix, we denote by L U and R U the linear operatorson M N ( C ) of left and right multiplication by U respectively. These operatorscommute and they are normal with respect to the scalar product (cid:104) A, B (cid:105) = N Tr( A ∗ B ) on M N ( C ). In fact, L ∗ U = L U − and R ∗ U = R U − . Hence, if g is afunction on U × U , then g ( L U , R U ) is a well-defined endomorphism of M N ( C ).Even when f is only Lipschitz continuous, D f ( L U , R U ) is well-defined for almostall U ∈ U ( N ).Let us define a special orthonormal basis of u ( N ). We use the notation( E jk ) j,k ∈{ ,...,N } for the canonical basis of M N ( C ). For all j, k with 1 ≤ j Let ( X k ) k ∈{ ,...,N } be a orthonormal basis of u ( N ) . Let A, B beelements of M N ( C ) . Then the following equalities hold: N (cid:88) k =1 tr( AX k )tr( BX k ) = − N tr( AB ) , (9) N (cid:88) k =1 tr( AX k BX k ) = − tr( A )tr( B ) . (10) Proof. 1. For A, B ∈ u ( N ), this equality multiplied by N is indeed simply N (cid:88) k =1 (cid:104) A, X k (cid:105) u ( N ) (cid:104) B, X k (cid:105) u ( N ) = (cid:104) A, B (cid:105) u ( N ) . The general case follows thanks to the equality M N ( C ) = u ( N ) ⊕ i u ( N ) and thefact that the relations are C -bilinear in ( A, B ).2. Choose i, j, l, m ∈ { , . . . , N } . By taking A = E ji and B = E ml in thefirst relation, we find N (cid:88) k =1 ( X k ) ij ( X k ) lm = − N δ i,m δ j,l . The second relation follows by developing the trace.14 roposition 4.5. Let f : U → R be a differentiable function. Then tr f isdifferentiable and, for all U ∈ U ( N ) and all Y ∈ u ( N ) , we have ( L Y (tr f ))( U ) = − i tr( f (cid:48) ( U ) Y ) . (11) In particular, ∀ U ∈ U ( N ) , (cid:107)∇ (tr f )( U ) (cid:107) = N tr( f (cid:48) ( U ) ) .Proof. Since tr f is invariant by conjugation, we have for all U, V ∈ U ( N ) andall Y ∈ u ( N ) the equality ( L Y (tr f ))( U ) = ( L V Y V − (tr f ))( V U V − ). Hence, itsuffices to check (11) for all Y when U is diagonal. In this case, the result isa direct consequence of Proposition 4.3. The second assertion follows from thedefinition of the gradient and the identity (9). At the end of the proof of Proposition 3.1 (see Section 6.2), we will need toestimate the Lipschitz norm of a function of a unitary matrix of a special form.We state and prove this estimation below, although the reader might want toskip it now and jump to Section 5. Proposition 4.6. Let f be an element of C , ( U ) . Let V, W be two elementsof U ( N ) . Define a function F V,W : U ( N ) → C by setting F V,W ( U ) = tr ( f (cid:48) ( U V ) f (cid:48) ( U W )) . Then F is Lipschitz continuous and we have the estimate (cid:107) F V,W (cid:107) Lip ≤ πN (cid:107) f (cid:48) (cid:107) L ∞ (cid:107) f (cid:48)(cid:48) (cid:107) L ∞ . Proof. We prove that F V,W is differentiable almost everywhere on U ( N ) andestimate the L ∞ norm of its differential. According to Proposition 4.3, we have,for all X ∈ u ( N ) and almost all U ∈ U ( N ), the equality( L X F V,W )( U ) =tr (cid:0) V − D f (cid:48) ( L V U , R V U )( V U X ) V f (cid:48) ( U W ) (cid:1) +tr (cid:0) f (cid:48) ( U V ) W − D f (cid:48) ( L W U , R W U )( W U X ) W (cid:1) . We have used the fact that ddt | t =0 f (cid:48) ( U e tX V ) = V − ddt | t =0 f (cid:48) ( V U e tX ) V . Let usfocus on the first term of the right-hand side, the second being similar. By theCauchy-Schwarz inequality, (cid:12)(cid:12) tr (cid:0) V − D f (cid:48) ( L V U , R V U )( V U X ) V f (cid:48) ( U W ) (cid:1)(cid:12)(cid:12) ≤ tr( M ∗ M )tr ( f (cid:48) ( U W ) ∗ f (cid:48) ( U W )) , where we have set M = D f (cid:48) ( L V U , R V U )( V U X ).Recall that M N ( C ) is endowed with the scalar product (cid:104) A, B (cid:105) = N Tr( A ∗ B ).We claim that the operator norm of the endomorphism D f (cid:48) ( L V U , R V U ) of M N ( C ) with respect to this norm is bounded above by π (cid:107) f (cid:48)(cid:48) (cid:107) L ∞ . Indeed, thisoperator is normal with respect to this scalar product, so that its operator norm15quals its spectral radius, which is smaller than the L ∞ norm of D f (cid:48) . Hence,we find tr( M ∗ M ) ≤ π (cid:107) f (cid:48)(cid:48) (cid:107) L ∞ tr( X ∗ X ) . It follows that (cid:107)L X F V,W (cid:107) L ∞ ≤ π (cid:107) f (cid:48)(cid:48) (cid:107) L ∞ (cid:107) X (cid:107) u ( N ) N (cid:107) f (cid:48) (cid:107) L ∞ , from which the result follows easily. 5. Convergence of the bracket In this section, we prove the first two assertions of Proposition 3.1. Let usfirst prove a fundamental property of the generator of the Brownian motionon U ( N ). The action of U ( N ) on u ( N ) by conjugation is an isometric action.Hence, for all V ∈ U ( N ), the processes U N and V U N V − satisfy two stochasticdifferential equations (see (1)) driven by two processes in u ( N ) with the samedistribution, so that they have the same distribution. Lemma 5.1. Let F : U ( N ) → R be a Lipschitz continuous function. Let Y bean element of u ( N ) . Let t ≥ be a real number. Then L Y ( P t F ) = P t ( L Y F ) .Proof. Since F is Lipschitz continuous, L Y F is well-defined as an element of L ∞ ( U ( N )). The result amounts simply to the interversion of an integrationand a derivation: for all U ∈ U ( N ), L Y ( P t F )( U ) = dds | s =0 E (cid:2) F ( U e sY U N ( t )) (cid:3) = dds | s =0 E (cid:2) F ( U U N ( t ) e sY ) (cid:3) = E (cid:20) dds | s =0 F ( U U N ( t ) e sY ) (cid:21) = P t ( L Y F )( U ) . We have used the fact that U N ( t ) has the same distribution as e − sY U N ( t ) e sY . The following result summarizes the applications of Itˆo formula that we willuse. The third assertion below implies, by polarization, the first assertion ofProposition 3.1. Proposition 5.2. Let F : U ( N ) → R be an integrable function. Define a real-valued martingale L F indexed by [0 , T ] by setting, for all t ∈ [0 , T ] , L F ( t ) = E [ F ( U N ( T )) |F N,t ] . Let ( X k ) k ∈{ ,...,N } be an orthonormal basis of u ( N ) . Thenthe following equalities hold for all t ∈ [0 , T ] . L F ( t ) = ( P T − t F )( U N ( t )) . L F ( t ) = L F (0) + (cid:90) t N (cid:88) k =1 L X k ( P T − s F )( U N ( s )) d (cid:104) X k , K N (cid:105) u ( N ) ( s ) . (cid:104) L F (cid:105) ( t ) = (cid:90) t (cid:107) ( ∇ ( P T − s F ))( U N ( s )) (cid:107) ds . If F is Lipschitz continuous, then (cid:104) L F (cid:105) ( t ) = (cid:90) t N (cid:88) k =1 [ P T − s ( L X k F )( U N ( s ))] ds .Proof. 1. Choose t ∈ [0 , T ]. Since the unitary Brownian motion has independentmultiplicative increments, L F ( t ) can be rewritten as L F ( t ) = E [ F ( U N ( T )) |F N,t ] = E [ F ( U N ( t ) U ∗ N ( t ) U N ( T )) |F N,t ]= E [ F ( U N ( t ) V N ( T − t )) |F N,t ] , where V N is a Brownian motion on U ( N ) with the same distribution as U N andindependent of U N . The result follows.2. Let us apply (3) to the function G : [0 , T ] × U ( N ) → R defined by G ( t, U ) = ( P T − t F )( U ). It follows from the definition of the semigroup ( P t ) t ≥ that G satisfies the time-reversed heat equation ∆ G + ∂ t G = 0. Hence, Itˆo’sformula reads L F ( t ) = L F (0) + N (cid:88) k =1 (cid:90) t ( L X k ( P T − s F ))( U N ( s )) d (cid:104) X k , K N (cid:105) u ( N ) ( s ) . 