arXiv subjects

Mireille Capitaine

Publications and source records attributed to Mireille Capitaine.

At least 19 recordsLinked to original sources

arXiv2025

Strong Convergence of Multiplicative Brownian Motions on the General Linear Group

We consider the family of multiplicative Brownian motions $G_{λ,τ}$ on the general linear group introduced by Driver-Hall-Kemp. They are parametrized by the real variance $λ\in \mathbb{R}$ and the complex covariance $τ\in \mathbb{C}$ of the underlying elliptic Brownian motion. We show the almost sure strong convergence of the finite-dimensional marginals of $G_{λ,τ}$ to the corresponding free multiplicative Brownian motion introduced by Hall-Ho: as the dimension tends to infinity, not only does the noncommutative distribution converge almost surely, but the operator norm does as well. This result generalizes the work of Collins-Dahlqvist-Kemp for the special case $(λ,τ)=(1,0)$ which corresponds to the Brownian motion on the unitary group. Actually, this strong convergence remains valid when the family of multiplicative Brownian motions $G_{λ,τ}$ is considered alongside a family of strongly converging deterministic matrices.

math.PR↗

arXiv2024

Outliers of perturbations of banded Toeplitz matrices

Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $σ_n^2 / n$ in the asymptotic of high dimension and $σ_n$ converging to $σ\geq 0$. Our results complement and provide new proofs on recent progresses in the case $σ= 0$. For any $σ\geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem.

math.PR↗

arXiv2024

Strong convergence of tensor products of independent G.U.E. matrices

Given tuples of properly normalized independent $N\times N$ G.U.E. matrices $(X_N^{(1)},\dots,X_N^{(r_1)})$ and $(Y_N^{(1)},\dots,Y_N^{(r_2)})$, we show that the tuple $(X_N^{(1)}\otimes I_N,\dots,X_N^{(r_1)}\otimes I_N,I_N\otimes Y_N^{(1)},\dots,I_N\otimes Y_N^{(r_2)})$ of $N^2\times N^2$ random matrices converges strongly as $N$ tends to infinity. It was shown by Ben Hayes that this result implies that the Peterson-Thom conjecture is true.

math.OA↗

arXiv2021

Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices

We investigate the fluctuations around the mean of the Stieltjes transform of the empirical spectral distribution of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix. We obtain the convergence in distribution to a centred complex Gaussian process whose covariance is expressed in terms of operator-valued subordination functions.

math.PR↗

arXiv2020

Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked $N\times N$ complex Deformed Wigner matrix $M_N$: $M_N =W_N/\sqrt{N} + A_N$ where $W_N$ is an $N \times N$ Hermitian Wigner matrix whose entries have a law $μ$ satisfying a Poincaré inequality and the matrix $A_N$ is a block diagonal matrix, with an eigenvalue $θ$ of multiplicity one, generating an outlier in the spectrum of $M_N$. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of $M_N$ onto a unit eigenvector corresponding to $θ$ are not universal.

math.PR↗

arXiv2019

Outlier eigenvalues for non-Hermitian polynomials in independent i.i.d. matrices and deterministic matrices

We consider a square random matrix of size $N$ of the form $P(Y,A)$ where $P$ is a noncommutative polynomial, $A$ is a tuple of deterministic matrices converging in $\ast$-distribution, when $N$ goes to infinity, towards a tuple $a$ in some $\mathcal{C}^*$-probability space and $Y$ is a tuple of independent matrices with i.i.d. centered entries with variance $1/N$. We investigate the eigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where $c$ is a circular system which is free from $a$. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with those of $P(0,A)$.

math.PR↗

arXiv2019

Non universality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix

We study the fluctuations associated to the a.s. convergence, established by Belinschi-Bercovici-Capitaine, of the outliers of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the non universality phenomenon previously established for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.

math.PR↗

arXiv2018

On the outlying eigenvalues of a polynomial in large independent random matrices

Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A\_N,B\_N)$, where $A\_N$ and $B\_N$ are independent Hermitian random matrices and the distribution of $B\_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A\_N$ and $B\_N$ converge almost surely to deterministic probability measures $μ$ and $ν$, respectively. In addition, the eigenvalues of $A\_N$ and $B\_N$ are assumed to converge uniformly almost surely to the support of $μ$ and $ν,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A\_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A\_N,B\_N)$ converges to a certain deterministic probability measure $η$ (sometimes denoted $η=P^\square(μ,ν)$) and, when there are no spikes, the eigenvalues of $P(A\_N,B\_N)$ converge uniformly almost surely to the support of $η$. When spikes are present, we show that the eigenvalues of $P(A\_N,B\_N)$ still converge uniformly to the support of $η$, with the possible exception of certain isolated outliers whose location can be determined in terms of $μ,ν,P$, and the spikes of $A\_N$. We establish a similar result when $B\_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

math.OA↗

arXiv2017

Limiting eigenvectors of outliers for Spiked Information-Plus-Noise type matrices

We consider an Information-Plus-Noise type matrix where the Information matrix is a spiked matrix. When some eigenvalues of the random matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the spikes. Note that, in an Appendix, we present alternative versions of the earlier results of Bai and Silverstein about the lack of eigenvalues outside the support of the deterministic equivalent measure and of Capitaine about the exact separation phenomenon, where we remove some technical assumptions.

math.PR↗

arXiv2017

Spectral properties of polynomials in independent Wigner and deterministic matrices

