### A free analogue of the transportation cost inequality on the circle

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We present an analogue of the Harer–Zagier recursion formula for the moments of the gaussian Orthogonal Ensemble in the form of a five term recurrence equation. The proof is based on simple gaussian integration by parts and differential equations on Laplace transforms. A similar recursion formula holds for the gaussian Symplectic Ensemble. As in the complex case, the result is interpreted as a recursion formula for the number of 1-vertex maps in locally orientable surfaces with a given number of...

Linear relations, containing measurement errors in input and output data, are taken into account in this paper. Parameters of these so-called errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances. Such an estimate is highly non-linear. Moreover in some realistic situations, the errors cannot be considered as independent by nature. Weakly dependent ($\alpha $- and $\varphi $-mixing) disturbances, which are not necessarily stationary nor identically...

We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential. Our approach is based on a deformation of potential and on the use of the underlying integrable structures of the matrix model. The main results include the existence of a full asymptotic expansion in even powers of $N$ of the recurrence coefficients of the related orthogonal polynomials for a one-cut regular potential and the double scaling asymptotics of the free...

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the...

In this paper, we are concerned with the large $n$ limit of the distributions of linear combinations of the entries of a Brownian motion on the group of $n\times n$ unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic...

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given...

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of “symmetric exponential” processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity ${\Gamma}_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates...

Let $\left\{{X}_{ij}\right\}$, $i,j=\cdots $, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu $, $E|{X}_{11}{-\mu |}^{2}=1$ and $E|{X}_{11}{|}^{4}lt;\infty $. Consider sample covariance matrices (with/without empirical centering) $\mathcal{S}=\frac{1}{n}{\sum}_{j=1}^{n}({\mathbf{s}}_{j}-\overline{\mathbf{s}}){({\mathbf{s}}_{j}-\overline{\mathbf{s}})}^{T}$ and $\mathbf{S}=\frac{1}{n}{\sum}_{j=1}^{n}{\mathbf{s}}_{j}{\mathbf{s}}_{j}^{T}$, where $\overline{\mathbf{s}}=\frac{1}{n}{\sum}_{j=1}^{n}{\mathbf{s}}_{j}$ and ${\mathbf{s}}_{j}={\mathbf{T}}_{n}^{1/2}{({X}_{1j},...,{X}_{pj})}^{T}$ with ${\left({\mathbf{T}}_{n}^{1/2}\right)}^{2}={\mathbf{T}}_{n}$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $\mathcal{S}$ and $\mathbf{S}$ are different as $n\to \infty $ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u\u2099+{u}_{\infty}$ where uₙ and ${u}_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form ${S}_{\alpha}+i{S}_{\beta}$ where ${S}_{\alpha}$ and ${S}_{\beta}$ are free semicircular elements of variance α and β.