3. The equality follows immediately from the equality 2 and the fact thatthe processes {(cid:104) X k , K N (cid:105) u ( N ) : k ∈ { , . . . , N }} are independent standard realBrownian motions.4. This equality follows from the previous one by applying Lemma 5.1. We can now prove the second assertion of Proposition 3.1. Recall thatwe use the notation E f,gN ( s, U ) = N (cid:104)∇ ( P T − s (tr f ))( U ) , ∇ ( P T − s (tr g ))( U ) (cid:105) u ( N ) .We will use the fact, which is a consequence of Jensen’s inequality, that for anysquare-integrable function G : U ( N ) → R , and for all t ≥ 0, ( P t G ) ≤ P t ( G ). Proof of the second assertion of Proposition 3.1. Let f : U → R be Lipschitzcontinuous. By definition and by Lemma 5.1 E f,fN ( s, U ) = N N (cid:88) k =1 ( P T − s ( L X k (tr f )))( U ) ≤ N N (cid:88) k =1 P T − s (( L X k tr f ) )( U ) = N P T − s ( (cid:107)∇ (tr f ) (cid:107) )( U ) . 17y Proposition 4.5 and the fact that P T − s does not increase the uniform norm,this implies that | E f,fN ( s, U ) | ≤ (cid:107) f (cid:48) (cid:107) L ∞ . By polarization, the estimation of | E f,gN ( s, U ) | follows.Now, let us consider two independent copies V N and W N of the unitaryBrownian motion U N . Then, denoting by E V N ,W N the expectation with respectto V N and W N only, we have E f,fN ( s, U N ( s )) = N N (cid:88) k =1 ( P T − s ( L X k (tr f )))( U N ( s )) = N N (cid:88) k =1 E V N ,W N [( L X k tr f )( U N ( s ) V N ( T − s ))( L X k tr f )( U N ( s ) W N ( T − s ))] . Using successively Proposition 4.5 and Lemma 4.4, we find E f,fN ( s, U N ( s )) = E V N ,W N [tr ( f (cid:48) ( U N ( s ) V N ( T − s )) f (cid:48) ( U N ( s ) W N ( T − s )))] . Taking the expectation with respect to U N , we find finally E [ E f,fN ( s, U N ( s ))] = E [tr ( f (cid:48) ( U N ( s ) V N ( T − s )) f (cid:48) ( U N ( s ) W N ( T − s )))] . Let ( A , τ ) be a C ∗ -probability space which carries three free mutliplica-tive brownian motions u, v, w which are mutually free. According to Theo-rem 2.3, the family ( U N ( s ) , V N ( t ) , W N ( u )) s,t,u ≥ , seen as a collection of non-commutative random variables in the non-commutative probability space ( L ∞ ⊗ M N ( C ) , E ⊗ tr), converges in distribution to ( u s , v t , w u ) s,t,u ≥ as N tends to in-finity. This implies in particular that for all non-commutative polynomial p inthree variables and their adjoints, and for all s, t, u ≥ E [tr p ( U N ( s ) , V N ( t ) , W N ( u ))] −→ N →∞ τ ( p ( u s , v t , w u )) . Let us fix s ∈ [0 , T ). Since A is a C ∗ -algebra, there is a continuous functionalcalculus on normal elements, hence on unitary elements, and f (cid:48) ( u s v T − s ) f (cid:48) ( u s w T − s )is a well-defined element of A . On the other hand, choose ε > q ( z, w ) bea polynomial function in z, w and their adjoints such that sup z,w ∈ U | f (cid:48) ( z ) f (cid:48) ( w ) − q ( z, w ) | < ε . Then | E [tr ( f (cid:48) ( U N ( s ) V N ( T − s )) f (cid:48) ( U N ( s ) W N ( T − s )))] − τ ( f (cid:48) ( u s v T − s ) f (cid:48) ( u s w T − s )) |≤ | E [tr ( f (cid:48) ( U N ( s ) V N ( T − s )) f (cid:48) ( U N ( s ) W N ( T − s ))) − tr q ( U N ( s ) V N ( T − s ) , U N ( s ) W N ( T − s ))] | + | E [tr q ( U N ( s ) V N ( T − s ) , U N ( s ) W N ( T − s ))] − τ ( q ( u s v T − s , u s w T − s )) | + | τ ( q ( u s v T − s , u s w T − s )) − τ ( f (cid:48) ( u s v T − s ) f (cid:48) ( u s w T − s )) | . q ( · , · ) and f (cid:48) ( · ) f (cid:48) ( · ), hence smaller than ε . The middle term tends to 0 as N tends to infinity. Altogether, this proves that E [ E f,fN ( s, U N ( s ))] −→ N →∞ τ ( f (cid:48) ( u s v T − s ) f (cid:48) ( u s w T − s )) , from which the expected result follows by polarization. 6. Convergence of the variance of the bracket This section is devoted to the proof of the third assertion of Proposition 3.1. Consider a function F : U ( N ) → R . If F is Lipschitz continuous, then theequality (cid:107) F (cid:107) Lip = (cid:107)∇ F (cid:107) L ∞ holds. The goal of this paragraph is to prove thefollowing inequality. Proposition 6.1. Let F : U ( N ) → R be a Lipschitz continuous function. Forall T ≥ , one has the following inequality: Var[ F ( U N ( T ))] ≤ T (cid:107) F (cid:107) . Note that this inequality is preserved by rescaling of the Riemannian metricon U ( N ), that is, by rescaling of the scalar product on u ( N ). Indeed, let λ bea positive real and let us consider the scalar product (cid:104)· , ·(cid:105) (cid:101) u = λ (cid:104)· , ·(cid:105) u on u ( N ).Then, putting a tilda to the quantities associated with this new scalar product,we have on one hand ˜ d = λ d and (cid:107) F (cid:107) (cid:103) Lip = λ − (cid:107) F (cid:107) Lip , and on the other hand (cid:101) ∆ = λ − ∆ and (cid:101) U N ( T ) has the distribution of U N ( λ − T ). Proof. Recall the definition of the martingale L F (see Proposition 5.2). Theleft-hand side is equal to E [ (cid:104) L F (cid:105) ( T )], thus, by the third assertion of Proposition5.2, to E (cid:90) T (cid:107) ( ∇ ( P T − s F ))( U N ( s )) (cid:107) ds ≤ T sup s ∈ [0 ,T ) (cid:107)∇ ( P T − s F ) (cid:107) L ∞ = T sup s ∈ [0 ,T ) (cid:107) P T − s F (cid:107) . On the other hand, since F is Lipschitz continuous, for all t ≥ (cid:107) P t F (cid:107) Lip ≤(cid:107) F (cid:107) Lip . The result follows. 19 .2. An estimate of a Lipschitz norm With Proposition 6.1 in mind, we are going to study the Lipschitz norm of U (cid:55)→ E f,fN ( s, U ) in order to estimate the variance of E f,fN ( s, U N ( s )). Proposition 6.2. Assume that f is of class C , ( U ) . Then sup s ∈ [0 ,T ] (cid:107) E f,fN ( s, · ) (cid:107) Lip = O ( N − ) . Proof. The proof relies on the identity E f,fN ( s, U N ( s )) = E V N ,W N [tr ( f (cid:48) ( U N ( s ) V N ( T − s )) f (cid:48) ( U N ( s ) W N ( T − s )))] . By Proposition 4.6, the expression between the brackets is a Lipschitz contin-uous function of U N ( s ) for all values of V N ( T − s ) and W N ( T − s ), with aLipschitz norm which does not depend on V N ( T − s ) and W N ( T − s ) and is O ( N − ). Hence, the same estimate holds for the expectation. Proof of the third assertion of Proposition 3.1. It suffices to combine Proposi-tion 6.2 and Proposition 6.1 to find that that sup s ∈ [0 ,T ) Var[ E f,fN ( s, U N ( s ))] = O ( N − ). The same result for E f,gN follows easily.This concludes the proof of Proposition 3.1 and thus of Theorem 2.6. 7. Other Brownian motions, on unitary and special unitary groups In this section, we explain how Theorem 2.6 can be extended to other Brow-nian motions on the unitary group and to the Brownian motion on the specialunitary group.In this paper so far, we have considered the Brownian motion U N on U ( N )associated to the scalar product on u ( N ) given by (cid:104) X, Y (cid:105) u ( N ) = N Tr( X ∗ Y ) , for any X, Y ∈ u ( N ). The crucial property of this scalar product is its in-variance under the action of U ( N ) on u ( N ) by conjugation. There is in fact atwo-parameter family of scalar products with this invariance property, namely a Tr (( X − tr X ) ∗ ( Y − tr Y )) + b Tr( X ∗ )Tr( Y ) with a, b > 0. Multiplying the twoparameters a and b by the same constant simply affects the Brownian motionby a global rescaling of time, indeed dividing time by this constant, so that wemay choose the value of one of them. We take a = N in order to have correctasymptotics as N tends to infinity. This choice being made, varying b reallyyields different Brownian motions. It turns out to be more convenient to take α = b − as the parameter: we define, for all α > 0, the scalar product (cid:104) X, Y (cid:105) ( α ) u ( N ) = N Tr(( X − tr X ) ∗ ( Y − tr Y )) + 1 α Tr( X ∗ )Tr( Y )on u ( N ). In particular, the scalar product considered in the rest of this papercorresponds to α = 1. 