249 research outputs found
The circular law for random matrices
We consider the joint distribution of real and imaginary parts of eigenvalues
of random matrices with independent entries with mean zero and unit variance.
We prove the convergence of this distribution to the uniform distribution on
the unit disc without assumptions on the existence of a density for the
distribution of entries. We assume that the entries have a finite moment of
order larger than two and consider the case of sparse matrices. The results are
based on previous work of Bai, Rudelson and the authors extending those results
to a larger class of sparse matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOP522 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the Distribution of Complex Roots of Random Polynomials with Heavy-tailed Coefficients
Consider a random polynomial with i.i.d.
complex-valued coefficients. Suppose that the distribution of
has a slowly varying tail. Then the distribution of
the complex roots of concentrates in probability, as , to two
centered circles and is uniform in the argument as . The radii of
the circles are and
, where denotes the coefficient with
the maximum modulus.Comment: 8 page
Preferred attachment model of affiliation network
In an affiliation network vertices are linked to attributes and two vertices
are declared adjacent whenever they share a common attribute. For example, two
customers of an internet shop are called adjacent if they have purchased the
same or similar items. Assuming that each newly arrived customer is linked
preferentially to already popular items we obtain a preferred attachment model
of an evolving affiliation network. We show that the network has a scale-free
property and establish the asymptotic degree distribution.Comment: 9 page
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
Limit Correlation Functions for Fixed Trace Random Matrix Ensembles
Universal limits for the eigenvalue correlation functions in the bulk of the
spectrum are shown for a class of nondeterminantal random matrices known as the
fixed trace ensemble.Comment: 32 pages; Latex; result improved; proofs modified; reference added;
typos correcte
Lattice point problems and distribution of values of quadratic forms
For d-dimensional irrational ellipsoids E with d >= 9 we show that the number
of lattice points in rE is approximated by the volume of rE, as r tends to
infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier
authors' bound of order O(r^{d-2}) which holds for arbitrary ellipsoids, and is
optimal for rational ellipsoids. As an application we prove a conjecture of
Davenport and Lewis that the gaps between successive values, say s<n(s), s,n(s)
in Q[Z^d], of a positive definite irrational quadratic form Q[x], x in R^d, are
shrinking, i.e., that n(s) - s -> 0 as s -> \infty, for d >= 9. For comparison
note that sup_s (n(s)-s) 0, for rational Q[x] and
d>= 5. As a corollary we derive Oppenheim's conjecture for indefinite
irrational quadratic forms, i.e., the set Q[Z^d] is dense in R, for d >= 9,
which was proved for d >= 3 by G. Margulis in 1986 using other methods.
Finally, we provide explicit bounds for errors in terms of certain
characteristics of trigonometric sums.Comment: 51 pages, published versio
Estimates for the closeness of convolutions of probability distributions on convex polyhedra
The aim of the present work is to show that the results obtained earlier on
the approximation of distributions of sums of independent summands by the
accompanying compound Poisson laws and the estimates of the proximity of
sequential convolutions of multidimensional distributions may be transferred to
the estimation of the closeness of convolutions of probability distributions on
convex polyhedra.Comment: 8 page
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