178 research outputs found
On high moments of strongly diluted large Wigner random matrices
We consider a dilute version of the Wigner ensemble of nxn random matrices
and study the asymptotic behavior of their moments in the limit of
infinite , and , where is the dilution parameter. We show
that in the asymptotic regime of the strong dilution, the moments with
depend on the second and the fourth moments of the random entries
and do not depend on other even moments of . This fact can be
regarded as an evidence of a new type of the universal behavior of the local
eigenvalue distribution of strongly dilute random matrices at the border of the
limiting spectrum. As a by-product of the proof, we describe a new kind of
Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other
minor modifications
On Eigenvalues of the sum of two random projections
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N
are two N -by-N random orthogonal projections. We relate the joint eigenvalue
distribution of this matrix to the Jacobi matrix ensemble and establish the
universal behavior of eigenvalues for large N. The limiting local behavior of
eigenvalues is governed by the sine kernel in the bulk and by either the Bessel
or the Airy kernel at the edge depending on parameters. We also study an
exceptional case when the local behavior of eigenvalues of P_N + Q_N is not
universal in the usual sense.Comment: 14 page
On universality of local edge regime for the deformed Gaussian Unitary Ensemble
We consider the deformed Gaussian ensemble in which
is a hermitian matrix (possibly random) and is the Gaussian
unitary random matrix (GUE) independent of . Assuming that the
Normalized Counting Measure of converges weakly (in probability if
random) to a non-random measure with a bounded support and assuming
some conditions on the convergence rate, we prove universality of the local
eigenvalue statistics near the edge of the limiting spectrum of .Comment: 25 pages, 2 figure
Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices
We study the fluctuations of eigenvalues from a class of Wigner random
matrices that generalize the Gaussian orthogonal ensemble. We begin by
considering an matrix from the Gaussian orthogonal ensemble (GOE)
or Gaussian symplectic ensemble (GSE) and let denote eigenvalue number
. Under the condition that both and tend to infinity with , we
show that is normally distributed in the limit. We also consider the
joint limit distribution of eigenvalues from the GOE or GSE with similar
conditions on the indices. The result is an -dimensional normal
distribution. Using a recent universality result by Tao and Vu, we extend our
results to a class of Wigner real symmetric matrices with non-Gaussian entries
that have an exponentially decaying distribution and whose first four moments
match the Gaussian moments.Comment: 21 pages, to appear, J. Stat. Phys. References and other corrections
suggested by the referees have been incorporate
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble
The paper studies the spectral properties of large Wigner, band and sample
covariance random matrices with heavy tails of the marginal distributions of
matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at
Giens, France on September 13-17, 200
Stein's Method and Characters of Compact Lie Groups
Stein's method is used to study the trace of a random element from a compact
Lie group or symmetric space. Central limit theorems are proved using very
little information: character values on a single element and the decomposition
of the square of the trace into irreducible components. This is illustrated for
Lie groups of classical type and Dyson's circular ensembles. The approach in
this paper will be useful for the study of higher dimensional characters, where
normal approximations need not hold.Comment: 22 pages; same results, but more efficient exposition in Section 3.
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d.
complex Gaussian coefficients a_n. We show that these zeros form a
determinantal process: more precisely, their joint intensity can be written as
a minor of the Bergman kernel. We show that the number of zeros of f in a disk
of radius r about the origin has the same distribution as the sum of
independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover,
the set of absolute values of the zeros of f has the same distribution as the
set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1].
The repulsion between zeros can be studied via a dynamic version where the
coefficients perform Brownian motion; we show that this dynamics is conformally
invariant.Comment: 37 pages, 2 figures, updated proof
Some Universal Properties for Restricted Trace Gaussian Orthogonal, Unitary and Symplectic Ensembles
Consider fixed and bounded trace Gaussian orthogonal, unitary and symplectic
ensembles, closely related to Gaussian ensembles without any constraint. For
three restricted trace Gaussian ensembles, we prove universal limits of
correlation functions at zero and at the edge of the spectrum edge. In
addition, by using the universal result in the bulk for fixed trace Gaussian
unitary ensemble, which has been obtained by Gtze and Gordin, we also
prove universal limits of correlation functions for bounded trace Gaussian
unitary ensemble.Comment: 19pages,bounded trace Gaussian ensembles are adde
Deformations of the Tracy-Widom distribution
In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes
the behavior of the largest eigenvalue. We consider here two models in which TW
undergoes transformations. In the first one disorder is introduced in the
Gaussian ensembles by superimposing an external source of randomness. A
competition between TW and a normal (Gaussian) distribution results, depending
on the spreading of the disorder. The second model consists in removing at
random a fraction of (correlated) eigenvalues of a random matrix. The usual
formalism of Fredholm determinants extends naturally. A continuous transition
from TW to the Weilbull distribution, characteristc of extreme values of an
uncorrelated sequence, is obtained.Comment: 9 pages, 1 figur
Non-colliding Brownian Motions and the extended tacnode process
We consider non-colliding Brownian motions with two starting points and two
endpoints. The points are chosen so that the two groups of Brownian motions
just touch each other, a situation that is referred to as a tacnode. The
extended kernel for the determinantal point process at the tacnode point is
computed using new methods and given in a different form from that obtained for
a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the
extended kernel is also different from that obtained for the extended tacnode
kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the
correlation kernel for a finite number of non-colliding Brownian motions
starting at two points and ending at arbitrary points.Comment: 38 pages. In the revised version a few arguments have been expanded
and many typos correcte
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