20n order to understand the Brownian motions associated to the scalar prod-ucts (cid:104)· , ·(cid:105) ( α ) u ( N ) , we start by defining the Brownian motion on SU ( N ), whichcorresponds to the limit where α tends to 0.Let us denote by su ( N ) the hyperplane of u ( N ) consisting of traceless ma-trices, which is also the Lie algebra of the special unitary group SU ( N ), and let K N be the linear Brownian motion on su ( N ) corresponding to the scalar prod-uct induced by (cid:104)· , ·(cid:105) u ( N ) . Let V N be the solution of the stochastic differentialequation dV N ( t ) = V N ( t ) dK N ( t ) − (cid:18) − N (cid:19) V N ( t ) dt. (12)One can check that if the initial condition is in the special unitary group, thenthe process V N stays in it: the constant 1 − N is designed for that purpose.We call V N the Brownian motion on SU ( N ).Now, for all α ≥ 0, let us consider the following process with values in U ( N ): V ( α ) N ( t ) = e iαBtN V N ( t ) , where ( B t ) t ≥ is a standard real Brownian motion independent of V N . Let( Y , . . . , Y N − ) be an orthonormal basis of su ( N ). For all α ≥ 0, the generatorof V ( α ) N is given by 12 ∆ ( α ) = 12 N − (cid:88) k =1 L Y i + α L iN I N , and we call V ( α ) N the α -Brownian motion on U ( N ).For each α > 0, the process V ( α ) N is naturally associated with the scalarproduct (cid:104) X, Y (cid:105) ( α ) u ( N ) on u ( N ). Indeed, let K ( α ) N be the linear Brownian motionon u ( N ) corresponding to this scalar product. It can be expressed as K ( α ) N = K N + iαN B. Then the process V ( α ) N satisfies the stochastic differential equation dV ( α ) N ( t ) = V ( α ) N ( t ) dK ( α ) N ( t ) − (cid:18) α − N (cid:19) V ( α ) N ( t ) dt. (13)In particular, V (1) N has the same distribution as U N .The main feature of the Brownian motion on U ( N ) which we have usedextensively in the proof of Theorem 2.6 is that its generator commutes with allLie derivatives. Since the Lie derivative in the direction of iI N commutes withall Lie derivatives, this is also the case for the generator of V N and of all theprocesses V ( α ) N , α > α ≥ 0, the process V ( α ) N converges as N tends to infinity to a free multiplicative Brownian motion.Let us now define a modified version of the covariance σ T .21 efinition 7.1. With all the notation of Definition 2.4, we define, for all α ≥ , σ ( α ) T ( f, g ) = (cid:90) T τ ( f (cid:48) ( u s v T − s ) g (cid:48) ( u s w T − s ))+( α − τ ( f (cid:48) ( u s v T − s )) τ ( g (cid:48) ( u s w T − s )) ds. Following step by step the proof of Theorem 2.6, one finds the followingresult. Theorem 7.2. Let T ≥ be a real number. Let n ≥ be an integer. Let f , . . . , f n : U → R be n functions of C , ( U ) . Let us define a n × n real non-negative symmetric matrix by setting Σ ( α ) T ( f , . . . , f n ) = ( σ ( α ) T ( f i , f j )) i,j ∈{ ,...,n } .Then, as N tends to infinity, the following convergence of random vectors in R n holds in distribution: N (cid:16) tr f i ( V ( α ) N ( T )) − E (cid:104) tr f i ( V ( α ) N ( T )) (cid:105)(cid:17) i ∈{ ,...,n } ( d ) −→ N →∞ N (0 , Σ ( α ) T ( f , . . . , f n )) . (14)We leave the details to the reader, since every step can be adapted in astraightforward way. The only substantial change is in Lemma 4.4, which nowwill take the following form. Lemma 7.3. Let ( X k ) k ∈{ ,...,N } be an orthonormal basis of ( u ( N ) , (cid:104)· , ·(cid:105) ( α ) u ( N ) ) .Let A, B be elements of M N ( C ) . Then the following equality holds: N (cid:88) k =1 tr( AX k )tr( BX k ) = − N (cid:0) tr( AB ) + ( α − A )tr( B ) (cid:1) . (15) Assume that ( X , . . . , X N − ) form an orthonormal basis of su ( N ) endowed withthe scalar product induced by (cid:104)· , ·(cid:105) u ( N ) . Then N − (cid:88) k =1 tr( AX k )tr( BX k ) = − N (tr( AB ) − tr( A )tr( B )) . (16)It is this modification which gives rise to the new covariance introduced inDefinition 7.1. 8. Joint fluctuations of the unitary Brownian motion at differenttimes A natural generalization of our main result consists in considering severalBrownian motions stopped at possibly different times. The goal of this sectionis to establish an analogue of Theorem 2.6 in this case. In order to state theresult, we define a new covariance function.22 efinition 8.1. Let ( A , τ ) be a C ∗ -probability space which carries three freemultiplicative Brownian motions u, v, w which are mutually free. Let T , T ≥ be real numbers. Let f, g : U → R be two functions of C ( U ) . For all s ∈ [0 , T ∧ T ] , we set σ T ,T ,s ( f, g ) = τ ( f (cid:48) ( u s v T − s ) g (cid:48) ( u s w T − s )) . Then, we define σ T ,T ( f, g ) = (cid:90) T ∧ T σ T ,T ,s ( f, g ) ds = (cid:90) T ∧ T τ ( f (cid:48) ( u s v T − s ) g (cid:48) ( u s w T − s )) ds. We have the following result. Theorem 8.2. Let n ≥ be an integer. Let T , . . . , T n ≥ be real num-bers. Let f , . . . , f n : U → R be n functions of C , ( U ) . Let us define a n × n real non-negative symmetric matrix by setting Σ T ,...,T n ( f , . . . , f n ) =( σ T i ,T j ( f i , f j )) i,j ∈{ ,...,n } . Then, as N tends to infinity, the following conver-gence of random vectors in R n holds in distribution: N (tr f i ( U N ( T i )) − E [tr f i ( U N ( T i ))]) i ∈{ ,...,n } ( d ) −→ N →∞ N (0 , Σ T ,...,T n ( f , . . . , f n )) . (17)The proof of this result is very similar to the proof of Theorem 2.6 and, asin the previous section, we simply point out the small differences between thetwo.For the sake of convenience, let us assume T ≤ . . . ≤ T n . Let f , . . . , f n : U → R be n functions of C , ( U ). We define for each i ∈ { , . . . , n } a martingaleindexed by [0 , T n ] by setting M f i N ( t ) = E (tr f i ( U N ( T i )) |F N,t ) . Observe that the martingale M f i N is constant on the interval [ T i , T n ]. Let us nowdefine the vector-valued martingale Q N ( t ) = N (cid:16) M f i N ( t ) − M f i N (0) (cid:17) i ∈{ ,...,n } , sothat the left hand-side of (17) is equal to Q N ( T n ). The proof of Theorem 8.2relies on an analogue of Proposition 3.1, for which we introduce the followingnotation : for all i, j ∈ { , . . . , n } with i ≤ j and all s ∈ [0 , T i ), we set E f i ,f j N ( s, U ) = N (cid:104)∇ ( P T i − s (tr f i ))( U ) , ∇ ( P T j − s (tr f j ))( U ) (cid:105) u ( N ) . We state the following result for the two functions f and f . Proposition 8.3. With the notation introduced above, the following propertieshold. For all t ∈ [0 , T ] , the quadratic covariation of the martingales N M f N and N M f N is given by (cid:104) N M f N , N M f N (cid:105) t = (cid:90) t ∧ T E f ,f N ( s, U