Operator-valued spectral measures and large deviations
We study large deviation properties of random matricial spectral measures.
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arXiv subjects
Alain Rouault
Publications and source records attributed to Alain Rouault.
Bridges and random truncations of random matrices
We continue to study the squared Frobenius norm of a submatrix of a $n \times n$ random unitary matrix. When the choice of the submatrix is deterministic and its size is $[ns] \times [nt]$, we proved in a previous paper that, after centering and without any rescaling, the two-parameter process converges in distribution to a bivariate Brownian bridge. Here, we consider Bernoulli independent choices of rows and columns with respective parameters $s$ and $t$. We prove by subordination that after centering and rescaling by $n^{-1/2}$, the process converges to another Gaussian process.
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Limit theorems for orthogonal polynomials related to circular ensembles
For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t in [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations.
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Random truncations of Haar distributed matrices and bridges
Let $U$ be a Haar distributed matrix in $\mathbb U(n)$ or $\mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process \[T^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2\] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.
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Large Deviations for Random Matricial Moment Problems
We consider the moment space $\mathcal{M}_n^{K}$ corresponding to $p \times p$ complex matrix measures defined on $K$ ($K=[0,1]$ or $K=\D$). We endow this set with the uniform law. We are mainly interested in large deviations principles (LDP) when $n \rightarrow \infty$. First we fix an integer $k$ and study the vector of the first $k$ components of a random element of $\mathcal{M}_n^{K}$. We obtain a LDP in the set of $k$-arrays of $p\times p$ matrices. Then we lift a random element of $\mathcal{M}_n^{K}$ into a random measure and prove a LDP at the level of random measures. We end with a LDP on Carthéodory and Schur random functions. These last functions are well connected to the above random measure. In all these problems, we take advantage of the so-called canonical moments technique by introducing new (matricial) random variables that are independent and have explicit distributions.
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Truncations of Haar distributed matrices, traces and bivariate Brownian bridges
Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = \sum_{i \leq \lfloor ns \rfloor, j \leq \lfloor nt\rfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.
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Large Deviations for Random Spectral Measures and Sum Rules
We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $β$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit.
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Generalized Dirichlet distributions on the ball and moments
The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit some known results of this flavour with a new point of view. Roughly speaking, we will endow the ball with some kind of Dirichlet distribution that generalizes the uniform one and will follow the method developed in Skibinsky (1967), Chang et al. (1993) in the context of the randomized moment space. The main idea is to build a suitable coordinate change involving independent random variables. Moreover, we will shed light on a nice connection between the randomized balls and the randomized moment space.
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Martingales and Rates of Presence in Homogeneous Fragmentations
The main focus of this work is the asymptotic behavior of mass-conservative homogeneous fragmentations. Considering the logarithm of masses makes the situation reminiscent of branching random walks. The standard approach is to study {\bf asymptotical} exponential rates. For fixed $v > 0$, either the number of fragments whose sizes at time $t$ are of order $\e^{-vt}$ is exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the probability of presence of such fragments is exponentially decreasing with rate $C(v) < 0$, for some concave function $C$. In a recent paper, N. Krell considered fragments whose sizes decrease at {\bf exact} exponential rates, i.e. whose sizes are confined to be of order $\e^{-vs}$ for every $s \leq t$. In that setting, she characterized the effective rates. In the present paper we continue this analysis and focus on probabilities of presence, using the spine method and a suitable martingale.
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Ewens measures on compact groups and hypergeometric kernels
On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as $n$ tends to infinity to a limit kernel at the singularity.
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Circular Jacobi Ensembles and deformed Verblunsky coefficients
Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{δ,β}^{(n)} \prod_{1\leq k -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(α_0, ..., α_{n-1})$. We introduce here a deformation $(γ_0, >..., γ_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(γ_0)... r(γ_{n-1})$ of elementary reflections parameterized by these coefficients. If $γ_0, ..., γ_{n-1}$ are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow to prove that, in the regime $δ= δ(n)$ with $δ(n)/n \to \dd$, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution.
