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 $\beta$ models, if we define the determinant as the product of charges.Comment: 51 pages ; it replaces and extends arXiv:math/0607767 and arXiv:math/0509021 Third version: corrected constants in Theorem 3.

Asymptotical behaviour of the presence probability in branching random walks and fragmentations

For a subcritical Galton-Watson process $(\zeta_n)$, it is well known that under an $X \log X$ condition, the quotient $P(\zeta_n > 0)/ E\zeta_n$ 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.Comment: 15 pages, companion paper of math.PR/040954

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 $\beta$- 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

Discretization methods for homogeneous fragmentations

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations. On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\N$. Spatial discretization is especially well-suited to develop directly for continuous times the conceptual method of probability tilting of Lyons, Pemantle and Peres.Comment: 21 page

Large deviations for near-extreme eigenvalues in the beta-ensembles

For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.Comment: We fixed typos and changed Remarks 2.13 and 2.1

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.Comment: Random matrices: Theory and Applications (RMTA) To appear (2012) http://www.editorialmanager.com/rmta

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_{\delta,\beta}^{(n)} \prod_{1\leq k with $\Re \delta > -1/2$. 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 $(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation $(\gamma_0, >..., \gamma_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients. If $\gamma_0, ..., \gamma_{n-1}$ 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 $\delta = \delta(n)$ with \delta(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.Comment: New section on large deviations for the empirical spectral distribution, Corrected value for the limiting free energ