N ( s )) ds. Assume that f and f are Lipschitz continuous. Then for all s ∈ [0 , T ) and all U ∈ U ( N ) , | E f ,f N ( s, U ) | ≤ ( (cid:107) f (cid:107) Lip + (cid:107) f (cid:107) Lip ) . Moreover, if f and f belong to C ( U ) , then the following convergence holds: E [ E f ,f N ( s, U N ( s ))] −→ N →∞ σ T ,T ,s ( f , f ) . Assume that f and f belong to C , ( U ) . Then the following estimateholds: sup s ∈ [0 ,T ) Var( E f ,f N ( s, U N ( s ))) = O ( N − ) . The proof of this Proposition is in no way different from the proof of Propo-sition 3.1. The unique novelty is the fact that M f N is constant on the interval[ T , T ] so that the quadratic covariation (cid:104) M f N , M f N (cid:105) vanishes on this interval.Then, one deduces Theorem 8.2 from Proposition 8.3 just as one deducesTheorem 2.6 from Proposition 3.1.Let us mention that, in the case where the functions f , . . . , f n are polyno-mial, and given Theorem 2.6, the Gaussian character of the fluctuations in thecase where the Brownian motions are stopped at different times is a consequenceof the work of J. Mingo, R. Speicher and P. ´Sniady [24, 23] on the notion of sec-ond order freeness and its specialization to the case of unitary matrices. Theirwork also provides one with a covariance function and it could be interestingto investigate the relation between our expression of what we call σ T ,T andtheirs.Another natural question which is answered by the theory of second orderfreeness is that of the asymptotic fluctuations of random variables of the formtr p ( U N ( T ) , . . . , U N ( T k )) where p is a non-commutative polynomial. It seemsmore difficult, although not hopeless, to apply our techniques to such function-als. 9. Behaviour of the covariance for large time For any fixed N , the Markov process ( U N ( T )) T ≥ converges in distribution,as T goes to infinity, to its invariant measure, which is the Haar measure on U ( N ) . In [8], P. Diaconis and S. Evans established a central limit theorem forHaar distributed unitary random matrices. In this section, we relate our resultto theirs by comparing the limit as T tends to infinity of the covariance σ T withthe covariance which they have found. In order to state the result of Diaconis and Evans, we need to introducesome notation. Definition 9.1. Let H ( U ) denote the space of functions that are square-integrable on U and such that (cid:107) f (cid:107) := 116 π (cid:90) [0 , π ] (cid:12)(cid:12) f ( e iϕ ) − f ( e iθ ) (cid:12)(cid:12) sin (cid:16) ϕ − θ (cid:17) dϕdθ < ∞ . e denote by (cid:104)· , ·(cid:105) the inner product associated to this Hilbertian semi-norm. For all f : U → C which is square-integrable and all j ∈ Z , we denote by a j ( f ) = π (cid:82) U f ( ξ ) e − ijξ dξ the j -th Fourier coefficient of f . One can check that f ∈ H ( U ) if and only if (cid:80) j ∈ Z | j || a j ( f ) | is finite and that, in this case, (cid:107) f (cid:107) = (cid:88) j ∈ Z | j || a j ( f ) | . The result of Diaconis and Evans states as follows. Theorem 9.2. (5.1 in [8]) For all N ∈ N , let M N be a N × N unitary matrixdistributed according to the Haar measure on U ( N ) . Let n ≥ be an integer.For all f , . . . , f n ∈ H ( U ) , let Σ( f , . . . , f n ) be the n × n real non-negativesymmetric matrix defined by Σ( f , . . . , f n ) = (cid:16) (cid:104) f i , f j (cid:105) (cid:17) i,j =1 ,...,n . As N goes toinfinity, the following convergence of random vectors in R n holds in distribution: N (tr f i ( M N ) − E [tr f i ( M N )]) i ∈{ ,...,n } ( d ) −→ N →∞ N (0 , Σ( f , . . . , f n )) . In view of this result, it is natural to expect the covariance that we haveintroduced in Definition 2.4 to converge, as T tends to infinity, to the covariancegiven by the H -scalar product. This is what the following result expresses. Theorem 9.3. For all n ≥ and all f , . . . , f n ∈ H ( U ) , Σ T ( f , . . . , f n ) −→ T →∞ Σ( f , . . . , f n ) . Let us emphasize that Σ T ( f , . . . , f n ) has only been defined so far for func-tions in C , ( U ). From this point on, we focus on extending the definition ofthe covariance to functions of the space H ( U ) and proving Theorem 9.3. In the sequel, ( u t ) t ≥ , ( v t ) t ≥ and ( w t ) t ≥ will be three multiplicative freeBrownian motions, that are mutually free. For all T ≥ k ∈ Z , let usdenote by µ k ( T ) = τ ( u kT ) the k -th moment of u T . Recall that, since u T has thesame law as u ∗ T , one has, for all k ∈ Z , the equality µ k ( T ) = µ − k ( T ). For each k ≥ 1, according to [3], µ k ( T ) is given by µ k ( T ) = e − kT k − (cid:88) l =0 ( − T ) l l ! (cid:18) kl + 1 (cid:19) k l − . (18) Lemma 9.4. For all ε > , all T ≥ T ( ε ) = ε log(1 + ε ) and all k ∈ Z , onehas | µ k ( T ) | ≤ e −| k | T ( − ε ) . roof. If k = 0 or ε ≥ , the inequality is trivial. Moreover, since µ k ( T ) = µ − k ( T ), it suffices to prove the inequality for k > 0. So, let us assume that ε ≤ and k ∈ N ∗ . It is easy to check that the expression (18) of µ k ( T ) isequivalent to the following: µ k ( T ) = e − kT ikπ (cid:73) e − kT z (cid:18) z (cid:19) k dz, where we integrate over a closed path of index 1 around the origin of the complexplane. If we choose as our contour the circle of radius ε centered at the origin,we get µ k ( T ) = e − kT ikπ (cid:90) π e − kT ε e iθ (cid:18) εe iθ (cid:19) k i ε e iθ dθ, so that, provided T ≥ T ( ε ) , | µ k ( T ) | ≤ ε k e − kT e kT ε (cid:18) ε (cid:19) k ≤ e − kT ( − ε ) , as expected.We will denote by T a real large enough such that for all T ≥ T and all k ∈ Z , the inequality | µ k ( T ) | ≤ e −| k | T holds. One can check that 31 is largeenough but we choose T = 32 for reasons which will soon become apparent.For all j, k ∈ Z and T > , we define τ j,k ( T ) = (cid:90) T τ (cid:0) ( u s v T − s ) j ( u s w T − s ) k (cid:1) ds. (19) Proposition 9.5. Set T = 32 . For all T ≥ T and all ( j, k ) (cid:54) = (0 , , thefollowing inequality holds: | τ j,k ( T ) | ≤ e − | j + k | T | j | + | k | + ( | j | + | k | ) T e − | j | + | k | ( T − T ) . (20) Moreover, if j (cid:54) = 0 , then (cid:12)(cid:12)(cid:12)(cid:12) τ j, − j ( T ) − | j | (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − T | j | + 2 | j | T e − | j | ( T − T ) . (21) In particular, for all ( j, k ) (cid:54) = (0 , , the following convergence holds : lim T →∞ τ j,k ( T ) = δ j + k, | j | . The proof of these estimates relies on a differential system satisfied by thefunctions τ j,k . This differential system is a consequence of the free Itˆo calculusfor free multiplicative Brownian motions. We state the form that we use, whichis of interest on its own. 26 roposition 9.6. Let ( u t ) t ≥ be a free multiplicative Brownian motion on somenon-commutative ∗ -probability space ( A , τ ) . Let a , . . . , a n ∈ A be random vari-ables such that the two families { u t : t ≥ } and { a , . . . , a n } are free. Finally,choose ε , . . . , ε n ∈ { , ∗} . Then ddt τ ( u ε t a . . . u ε n t a n ) = − n τ ( u ε t a . . . u ε n t a n ) − (cid:88) ≤ i This differential system follows easily from an application of Proposition9.6 to the expression (19).Before we turn to the proof of Proposition 9.5, let us state some elementaryproperties of the functions τ j,k . For all k ≥ 0, define the polynomial P k by therelation µ k ( T ) = e − kT P k ( T ). For k < 0, define P k = P − k . Lemma 9.8. For all j, k ∈ Z , the function τ j,k is real-valued and satisfies τ j,k = τ k,j = τ − j, − k . Moreover, there exists a family of polynomials ( R j,k ) j,k ∈ Z with rational coefficients such that the following equality holds : ∀ j, k ∈ Z , τ j,k ( T ) = j (cid:54) =0 | j | δ j + k, + e − | j | + | k | T R j,k ( T ) . (22) These polynomials are characterized by the fact that for all j, k ∈ Z , R j,k (0) = 0 and ˙ R j,k = jk ≥ P j + k − | j |− (cid:88) l =1 ( | j |− l ) P l R sgn( j )( | j |− l ) ,k − | k |− (cid:88) m =1 ( | k |− m ) P m R j, sgn( k )( | k |− m ) . (23) Proof. The equalities τ j,k = τ − j, − k = τ k,j follow from the definition of τ j,k , usingthe unitarity of u, v, w , the traciality of τ , and the fact that the families ( u, v, w )and ( u, w, v ) have the same joint distribution. The fact that τ j,k is real-valuedcan be proved by induction using the differential system stated in Lemma 9.7,or directly using the definition and the fact that ( u, v, w ) and ( u ∗ , v ∗ , w ∗ ) havethe same distribution.The functions R j,k defined by (22) are easily checked to satisfy the differen-tial system (23) and, by induction, to be polynomial. Proof of Proposition 9.5. Since the differential equation for τ j,k expressed byLemma 9.7 involves only indices ( j (cid:48) , k (cid:48) ) such that | j (cid:48) | + | k (cid:48) | ≤ | j | + | k | , we willprove the conjunction of (20) and (21) by induction on | j | + | k | . It is understoodthat k = − j in (21). 28he symmetry properties of τ j,k allow us to restrict ourselves to the twocases where j, k ≥ j > , k < 0. We may also assume that j + k ≥ | j | + | k | is 1. So, we start with τ , ( T ) = T µ ( T ) = T e − T , which is smaller than e − T for T larger than T . Hence, if | j | + | k | = 1 and T ≥ T , then | τ j,k ( T ) | ≤ e − T . This proves the result when | j | + | k | = 1.Let us consider now j and k and assume that (20) and (21) have been provedfor all j (cid:48) , k (cid:48) such that | j (cid:48) | + | k (cid:48) | < | j | + | k | . Let us first assume that j + k (cid:54) = 0.In this case, define ρ j,k ( T ) = e | j | + | k | T τ j,k ( T ) . Then Lemmas 9.4 and 9.7 and the induction hypothesis imply the inequality | ˙ ρ j,k ( T ) | ≤ e | j | + | k | T e − | j + k | T + 4 e | j | + | k | T | j |− (cid:88) l =1 ( | j | − l ) e − l T e −| sgn( j )( | j |− l )+ k | T | j | − l + | k | + ( | j | + | k | − T e | j | + | k | ( T + T ) | j |− (cid:88) l =1 ( | j | − e − l T e l T − T + 4 e | j | + | k | T | k |− (cid:88) m =1 ( | k | − m ) e − m T e −| j +sgn( k )( | k |− m ) | T | j | + | k | − m + ( | j | + | k | − T e | j | + | k | ( T + T ) | k |− (cid:88) m =1 ( | k | − e − m T e m T − T . Since | j | − l ≤ | j | − l + | k | , | k | − m ≤ | j | + | k | − m and e − l T ≤ 1, we find | ˙ ρ j,k ( T ) | ≤ e | j | + | k | T e − | j + k | T + 4 e | j | + | k | T | j |− (cid:88) l =1 e − l T e −| sgn( j )( | j |− l )+ k | T + 4 e | j | + | k | T | k |− (cid:88) m =1 e − m T e −| j +sgn( k )( | k |− m ) | T + 2( | j | + | k | − T e | j | + | k | ( T + T ) ∞ (cid:88) l =1 e − l T . (24)If we are in the case where j, k ≥ 0, then we obtain immediately the estimate | ˙ ρ j,k ( T ) | ≤ e | j | + | k | T (cid:32) e − | j + k | T + e − T − e − T (cid:16) e − | j + k | T + 2( | j | + | k | − T e − | j | + | k | ( T − T ) (cid:17)(cid:33) . (25)In the case where j > k < 0, the computation is slightly more complicated.In this case, let us also assume that j + k > 0, as we have indicated that it is29ossible to do. Then the estimation of the sum over m in (24) is the same asbefore, since j + sgn( k )( | k | − m ) is positive for all values of m . However, thesign of sgn( j )( | j | − l ) + k now depends on l . Thus, we bound the first sum over l by e − | j + k | T j + k (cid:88) l =1 e − l T + + ∞ (cid:88) l = j + k +1 e − l T e − ( l − ( j + k )) T . In the first term, we could actually have e − l T instead of e − l T but we are notseeking any optimality. In the second term, we write e − ( l − ( j + k )) T = e − (2 l − ( j + k )) T e l T ≤ e − ( j + k ) T e l T , and we find that the first sum over l in (24) is bounded by 2 e − | j + k | T e − T − e − T .Finally, we have established that, when j > , k < j + k > | ˙ ρ j,k ( T ) | ≤ e | j | + | k | T (cid:32) e − | j + k | T + e − T − e − T (cid:16) e − | j + k | T + 2( | j | + | k | − T e − | j | + | k | ( T − T ) (cid:17)(cid:33) . In view of (25), the last estimate holds for all values of j and k . Our choice of T guarantees that for T ≥ T , the inequalities e − T + 12 e − T − e − T ≤ e − T − e − T ≤ | ˙ ρ j,k ( T ) | ≤ e | j | + | k | T (cid:18) e − | j + k | T + 14 ( | j | + | k | − T e − | j | + | k | ( T − T ) (cid:19) . Integrating the last inequality from T on and using the fact that | j | + | k | − | j + k | ≥ | j | + | k | , we find | ρ j,k ( T ) | ≤ T e | j | + | k | T + e | j | + | k | T (cid:32) e − | j + k | T | j | + | k | + ( | j | + | k | − T e − | j | + | k | ( T − T ) (cid:33) , from which it follows immediately that | τ j,k ( T ) | ≤ e − | j + k | T | j | + | k | + ( | j | + | k | ) T e − | j | + | k | ( T − T ) , which is the expected equality.Let us now treat the case where k = − j . As before, we can assume that j > 0. Setting ρ j ( T ) = e | j | T (cid:16) τ j, − j ( T ) − | j | (cid:17) , we find, using the same estimatesas before, that | ˙ ρ j ( T ) | ≤ e | j | T ∞ (cid:88) l =1 e − l T + 2(2 | j | − e | j | ( T + T ) T ∞ (cid:88) l =1 e − l T . 30t follows that | ρ j ( T ) | ≤ T e | j | T + e | j | T e − T (cid:0) | j | − (cid:1) + (2 | j | − T e | j | ( T + T ) , so that (cid:12)(cid:12)(cid:12)(cid:12) τ j, − j ( T ) − | j | (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − T | j | + 2 | j | T e − | j | ( T − T ) , which is the expected inequality. This concludes the proof. Proposition 9.9. Let f ∈ H ( U ) be real-valued. The following properties hold.1. For all T > T , (cid:88) j,k ∈ Z | jka j ( f ) a k ( f ) τ j,k ( T ) | < ∞ . lim T →∞ (cid:88) j,k ∈ Z jka j ( f ) a k ( f ) τ j,k ( T ) = −(cid:107) f (cid:107) . Proof. Choose an integer n ≥ 1. Then for all T ≥ T , Proposition 9.5 implies (cid:88) | j | , | k |≤ n | jka j ( f ) a k ( f ) τ j,k ( T ) | ≤ (cid:88) | j | , | k |≤ n | jk || j | + | k | | a j ( f ) a k ( f ) | e − | j + k | T + T (cid:88) | j | , | k |≤ n | jk | ( | j | + | k | ) | a j ( f ) a k ( f ) | e − | j | + | k | ( T − T ) ≤ (cid:88) | j | , | k |≤ n (cid:112) | jk || a j ( f ) a k ( f ) | e − | j + k | T + 2 (cid:88) | j | , | k |≤ n | j | | a j ( f ) | e − | j | ( T − T ) | k | | a k ( f ) | e − | k | ( T − T ) ≤ (cid:88) l ∈ Z e −| l | T (cid:88) | j | , | k |≤ n,j + k = l (cid:112) | jk || a j ( f ) a k ( f ) | + 2 (cid:88) | j |≤ n | j | | a j ( f ) | e − | j | ( T − T ) ≤ (cid:107) f (cid:107) (cid:88) l ∈ Z e −| l | T + 2 (cid:88) j ∈ Z | j | e − | j | ( T − T ) . The first assertion follows. The second is a consequence of the second statementin Proposition 9.5 and the theorem of dominated convergence.Proposition 9.9 above allows us to give a new definition of the covariance σ T when T is large