Particle Systems with Repulsion Exponent β\beta and Random Matrices

We consider a class of particle systems generalizing the $\beta$-Ensembles from random matrix theory. In these new ensembles, particles experience repulsion of power $\beta>0$ when getting close, which is the same as in the $\beta$-Ensembles. For distances larger than zero, the interaction is allowed to differ from those present for random eigenvalues. We show that the local bulk correlations of the $\beta$-Ensembles, universal in random matrix theory, also appear in these new ensembles.Comment: 12 pages; has been rewritten as a not

Universality at weak and strong non-Hermiticity beyond the elliptic Ginibre ensemble

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.Comment: v1: 22 pages; v2: 30 pages, substantially extended to include fixed trace ensemble

Local universality of repulsive particle systems and random matrices

We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results may provide an explanation for the appearance of sine kernel correlation statistics in a number of situations which do not have an obvious interpretation in terms of random matrices.Comment: Published in at http://dx.doi.org/10.1214/13-AOP844 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Moment Problems under Constraints

We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order $n$. By equipping these sections with the uniform or more general probability distributions, we manage to give for large $n$ precise results on the (probabilistic) barycenters of moment space sections and the fluctuations of random moments around these barycenters. The measures associated to the barycenters belong to the Bernstein-Szeg\H{o} class and show strong universal behavior. We prove Gaussian fluctuations and moderate and large deviations principles. Furthermore, we demonstrate how fixing moments by a constraint leads to breaking the connection between random moments and random matrices.Comment: 43 page

Experiences with the silicone T‐tube in man and dog

Silicone T‐tubes are particularly useful for long‐term stenting of the laryngotracheal airway. Silicone is useful when non‐autogenous non‐reactive soft material is wanted for use in the human and animal body. Two case reports illustrate the use of the silicone T‐tube