98 research outputs found
Eigenvalue confinement and spectral gap for random simplicial complexes
We consider the adjacency operator of the Linial-Meshulam model for random
simplicial complexes on vertices, where each -cell is added
independently with probability to the complete -skeleton. Under the
assumption , we prove that the spectral gap between the
smallest eigenvalues and the remaining
eigenvalues is with high probability.
This estimate follows from a more general result on eigenvalue confinement. In
addition, we prove that the global distribution of the eigenvalues is
asymptotically given by the semicircle law. The main ingredient of the proof is
a F\"uredi-Koml\'os-type argument for random simplicial complexes, which may be
regarded as sparse random matrix models with dependent entries.Comment: 29 pages, 6 figure
The outliers of a deformed Wigner matrix
We derive the joint asymptotic distribution of the outlier eigenvalues of an
additively deformed Wigner matrix . Our only assumptions on the deformation
are that its rank be fixed and its norm bounded. Our results extend those of
[The isotropic semicircle law and deformation of Wigner matrices. Preprint] by
admitting overlapping outliers and by computing the joint distribution of all
outliers. In particular, we give a complete description of the failure of
universality first observed in [Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri
Poincar\'{e} Probab. Stat. 48 (1013) 107-133; Free convolution with a
semi-circular distribution and eigenvalues of spiked deformations of Wigner
matrices. Preprint]. We also show that, under suitable conditions, outliers may
be strongly correlated even if they are far from each other. Our proof relies
on the isotropic local semicircle law established in [The isotropic semicircle
law and deformation of Wigner matrices. Preprint]. The main technical
achievement of the current paper is the joint asymptotics of an arbitrary
finite family of random variables of the form
.Comment: Published in at http://dx.doi.org/10.1214/13-AOP855 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Mesoscopic eigenvalue statistics of Wigner matrices
We prove that the linear statistics of the eigenvalues of a Wigner matrix
converge to a universal Gaussian process on all mesoscopic spectral scales,
i.e. scales larger than the typical eigenvalue spacing and smaller than the
global extent of the spectrum.Comment: 34 page
The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case
We consider the spectral statistics of large random band matrices on
mesoscopic energy scales. We show that the two-point correlation function of
the local eigenvalue density exhibits a universal power law behaviour that
differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in
the physics literature by Altshuler and Shklovskii [4]; it describes the
correlations of the eigenvalue density in general metallic samples with weak
disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas
for band matrices. In two dimensions, where the leading term vanishes owing to
an algebraic cancellation, we identify the first non-vanishing term and show
that it differs substantially from the prediction of Kravtsov and Lerner [33]
Eigenvector Distribution of Wigner Matrices
We consider Hermitian or symmetric random matrices with
independent entries. The distribution of the -th matrix element is given
by a probability measure whose first two moments coincide with those
of the corresponding Gaussian ensemble. We prove that the joint probability
distribution of the components of eigenvectors associated with eigenvalues
close to the spectral edge agrees with that of the corresponding Gaussian
ensemble. For eigenvectors associated with bulk eigenvalues, the same
conclusion holds provided the first four moments of the distribution
coincide with those of the corresponding Gaussian ensemble. More generally, we
prove that the joint eigenvector-eigenvalue distributions near the spectral
edge of two generalized Wigner ensembles agree, provided that the first two
moments of the entries match and that one of the ensembles satisfies a level
repulsion estimate. If in addition the first four moments match then this
result holds also in the bulk
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