enough. Definition 9.10. For all T > T and all f ∈ H ( U ) , we define σ T ( f, f ) = − (cid:88) j,k ∈ Z jka j ( f ) a k ( f ) τ j,k ( T ) . emma 9.11. Let f be a function of C , ( U ) . For all T > T , the two defini-tions (Definition 2.4 and Definition 9.10) of σ T ( f, f ) coincide.Proof. The series (cid:80) j ∈ Z | a j ( f (cid:48) ) | is convergent, so that S n ( f (cid:48) )( e iξ ) = i (cid:80) | j |≤ n ja j ( f ) e ijξ converges uniformly to f (cid:48) on U as n tends to infinity. Therefore, starting fromDefinition 2.4, σ T ( f, f ) = − (cid:90) T τ (cid:88) j,k ∈ Z jka j ( f ) a k ( f )( u s v T − s ) j ( u s w T − s ) k ds. As the processes are unitary and (cid:80) j ∈ Z | j || a j ( f ) | < ∞ , we get by dominatedconvergence that, for all T ≥ σ T ( f, f ) = − (cid:88) j,k ∈ Z jka j ( f ) a k ( f ) τ j,k ( T ) , as expected.Theorem 9.3 is now a straightforward consequence of the polarisation ofDefinition 9.10 and Proposition 9.9. Remark 9.12. Let us emphasize that Proposition 9.5 implies that, for all ε > and all T > T , the following series converges: (cid:88) j,k ∈ Z ( | j | + | k | ) − ε | τ j,k ( T ) | < + ∞ . Hence, for all T > T , the equality K T ( e iθ , e iϕ ) = (cid:88) ( j,k ) ∈ Z \{ (0 , } e ijθ e ikϕ τ j,k ( T ) defines K T as a square-integrable real-valued function on U and, for all ε > and f, g ∈ H + ε ( U ) , one has the equality σ T ( f, g ) = (cid:90) [0 , π ] f (cid:48) ( e iθ ) K T ( e iθ , e iϕ ) g (cid:48) ( e iϕ ) dθdϕ π . We conclude this study of the covariance by showing some puzzling numericalexperiments (see Figure 1). It is striking on these pictures that the behaviourof the covariance σ T ( f, g ) is complicated and interesting for small T , and muchsimpler for large T . It is thus not surprising that we have been only able toanalyse σ T for large T . 32 (cid:45) (cid:45) (cid:45) Figure 1: For all k ≥ 1, let us define s k ( e iθ ) = sin( kθ ) and c k ( e iθ ) = cos( kθ ). The picturesabove are the graphs of the following functions of T for T ∈ [0 , σ T ( s k , s k )and σ T ( c k , c k ) for k ∈ { , . . . , } . Bottom left : µ k ( T ) for k ∈ { , . . . , } . Top center : σ T ( s k , s k +1 ) for k ∈ { , . . . , } . Bottom center : σ T ( c k , c k +1 ) for k ∈ { , . . . , } . Top right: σ T ( s k , s k +3 ) for k ∈ { , , , , } . Bottom right : σ T ( s k , s k +2 ) for odd k ∈ { , . . . , } . 10. Combinatorial approaches τ j,k The differential system satisfied by the functions τ j,k (Lemma 9.7) can beinterpreted, at least when j and k have the same sign, in terms of enumerationof walks on the symmetric group, in the same vein as the computations madeby one of us in [19]. This is what we explain in this section.Fix j ≥ . We consider the Cayley graph on the symmetric group S j gen-erated by all transpositions. The vertices of this graph are the elements of S j and two permutations σ and σ are joined by an edge if and only if σ σ − is atransposition. A finite sequence ( σ , . . . , σ n ) of permutations such that σ i and σ i +1 are joined by an edge for all i ∈ { , . . . , n − } is called a path of length n. The distance between two permutations is the length of the shortest paththat joins them. We call defect of a path the number of steps in the path whichincrease the distance to identity. Heuristically, one can understand the defectas follows : each time we compose a permutation with a transposition, eitherwe cut a cycle into two pieces and this is a step which decreases the distanceto identity, or we coalesce two cycles into a bigger one and this is a step whichincreases the distance to identity. The defect counts the number of steps of thesecond kind.For any σ ∈ S j , and any two integers n, d ≥ , we denote by S ( σ, n, d ) thenumber of paths in the Cayley graph of S j starting from σ, of length n and withdefect d. The interested reader can find more details about those combinatorialobjects in [19].Let j, k ≥ . If σ ∈ S j and τ ∈ S k , we denote by σ × τ the concatenation of σ and τ, that is the permutation in S j + k such that σ × τ ( i ) = σ ( i ) if 1 ≤ i ≤ j and σ × τ ( i ) = τ ( i − j ) + j if j + 1 ≤ i ≤ j + k. T ≥ E (cid:2) tr( U N ( T ) j )tr( U N ( T ) k ) (cid:3) − E (cid:104) tr( U N ( T ) j ) (cid:105) E (cid:2) tr( U N ( T ) k ) (cid:3) = e − ( j + k ) T ∞ (cid:88) n,d =0 ( − T ) n n ! N d S ((1 . . . j ) × (1 . . . k ) , n, d ) − ∞ (cid:88) n ,n ,d ,d =0 ( − T ) n + n n ! n ! N d + d ) S ((1 . . . j ) , n , d ) S ((1 . . . k ) , n , d ) . (26)Moreover, for all T (cid:48) ≥ , we recall that all the expansions involved convergeuniformly on ( N, T ) ∈ N × [0 , T (cid:48) ].Using this equality, it is for example easy to check thatlim N →∞ (cid:16) E (cid:2) tr( U N ( T ) j tr( U N ( T ) k ) (cid:3) − E (cid:104) tr( U N ( T ) j ) (cid:105) E (cid:2) tr( U N ( T ) k ) (cid:3)(cid:17) = e − ( j + k ) T ∞ (cid:88) n =0 ( − T ) n n ! (cid:18) S ((1 . . . j ) × (1 . . . k ) , n, − n (cid:88) n =0 (cid:18) nn (cid:19) S ((1 . . . j ) , n , S ((1 . . . k ) , n − n , (cid:19) = 0 , where the last equality comes from Proposition 5.3 of [19]. Each term of the sumis indeed zero and heuristically, it means that a path without defect startingfrom (1 . . . j ) × (1 . . . k ) is simply obtained by “shuffling” two paths withoutdefect from each of the two cycles in their respective symmetric group.More interesting for us is the fact we can also deduce from (26) that κ j,k ( T ) (def) =lim N →∞ N (cid:16) E (cid:2) tr( U N ( T ) j tr( U N ( T ) k ) (cid:3) − E (cid:104) tr( U N ( T ) j ) (cid:105) E (cid:2) tr( U N ( T ) k ) (cid:3)(cid:17) = e − ( j + k ) T ∞ (cid:88) n =0 ( − T ) n n ! S (cid:48) ((1 . . . j ) × (1 . . . k ) , n, , (27)where, σ ∈ S j , τ ∈ S k and n ≥ S (cid:48) ( σ × τ, n, 1) = S ( σ × τ, n, − n (cid:88) n =0 (cid:18) nn (cid:19)(cid:18) S ( σ, n , S ( τ, n − n , 0) + S ( σ, n , S ( τ, n − n , (cid:19) . Thus defined, S (cid:48) ( σ × τ, n, 1) is the number of paths of length n starting from σ × τ such that the unique step which increases the distance to the identity isthe multiplication by a transposition which exchanges an element of { , . . . , j } { j + 1 , . . . , j + k } . Thus, heuristically, the unique step whichis a coalescence is a coalescence between σ and τ .Our goal is now to show the following combinatorial identity Proposition 10.1. For any integers j, k ≥ , and n ≥ , we have S (cid:48) ((1 . . . j ) × (1 . . . k ) , n + 1 , 1) = jk S ((1 . . . j + k ) , n, j j − (cid:88) l =1 n (cid:88) p =0 (cid:18) np (cid:19) S ((1 . . . l ) , p, S (cid:48) ((1 . . . j − l ) × (1 . . . k ) , n − p, k k − (cid:88) m =1 n (cid:88) q =0 (cid:18) nq (cid:19) S ((1 . . . m ) , q, S (cid:48) ((1 . . . j ) × (1 . . . k − m ) , n − q, . The combinatorial interpretation of this identity is the following : let usconsider a path of length n + 1 from (1 . . . j ) × (1 . . . k ) whose unique stepincreasing the distance to identity is a true coalescence between the two cycles.The first step of such a path can be of three kinds, corresponding respectivelyto the three terms of the right hand-side : • either it coalesces the cycles, creating a ( j + k )-cycle, and this can be doneby choosing an element in each cycle. Then the path can be