Average Characteristic Polynomials of Determinantal Point Processes

We investigate the average characteristic polynomial $\mathbb E\big[\prod_{i=1}^N(z-x_i)\big]$ where the $x_i$'s are real random variables which form a determinantal point process associated to a bounded projection operator. For a subclass of point processes, which contains Orthogonal Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as $N$ goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from a theorem of Kuijlaars and Van Assche a unified way to describe the almost sure convergence for classical Orthogonal Polynomial Ensembles. As another application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures.Comment: 26 page

Polynomial Ensembles and Recurrence Coefficients

Polynomial ensembles are determinantal point processes associated with (non necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of non-orthogonal projection kernels. A few open problems are also suggested.Comment: 23 page

A Note on Large Deviations for 2D Coulomb Gas with Weakly Confining Potential

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.Comment: 13 page

Time-frequency transforms of white noises and Gaussian analytic functions

A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros of the Gabor transform of white noise had a very regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. The mathematical link with GAFs provides a wealth of probabilistic results to inform the design of such signal processing procedures. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises on Hilbert spaces of signals. Our main observation is a conceptual correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This correspondence covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul analytic wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. All these transforms may thus be of interest to the research program `filtering with zeros'. We also identify the GAF whose zeros are the extrema of the Gabor transform of the white noise and derive their first intensity. Moreover, we discuss important subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces. Finally, we provide quantitative estimates concerning the finite-dimensional approximations of these white noises, which is of practical interest when it comes to implementing signal processing algorithms based on GAFs.Comment: to appear in Applied and Computational Harmonic Analysi

Energy of the Coulomb gas on the sphere at low temperature

We consider the Coulomb gas of $N$ particles on the sphere and show that the logarithmic energy of the configurations approaches the minimal energy up to an error of order $\log N$, with exponentially high probability and on average, provided the temperature is $\mathcal O(1/N)$.Comment: 11 pages; revised argument in the proof of Proposition 3.1, results unchange

Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge

We study the eigenvalue behaviour of large complex correlated Wishart matrices near an interior point of the limiting spectrum where the density vanishes (cusp point), and refine the existing results at the hard edge as well. More precisely, under mild assumptions for the population covariance matrix, we show that the limiting density vanishes at generic cusp points like a cube root, and that the local eigenvalue behaviour is described by means of the Pearcey kernel if an extra decay assumption is satisfied. As for the hard edge, we show that the density blows up like an inverse square root at the origin. Moreover, we provide an explicit formula for the $1/N$ correction term for the fluctuation of the smallest random eigenvalue.Comment: 40 pages, 6 figures. Accepted for publication in EJ

Concentration for Coulomb gases and Coulomb transport inequalities

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.Comment: Improvement on an assumption, and minor modification

DLR equations and rigidity for the Sine-beta process

We investigate Sine$_\beta$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature $\beta>0$. We adopt a statistical physics perspective, and give a description of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_\beta$ to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.Comment: 46 pages. To appear in Communications on Pure and Applied Mathematic

Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges.

International audienceWe study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the scale $N^2$. As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided