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Eigenvalue distribution of large dilute random matrices
We study the eigenvalue distribution of dilute N3N random matrices HN that in
the pure ~undiluted! case describe the Hopfield model. We prove that for the fixed
dilution parameter a the normalized counting function ~NCF! of HN converges as
N!` to a unique sa(l). We find the moments of this distribution explicitly,
analyze the 1/a correction, and study the asymptotic properties of sa(l) for large
ulu. We prove that sa(l) converges as a !` to the Wigner semicircle distribution
~SCD!. We show that the SCD is the limit of the NCF of other ensembles of dilute
random matrices. This could be regarded as evidence of stability of the SCD to
dilution, or more generally, to random modulations of large random matrices
Non-universality of compact support probability distributions in random matrix theory
The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with that of the canonical Gaussian ensemble by a nonuniversal part that is given explicitly for all monomial potentials V(M)=M2p. Moreover, we prove that for the generalized fixed and bounded trace ensemble all k-point resolvents agree in the large-n limit, despite their nonuniversality
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
Real symmetric random matrices and paths counting
Exact evaluation of is here performed for real symmetric
matrices of arbitrary order , up to some integer , where the matrix
entries are independent identically distributed random variables, with an
arbitrary probability distribution.
These expectations are polynomials in the moments of the matrix entries ;
they provide useful information on the spectral density of the ensemble in the
large limit. They also are a straightforward tool to examine a variety of
rescalings of the entries in the large limit.Comment: 23 pages, 10 figures, revised pape
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