Second order freeness and fluctuations of random matrices: II. Unitary random matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.     

Limit distributions of random matrices

We study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation. In particular, we are concerned with the ensemble of symmetric blocks of independent Hermitian random matrices which are asymptotically free, asymptotically free from diagonal deterministic matrices, and whose norms are uniformly bounded almost surely. This class contains symmetric blocks of unitarily invariant Hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line. Our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability. We show that the associated matricially free Gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices, including non-Hermitian Gaussian matrices and matrices of Wishart type.     

On a class of free Lévy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Lévy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.     

Cumulants for Random Matrices as Convolutions on the Symmetric Group,II

In a previous paper we defined some “cumulants of matrices” which naturally converge toward the free cumulants of the limiting non commutative random variables when the size of the matrices tends to infinity. Moreover these cumulants satisfied some of the characteristic properties of cumulants whenever the matrix model was invariant under unitary conjugation. In this paper we present the fitting cumulants for random matrices whose law is invariant under orthogonal conjugation. The symplectic case could be carried out in a similar way.     

Asymptotic properties of random matrices and pseudomatrices

We study the asymptotics of sums of matricially free random variables, called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called ‘matricially free Gaussian operators’. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are ‘asymptotically matricially free’ whereas the corresponding symmetric random blocks are ‘asymptotically symmetrically matricially free’, where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, block-lower-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.     

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.     

算子值酉随机矩阵及其渐近自由性

该文在算子值非交换概率空间上引入半标准酉随机矩阵的概念, 证明了它是算子值Haar酉元的矩阵模型,并给出了半标准酉随机矩阵的渐近自由判定定理.     

Limit theorems for random permanents with exchangeable structure

Permanents of random matrices extend the concept of U-statistics with product kernels. In this paper, we study limiting behavior of permanents of random matrices with independent columns of exchangeable components. Our main results provide a general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries. The method of the proofs is based on a Hoeffding-type orthogonal decomposition of a random permanent function. The decomposition allows us to relate asymptotic behavior of permanents to that of elementary symmetric polynomials based on triangular arrays of rowwise independent rv's.     

Precise asymptotics for beta ensembles

We consider the extremal (largest and smallest) eigenvalues of random matrices in the β-Hermite and β-Laguerre ensembles. Using the general β Tracy-Widom law together with Ledoux and Rider’s small deviation inequalities for β-ensembles, we obtain some precise asymptotic results in both settings. This complements Su’s results for the largest eigenvalue of Gaussian and Laguerre unitary ensembles.     

On the maximal domain of attraction of Tracy–Widom distribution for Gaussian unitary ensembles

The asymptotic distribution of the largest eigenvalue of various classes of random matrices has been shown to have the Tracy–Widom distribution. In this article, we prove that the standardised maximum of an independent and identically distributed sequence of random variables having the Tracy–Widom distribution arising from the Gaussian unitary ensemble belongs to the Gumbel domain of attraction.     

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

In this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices.     

Zariski closure and the dimension of the Gaussian low of the product of random matrices. I

Summary We prove the Central Limit Theorem for products of i.i.d. random matrices. The main aim is to find the dimension of the corresponding Gaussian law. It turns out that ifG is the Zariski closure of a group generated by the support of the distribution of our matrices, and ifG is semi-simple, then the dimension of the Gaussian law is equal to the dimension of the diagonal part of Cartan decomposition ofG.In this article we present a detailed exposition of results announced in [GGu]. For reasons explained in the introduction, this part is devoted to the case ofSL(m, ) group. The general semi-simple Lie group will be considered in the second part of the work.The central limit theorem for products of independent random matrices is our main topic, and the results obtained complete to a large extent the general picture of the subject.The proofs rely on methods from two theories. One is the theory of asymptotic behaviour of products of random matrices itself. As usual, the existence of distinct Lyapunov exponents is the most important fact here. The other is the theory of algebraic groups. We want to point out that algebraic language and methods play a very important role in this paper.In fact, this mixture of methods has already been used for the study of Lyapunov exponents in [GM1, GM2, GR3]. We believe that it is impossible to avoid the algebraic approach if one aims to obtain complete and effective answers to natural problems arising in the theory of products of random matrices.In order also to present the general picture of the subject we describe several results which are well known. Some of these can be proven for stationary sequences of matrices, others are true also for infinite dimensional operators (see e.g. [BL, O, GM2, L, R]). But our main concern is with independent matrices, in which case very precise and constructive statements can be obtained.     

Stochastic aspects of the unitary dual group

In this note, we study the asymptotic properties of the ?-distribution of traces of some matrices, with respect to the free Haar trace on the unitary dual group. The considered matrices are powers of the unitary matrix generating the Brown algebra. We proceed in two steps, first computing the free cumulants of any R-cyclic family, then characterizing the asymptotic ?-distributions of the traces of powers of the generating matrix, thanks to these free cumulants. In particular, we obtain that these traces are asymptotic ?-free circular variables.     

Random Block Matrices and Matrix Orthogonal Polynomials

In this paper we consider random block matrices, which generalize the general beta ensembles recently investigated by Dumitriu and Edelmann (J. Math. Phys. 43:5830–5847, 2002; Ann. Inst. Poincaré Probab. Stat. 41:1083–1099, 2005). We demonstrate that the eigenvalues of these random matrices can be uniformly approximated by roots of matrix orthogonal polynomials which were investigated independently from the random matrix literature. As a consequence, we derive the asymptotic spectral distribution of these matrices. The limit distribution has a density which can be represented as the trace of an integral of densities of matrix measures corresponding to the Chebyshev matrix polynomials of the first kind. Our results establish a new relation between the theory of random block matrices and the field of matrix orthogonal polynomials, which have not been explored so far in the literature.     

On the characteristic map of finite unitary groups

In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne–Lusztig theory and Ennola's efforts to generalize Green's work, and from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving a model of Deligne–Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group.     

Density of eigenvalues and its perturbation invariance in unitary ensembles of random matrices

We generally study the density of eigenvalues in unitary ensembles of random matrices from the recurrence coefficients with regularly varying conditions for the orthogonal polynomials. By using a new method, we calculate directly the moments of the density (which has been obtained in the work of Nevai and Dehesa, Van Assche and others on asymptotic zero distribution), and prove that scaling eigenvalues converge weakly, in probability and almost surely to the Nevai–Ullmann measure. Furthermore, we can prove that the density is invariant when the weight function is perturbed by a polynomial.     

Integrable operators and the squares of Hankel operators

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. This paper gives sufficient conditions for an integrable operator to be the square of a Hankel operator, and applies the condition to the Airy, associated Laguerre, modified Bessel and Whittaker functions.     

The characterizations of a semicircle law by the certain freeness in a C*-probability space

In usual probability theory, various characterizations of the Gaussian law have been obtained. For instance, independence of the sample mean and the sample variance of independently identically distributed random variables characterizes the Gaussian law and the property of remaining independent under rotations characterizes the Gaussian random variables. In this paper, we consider the free analogue of such a kind of characterizations replacing independence by freeness. We show that freeness of the certain pair of the linear form and the quadratic form in freely identically distributed noncommutative random variables, which covers the case for the sample mean and the sample variance, characterizes the semicircle law. Moreover we give the alternative proof for Nica's result that the property of remaining free under rotations characterizes a semicircular system. Our proof is more direct and straightforward one. Received: 12 February 1997 / Revised version: 16 June 1998     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Remark on the norm of random Hankel matrices

In recent years, the asymptotic properties of structured random matrices have attracted the attention of many experts involved in probability theory. In particular, R. Adamczak (J. Theor. Probab., Vol. 23, 2010) proved that, under fairly weak conditions, the squared spectral norms of large square Hankel matrices generated by independent identically distributed random variables grow with probability 1, as Nln(N), where N is the size of a matrix. On the basis of these results, by using the technique and ideas of Adamczak’s paper cited above, we prove that, under certain constraints, the squared spectral norms of large rectangular Hankel matrices generated by linear stationary sequences grow almost certainly no faster than Nln(N), where N is the number of different elements in a Hankel matrix. Nekrutkin (Stat. Interface, Vol. 3, 2010) pointed out that this result may be useful for substantiating (by using series of perturbation theory) so-called “signal subspace methods,” which are often used for processing time series. In addition to the main result, the paper contains examples and discusses the sharpness of the obtained inequality.