Determinantal Processes and Independence

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.Comment: Published at http://dx.doi.org/10.1214/154957806000000078 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Eigenvalue variance bounds for Wigner and covariance random matrices

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example, which needs to be investigated first, the main bounds are extended to families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd\"os, Yau and Yin. The case of real Wigner matrices is obtained from interlacing formulas. As an application, bounds on the expected 2-Wasserstein distance between the empirical spectral measure and the semicircle law are derived. Similar results are available for random covariance matrices

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.Comment: 44 page

Non-stoichiometric silicon nitride for future gravitational wave detectors

Silicon nitride thin films were deposited at room temperature employing a custom ion beam deposition (IBD) system. The stoichiometry of these films was tuned by controlling the nitrogen gas flow through the ion source and a process gas ring. A correlation is established between the process parameters, such as ion beam voltage and ion current, and the optical and mechanical properties of the films based on post-deposition heat treatment. The results show that with increasing heat treatment temperature, the mechanical loss of these materials as well as their optical absorption decreases producing films with an extinction coefficient as low as k = 6.2(±0.5) × 10−7 at 1064 nm for samples annealed at 900°C. This presents the lowest value for IBD SiNx within the context of gravitational wave detector applications. The mechanical loss of the films was measured to be Φ = 2.1(±0.6) × 10−4 once annealed post deposition to 900°C

Non-stoichiometric silicon nitride for future gravitational wave detectors

Silicon nitride thin films were deposited at room temperature employing a custom ion beam deposition (IBD) system. The stoichiometry of these films was tuned by controlling the nitrogen gas flow through the ion source and a process gas ring. A correlation is established between the process parameters, such as ion beam voltage and ion current, and the optical and mechanical properties of the films based on post-deposition heat treatment. The results show that with increasing heat treatment temperature, the mechanical loss of these materials as well as their optical absorption decreases producing films with an extinction coefficient as low as k=6.2(±0.5)×10−7 at 1064 nm for samples annealed at 900 ∘C. This presents the lowest value for IBD SiN x within the context of gravitational wave detector applications. The mechanical loss of the films was measured to be ϕ=2.1(±0.6)×10−4 once annealed post deposition to 900 ∘C