On the one hand, we prove that almost surely, for large dimension, there is no eigenvalue of a Hermitian polynomial in independent Wigner and deterministic matrices, in any interval lying at some distance from the supports of a sequence of deterministic probability measures, which is computed with the tools of free probability. On the other hand, we establish the strong asymptotic freeness of independent Wigner matrices and any family of deterministic matrices with strong limiting distribution.

math.PR↗

arXiv2015

Outliers in the spectrum of large deformed unitarily invariant models

In this paper we characterize the possible outliers in the spectrum of large deformed unitarily invariant additive and multiplicative models, as well as the eigenvectors corresponding to them. We allow both the non-deformed unitarily invariant model and the perturbation matrix to have non-trivial limiting spectral measures and spiked outliers in their spectrum. We uncover a remarkable new phenomenon: a single spike can generate asymptotically several outliers in the spectrum of the deformed model. The free subordination functions play a key role in this analysis.

math.PR↗

arXiv2014

Outlier eigenvalues for deformed i.i.d. random matrices

We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has iid entries with variance 1/N. Under mild assumptions, as N grows, the empirical distribution of the eigenvalues of A+Y converges weakly to a limit probability measure βon the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function.

math.PR↗

arXiv2014

Exact separation phenomenon for the eigenvalues of large Information-Plus-Noise type matrices. Application to spiked models

We consider large Information-Plus-Noise type matrices of the form $M_N=(σ\frac{X_N}{\sqrt{N}}+A_N)(σ\frac{X_N}{\sqrt{N}}+A_N)^*$ where $X_N$ is an $n \times N$ ($n\leq N)$ matrix consisting of independent standardized complex entries, $A_N$ is an $n \times N$ nonrandom matrix and $σ>0$. As $N$ tends to infinity, if $n/N \rightarrow c\in ]0,1]$ and if the empirical spectral measure of $A_N A_N^*$ converges weakly to some compactly supported probability distribution $ν\neq δ_0$, Dozier and Silverstein established that almost surely the empirical spectral measure of $M_N$ converges weakly towards a nonrandom distribution $μ_{σ,ν,c}$. Bai and Silverstein proved, under certain assumptions on the model, that for some closed interval in $]0;+\infty[$ outside the support of $μ_{σ,ν,c}$ satisfying some conditions involving $A_N$, almost surely, no eigenvalues of $M_N$ will appear in this interval for all $N$ large. In this paper, we carry on with the study of the support of the limiting spectral measure previously investigated by Dozier and Silverstein and later by Vallet, Loubaton and Mestre and Loubaton and P. Vallet, and we show that, under almost the same assumptions as Bai and Silvertein, there is an exact separation phenomenon between the spectrum of $M_N$ and the spectrum of $A_NA_N^*$: to a gap in the spectrum of $M_N$ pointed out by Bai and Silverstein, it corresponds a gap in the spectrum of $A_NA_N^*$ which splits the spectrum of $A_NA_N^*$ exactly as that of $M_N$. We use the previous results to characterize the outliers of spiked Information-Plus-Noise type models.

math.PR↗

arXiv2012

Outliers in the spectrum of large deformed unitarily invariant models

We investigate the asymptotic behavior of the eigenvalues of the sum A+U*BU, where A and B are deterministic N by N Hermitian matrices having respective limiting compactly supported distributions μ, ν, and U is a random N by N unitary matrix distributed according to Haar measure. We assume that A has a fixed number of fixed eigenvalues (spikes) outside the support of μ, whereas the distances between the other eigenvalues of A and the support of μ, and between the eigenvalues of B and the support of ν, uniformly go to zero as N goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of A+U*BU outside the support of the limiting spectral measure, called outliers. This phenomenon is fully described in terms of free probability involving the subordination function related to the free additive convolution of μ and ν. Only finite rank perturbations had been considered up to now.

math.PR↗

arXiv2012

Additive/multiplicative free subordination property and limiting eigenvectors of spiked additive deformations of Wigner matrices and spiked sample covariance matrices

When some eigenvalues of a spiked multiplicative resp. additive deformation model of a Hermitian Wigner matrix resp. a sample covariance matrix separate from the bulk, we study how the corresponding eigenvectors project onto those of the perturbation. We point out that the inverse of the subordination function relative to the free additive resp. multiplicative convolution plays an important part in the asymptotic behavior.

math.PR↗

arXiv2011

Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a Poincaré inequality. The perturbation matrix is a deterministic Hermitian matrix whose spectral measure converges to some probability measure with compact support. We assume that this perturbation matrix has a fixed number of fixed eigenvalues (spikes) outside the support of its limiting spectral measure whereas the distance between the other eigenvalues and this support uniformly goes to zero as the dimension goes to infinity. We establish that only a particular subset of the spikes will generate some eigenvalues of the deformed model which will converge to some limiting points outside the support of the limiting spectral measure. This phenomenon can be fully described in terms of free probability involving the subordination function related to the additive free convolution of the limiting spectral measure of the perturbation matrix by a semi-circular distribution. Note that up to now only finite rank perturbations had been considered (even in the deformed GUE case).

math.PR↗

arXiv2011

Central limit theorems for eigenvalues of deformations of Wigner matrices

In this paper, we explain the dependance of the fluctuations of the largest eigenvalues of a Deformed Wigner model with respect to the eigenvectors of the perturbation matrix. We exhibit quite general situations that will give rise to universality or non universality of the fluctuations.

math.PR↗

arXiv2011

The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix $A_N$ is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ almost surely exit the limiting semicircle compact support as the size $N$ becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of $W_N$. On the other hand, when $A_N$ is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of $W_N$.

math.PR↗