76 research outputs found
Average Characteristic Polynomials of Determinantal Point Processes
We investigate the average characteristic polynomial where the '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 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
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
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
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 particles on the sphere and show that the
logarithmic energy of the configurations approaches the minimal energy up to an
error of order , with exponentially high probability and on average,
provided the temperature is .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 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, the universal point process arising as the
thermodynamic limit of the microscopic scale behavior in the bulk of
one-dimensional log-gases, or -ensembles, at inverse temperature
. We adopt a statistical physics perspective, and give a description
of Sine using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving
that it satisfies the DLR equations: the restriction of Sine 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 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 . 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 . As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided
- …