completed byany path of length n without defect from a ( j + k )-cycle. • either it cuts the cycle (1 . . . j ) into two cycles, one of length l that willthen be cut p times without being affected by the coalescence and anotherof length j − l which contains the element which will be exchanged withan element of { j + 1 , . . . , j + k } during the coalescing step. • either, symmetrically, it cuts the cycle (1 . . . k ).We will hereafter propose a rigorous proof of this identity through the freestochastic calculus tools introduced above in the paper. It should be notedthat the combinatorics which we investigate here is related to that of annularnoncrossing partitions introduced by J. Mingo and A. Nica [22]. Proof. Let the integers j, k ≥ T ≥ κ j,k ( T ) as defined in (27), if we denote, for any r ∈ Z , by f r : U → C the function given by f r ( z ) = z r , then, from Definition 2.4 and Theorem 2.6,we get κ j,k ( T ) = σ T ( f j , f k ) and from (19), it can be reexpressed as κ j,k ( T ) = − jk τ j,k ( T ) . Now, from Lemma 9.7, we get immediately˙ κ j,k ( T ) = − jk µ j + k ( T ) − j + k κ j,k ( T ) − j j − (cid:88) l =1 µ l ( T ) σ j − l,k ( T ) − k k − (cid:88) m =1 µ m ( T ) κ j,k − m ( T ) , 35o that we get immediately the anounced result, as we know from [19] that, forany r ∈ N ∗ , µ r ( T ) = e − r T ∞ (cid:88) n =0 ( − T ) n n ! S ((1 . . . r ) , n, κ j,k ( T ) = − j + k κ j,k ( T ) − e − ( j + k ) T ∞ (cid:88) n =0 ( − T ) n n ! S (cid:48) ((1 . . . j ) × (1 . . . k ) , n + 1 , . In principle, any computation involving functions invariant by conjugationon the unitary group can be performed by using harmonic analysis, that is, therepresentation theory of the unitary group. In this section, we use this approachto prove the following formula, which yields for each N ≥ SU ( N ). Withthe help of Section 7, it is easy to deduce the analogous result for the Brownianmotion on U ( N ). Theorem 10.2. Let N ≥ be an integer. Consider, on SU ( N ) , the Brownianmotion ( V N ( t )) t ≥ associated with the scalar product (cid:104) X, Y (cid:105) su ( N ) = N Tr( X ∗ Y ) on su ( N ) . Let n and m be positive integers. Assume that N ≥ n + m + 1 . Then E (cid:104) Tr( V N ( t ) n )Tr( V N ( t ) m ) (cid:105) = nδ n,m + ( − n + m e − ( n + m ) t − n ( n − m ( m − N t − ( n − m )2 N t n − (cid:88) r =0 m − (cid:88) r =0 (cid:20) ( − r + r e − nr t (cid:18) n − r (cid:19)(cid:18) N + r n (cid:19) e − nr t (cid:18) m − r (cid:19)(cid:18) N + r m (cid:19) ( N + r + r + 1)( N − n − m + r + r + 1)( N − n + r + r + 1)( N − m + r + r + 1) (cid:21) . The basic strategy for the proof is to expand the heat kernel and the tracesin the basis of Schur functions, and then to use the multiplication rules forSchur functions and their orthogonality properties. The multiplication rules areexpressed by the Littlewood-Richardson formula and they are rather compli-cated. Fortunately, in the present situation, the Young diagrams which occurare simple enough for the computation to be tractable.Let us recall the fundamental facts about Schur functions. Details can befound in [10]. A Young diagram is a non-increasing sequence of non-negativeintegers. If λ = ( λ ≥ . . . . . . λ k > 0) is such a sequence, we call k the lengthof λ and denote it by (cid:96) ( λ ). The set of Young diagrams of length at most k isdenoted by N k ↓ . We draw Young diagrams downwards in rows, according to theconvention illustrated by the left part of Figure 2.36 − rr Figure 2: The Young diagram on the left is (7 , , , , η n,r = ( n − 1) 1 r . The Schur function s λ is a symmetric function which, when evaluated onstrictly less than (cid:96) ( λ ) variables, yields 0. Whenever (cid:96) ( λ ) ≤ N , the function s λ is well defined and non-zero on SU ( N ). Its value s λ ( I N ) at the identity matrixin particular is a positive integer, which is the dimension of the irreduciblerepresentation of SU ( N ) of which s λ is the character. Another number attachedto λ will play an important role for us, which is the non-negative real number c ( λ ) such that ∆ s λ = − c ( λ ) s λ .It happens that distinct Young diagrams yield the same function on SU ( N ) :if λ and µ are Young diagrams such that (cid:96) ( λ ) , (cid:96) ( µ ) ≤ N , then s λ = s µ if and onlyif there exists l ∈ Z such that λ = µ + ( l, . . . , l ) = µ + l N . In fact, if ρ λ and ρ µ are the representations of U ( N ) corresponding to λ and µ , then ρ λ = ρ µ ⊗ det ⊗ l and the restrictions of these representations to SU ( N ) are equal.Finally, we need to use the decomposition of the heat kernel and the function U (cid:55)→ Tr( U n ) in terms of Schur functions. For the latter, we introduce a class ofYoung diagrams called hooks. For all n ≥ r ∈ { , . . . , n − } , we define η n,r = ( n − r, , . . . , (cid:124) (cid:123)(cid:122) (cid:125) r ) = ( n − r ) 1 r , which is depicted on the right part of Figure 2.The heat kernel at time t on SU ( N ) is the density, denoted by Q t : SU ( N ) → R , of the distribution of V N ( t ) with respect to the Haar measure. Proposition 10.3. Choose N ≥ and U ∈ SU ( N ) . Then the following equali-ties hold.1. For all n ≥ , Tr( U n ) = n − (cid:88) r =0 ( − r s η n,r ( U ) .2. For all t ≥ , Q t ( U ) = (cid:88) λ ∈ N N − ↓ e − c ( λ )2 t s λ ( I N ) s λ ( U ) . The proof of the first equality can be found in [21], the proof of the secondin [20]. The expectation that we want to compute in order to prove Theorem370.2 is thus equal to E (cid:104) Tr( V N ( t ) n )Tr( V N ( t ) m ) (cid:105) = (cid:88) λ ∈ N N − ↓ e − c ( λ )2 t s λ ( I N ) n − (cid:88) r =0 m − (cid:88) r =0 ( − r + r (cid:90) SU ( N ) s λ ( U ) s η n,r ( U ) s η m,r ( U ) dU. The multiplication of Schur functions is governed by the Littlewood-Richardsonformula, which describes a non-negative integer N γα,β for each triple of Youngdiagrams α, β, γ , in such a way that s α s β = (cid:88) γ N γα,β s γ . Using these coefficients, the integral above can be rewritten as (cid:90) SU ( N ) s λ ( U ) s η n,r ( U ) s η m,r ( U ) dU = (cid:88) γ N γλ,η n,r (cid:90) SU ( N ) s γ ( U ) s η m,r ( U ) dU = (cid:88) γ N γλ,η n,r (cid:88) l ≥ γ = η m,r + l N = (cid:88) l ≥ N η m,r + l N λ,η n,r . Thus, we need to compute n − (cid:88) r =0 m − (cid:88) r =0 ( − r + r (cid:88) l ≥ N η m,r + l N λ,η n,r . (28)It turns out that a slightly more general computation is simpler to perform : wecompute the Littlewood-Richardson coefficient N βα,η n,r for all α, β and all n, r .Let us introduce some notation.Let α = ( α , . . . ) and β = ( β , . . . ) be two Young diagrams. Set | α | = (cid:80) i α i and | β | = (cid:80) i β i . We assume that α ⊂ β , that is, α i ≤ β i for all i . Then wedenote by β/α the set of boxes of the graphical representation of β which arenot contained in α . We say that a subset of β/α is connected if one can go fromany box to any other inside this subset by a path which jumps from a box toanother only when they share an edge.We denote by k ( β/α ) the number of connected components of β/α . Also,we define v ( β/α ) as the number of boxes of β/α which are such that the boxlocated immediately above also belongs to β/α . Alternatively, this is the numberof distinct occurrences of the motif formed by two consecutive boxes one abovethe other in β/α .Our main combinatorial result is the following.38 roposition 10.4. Let α and β be two Young diagrams. Let η n,r be a hook.Then N βα,η n,r is non-zero if and only if the following conditions are satisfied: α ⊂ β , | β | = | α | + n , β/α contains no × square, and v ( β/α ) ≤ r ≤ v ( β/α ) + k ( β/α ) − . In this case, N βα,η n,r = (cid:0) k ( β/α ) − r − v ( β/α ) (cid:1) .Proof. According to the Littlewood-Richardson rule, N βα,η n,r is the number ofstrict expansions of α by η n,r which yield β , that is, the number of fillings of β/α with the boxes of η n,r such that the following conditions are satisfied:1. for all s ≥ 1, the union of α and the boxes of β/α filled by the first s rows of η n,r is a Young diagram,2. no two boxes of the first row of η n,r are put in the same column of β/α ,3. if one goes through the boxes of β/α from right to left and from top tobottom, writing for each box the number of the row of η n,r from which is issuedthe box which has been used to fill it, one obtains a sequence which starts with1, and in which all other numbers 2 , . . . , r appear, not necessarily consecutively,in this order.It is important to notice that, according to the third rule, a strict expansionof α by a hook which yields β is completely characterized by the set of boxes of β/α which are filled by boxes issued from the first row of the hook. We say forshort that these boxes of β/α are filled by the first row .The first two conditions α ⊂ β and | β | = | α | + n are obviously implied bythis rule. A less trivial implication is that there cannot exist a strict expansionif β/α contains a 2 × η n,r . This contradicts the third rule.Let us assume that β/α contains no 2 × β/α is a “snake” (see Figure 3). Figure 3: White boxes must be filled by boxes issued from the first row of η n,r . Greyboxes cannot. The black box may or may not, except if this snake is the topmost connectedcomponent of β/α , in which case it must also be filled by a box issued from the first row of η n,r . Any box of such a snake which has a box on its right must be filled by thefirst row. These boxes are the white boxes in Figure 3. Any box located belowa white box cannot be filled by the first row. These boxes are the grey boxesin Figure 3. Only one box is not in one of these two cases, the top-right box ofthe snake. In the topmost connected component of β/α the third rule impliesthat this box must be filled by the first row.39inally, if the first three conditions are satisfied, then β/α contains one boxin each connected component, except the topmost one, which can either be filledby the first row or not. The minimal number of boxes which are not filled bythe first row is the number of grey boxes, which we have denoted by v ( β/α ).This is the minimal value of r for which there exists a strict expansion of α by η n,r which yields β . Moreover, for this value of r , the expansion is unique, sincethe boxes filled by the first row are completely determined. Similarly, the max-imal value of r is v ( β/α ) + k ( β/α ) − 1. For r between these two bounds, thereare exactly as many expansions as there are choices of which snakes have theirtop-right box filled by the first row. There are thus (cid:0) k ( β/α ) − r − v ( β/α ) (cid:1) such expansions. Corollary 10.5. Let α and β be two Young diagrams. Choose n ≥ . Then n − (cid:88) r =0 ( − r N βα,η n,r = ( − v ( β/α ) if α ⊂ β , | β | = | α | + n , β/α contains no × square and is connected. Otherwise,it is equal to 0.Proof. If the first three conditions are not satisfied, then N βα,η n,r = 0 for all r = 0 . . . n − 1. Let us assume that they are satisfied. Then, by the previousproposition, the sum above is equal to v ( β/α )+ k ( β/α ) − (cid:88) r = v ( β/α ) ( − r (cid:18) k ( β/α ) − r − v ( β/α ) (cid:19) , which is equal to 0 unless k ( β/α ) = 1. In this case, only one term of the sum isnon-zero, for r = v ( β/α ).We apply now this result when β is of the sum of a hook and a rectangle. Lemma 10.6. Consider n ≥ m ≥ , r ∈ { , . . . , m − } , and N ≥ m + n . Forall r ∈ { , . . . , n − } , define λ Nm,r ,n,r = ( n − r + m − r ) ( n − r + 1) r ( n − r ) N − r − r − ( n − r − r . Then, for all λ ∈ N N − ↓ and all l ≥ , n − (cid:88) r =0 ( − r N η m,r + l N λ,η n,r = (cid:26) ( − n − l if l ∈ { , . . . , n } and λ = λ Nm,r ,n,n − l , otherwise.Moreover, when l ∈ { , . . . , n } , the only non-zero term of the sum is the termcorresponding to r = n − l .Finally, if n = m , then N η m,r λ,η n,r = 1 if r = r and λ is the empty diagram,and otherwise. m − r r r n − r Figure 4: The diagram λ Nm,r ,n,r . Proof. Let us first consider the case n > m . In this case, according to Corollary10.5, in order for the sum to be non-zero, λ must be a Young diagram of lengthat most N − 1, contained in η m,r + l N , such that ( η m,r + l N ) /λ contains no2 × n > m , l must be positive, so that thediagram η m,r + l N has length N whereas λ has length at most N − 1. Thus, the N -th row of ( η m,r + l N ) /λ is not empty, it has actually length l . In particular, | η m,r + l N | − | λ | ≥ l . If l > n , all the Littlewood-Richardson coefficientsappearing in the sum are zero. Otherwise, if l ≤ n , there is exactly one way tochoose λ a subdiagram of η m,r + l N such that all conditions are satisfied : it is λ = λ Nm,r ,n,n − l .When n = m , nothing changes for l ≥ 1. However, the sum may be non-zeroeven for l = 0. The diagram λ must be the empty diagram and it is easy tocheck that N η m,r ∅ ,η n,r = δ n,m δ r ,r .We can now go on to compute (28). We find the following result. Proposition 10.7. Let N , n and m be three positive integers. Assume that n ≥ m and N ≥ n + m + 1 . Then E (cid:104) Tr( V N ( t ) n )Tr( V N ( t ) m ) (cid:105) = nδ n,m + n − (cid:88) r =0 m − (cid:88) r =0 ( − r + r e − c ( λNm,r ,n,r t s λ Nm,r ,n,r ( I N ) . Proof. We have n − (cid:88) r =0 m − (cid:88) r =0 ( − r + r (cid:88) l ≥ N η m,r + l N λ,η n,r = nδ m,n λ = ∅ + m − (cid:88) r =0 ( − r n (cid:88) l =1 ( − n − l λ = λ Nm,r ,n,n − l = nδ m,n λ = ∅ + n − (cid:88) r =0 m − (cid:88) r =0 ( − r + r λ = λ Nm,r ,n,r . c ( λ Nm,r ,n,r ) and s λ Nm,r ,n,r ( I N ). This is by no means complicated but slightly tedious. We recallthe general formulae, give the results in this particular case and invite the readerto check them if s/he feels inclined to do so. Lemma 10.8. Consider n, m ≥ , r ∈ { , . . . , n − } and r ∈ { , . . . , m − } .Then the following identities hold. c ( λ Nm,r ,n,r ) = n + n ( n − r − N + m + m ( m − r − N − ( n − m ) N ,s λ Nm,r ,n,r ( I N ) = ( N − r − r − N + n + m − r − r − N + n − r − r − N + m − r − r − × (cid:18) n − r (cid:19)(cid:18) N + n − r − n (cid:19)(cid:18) m − r (cid:19)(cid:18) N + m − r − m (cid:19) . Proof. The general formulae are the following : for all α ∈ N N ↓ , one has c ( α ) = 1 N N (cid:88) i =1 α i + (cid:88) ≤ i Anderson, G. W., and Zeitouni, O. A CLT for a band matrix model. Probab. Theory Related Fields 134 , 2 (2006), 283–338.[2] Benaych-Georges, F. Central limit theorems for the brownian motionon large unitary groups. Preprint, arXiv:0904.1681 (2009).[3] Biane, P. 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