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Canonical moments and random spectral measures
We study some connections between the random moment problem and the random matrix theory. A uniform draw in a space of moments can be lifted into the spectral probability measure of the pair (A,e) where A is a random matrix from a classical ensemble and e is a fixed unit vector. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix grows. The rate function for these large deviations involves the reversed Kullback information.
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Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles
We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If $n$ is the size of the sample, $r\leq n$ the number of variates and $X_{n,r}$ such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of $\det X_{n,r}$ into a product of $r$ independent gamma or beta random variables. For $n$ fixed, we study the evolution as $r$ grows, and then take the limit of large $r$ and $n$ with $r/n = t \leq 1$. We derive limit theorems for the sequence of {\it processes with independent increments} $\{n^{-1} \log \det X_{n, \lfloor nt\rfloor}, t \in [0, T]\}_n$ for $T \leq 1$.. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed $t$) with those obtained by the spectral method. Actually, all the results hold true for $β$ models, if we define the determinant as the product of charges.
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The characteristic polynomial on compact groups with Haar measure : some equalities in law
This note presents some equalities in law for $Z_N:=\det(\Id-G)$, where $G$ is an element of a subgroup of the set of unitary matrices of size $N$, endowed with its unique probability Haar measure. Indeed, under some general conditions, $Z_N$ can be decomposed as a product of independent random variables, whose laws are explicitly known. Our results can be obtained in two ways : either by a recursive decomposition of the Haar measure or by previous results by Killip and Nenciu on orthogonal polynomials with respect to some measure on the unit circle. This latter method leads naturally to a study of determinants of a class of principal submatrices.
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On the reduction of a random basis
For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra & Lovász}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r\_j^{(n)}-1)\_j$ tends in distribution in some sense to an explicit process $({\mathcal R}\_j -1)\_j$; some properties of this latter are provided.
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Pathwise asymptotic behavior of random determinants in the uniform Gram and Wishart ensembles
This paper concentrates on asymptotic properties of determinants of some random symmetric matrices. If B_{n,r} is a n x r rectangular matrix and B_{n,r}' its transpose, we study det (B_{n,r}'B_{n,r}) when n,r tends to infinity with r/n \to c\in (0,1). The r column vectors of B_{n,r} are chosen independently, with common distribution ν_n. The Wishart ensemble corresponds to ν_n = {\cal N}(0, I_n), the standard normal distribution. We call uniform Gram ensemble the ensemble corresponding to ν_n = σ_n, the uniform distribution on the unit sphere `S_{n-1}. In the Wishart ensemble, a well known Bartlett's theorem decomposes the above determinant into a product of chi-square variables. The same holds in the uniform Gram ensemble. This allows us to study the process \{\frac{1}{n}\log \det\big(B_{n,\lfloor nt\rfloor}'B_{n,\lfloor nt\rfloor}\big), t \in [0,1]\} and its asymptotic behavior as n\to \infty: a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed t) with those obtained by the spectral method.
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Martingales and Profile of Binary Search Trees
We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.
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Asymptotical behaviour of the presence probability in branching random walks and fragmentations
For a subcritical Galton-Watson process $(ζ_n)$, it is well known that under an $X \log X$ condition, the quotient $P(ζ_n > 0)/ Eζ_n$ has a finite positive limit. There is an analogous result for a (one-dimensional) supercritical branching random walk: when $a$ is in the so-called subcritical speed area, the probability of presence around $na$ in the $n$-th generation is asymptotically proportional to the corresponding expectation. In Rouault (1993) this result was stated under a natural $X \log X$ assumption on the offspring point process and a (unnatural) condition on the offspring mean. Here we prove that the result holds without this latter condition, in particular we allow an infinite mean and a dimension $d \geq 1$ for the state-space. As a consequence the result holds also for homogeneous fragmentations as defined in Bertoin (2001), using the method of discrete-time skeletons; this completes the proof of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an application to conditioning on the presence allows to meet again the probability tilting and the so-called additive martingale.
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