695 research outputs found
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 where is sampled in the GUE(N) and e is
a fixed unit vector (and more generally in the - 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
On the ubiquity of the Cauchy distribution in spectral problems
We consider the distribution of the values at real points of random functions
which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper
half plane into itself. It is shown that under mild stationarity assumptions
the individual values of HP functions with singular spectra have a Cauchy type
distribution. The statement applies to the diagonal matrix elements of random
operators, and holds regardless of the presence or not of level repulsion, i.e.
applies to both random matrix and Poisson-type spectra.Comment: Slightly revised version: presentation was made more explicit in
places, and additional references were provide
Random covariance matrices: Universality of local statistics of eigenvalues
We study the eigenvalues of the covariance matrix of a
large rectangular matrix
whose entries are i.i.d. random variables of mean zero, variance one, and
having finite th moment for some sufficiently large constant . The
main result of this paper is a Four Moment theorem for i.i.d. covariance
matrices (analogous to the Four Moment theorem for Wigner matrices established
by the authors in [Acta Math. (2011) Random matrices: Universality of local
eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--572])). We
can use this theorem together with existing results to establish universality
of local statistics of eigenvalues under mild conditions. As a byproduct of our
arguments, we also extend our previous results on random Hermitian matrices to
the case in which the entries have finite th moment rather than
exponential decay.Comment: Published in at http://dx.doi.org/10.1214/11-AOP648 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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