54,173 research outputs found
A Local Law for Singular Values from Diophantine Equations
We introduce the random matrices and a fixed integer. We prove that the distribution
of their singular values converges to the local Marchenko-Pastur law at scales
for an explicit, small , as long as . To
our knowledge, this is the first instance of a random matrix ensemble that is
explicitly defined in terms of only random variables exhibiting a
universal local spectral law. Our main technical contribution is to derive
concentration bounds for the Stieltjes transform that simultaneously take into
account stochastic and oscillatory cancellations. Important ingredients in our
proof are strong estimates on the number of solutions to Diophantine equations
(in the form of Vinogradov's main conjecture recently proved by
Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward
identity with an algebraic uniqueness condition for Diophantine equations
derived from the Newton-Girard identities.Comment: 30 page
Small ball probability for the condition number of random matrices
Let be an random matrix with i.i.d. entries of zero mean,
unit variance and a bounded subgaussian moment. We show that the condition
number satisfies the small ball probability estimate
where may only depend on the subgaussian moment.
Although the estimate can be obtained as a combination of known results and
techniques, it was not noticed in the literature before. As a key step of the
proof, we apply estimates for the singular values of , obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment
Smallest singular value of sparse random matrices
We extend probability estimates on the smallest singular value of random
matrices with independent entries to a class of sparse random matrices. We show
that one can relax a previously used condition of uniform boundedness of the
variances from below. This allows us to consider matrices with null entries or,
more generally, with entries having small variances. Our results do not assume
identical distribution of the entries of a random matrix and help to clarify
the role of the variances of the entries. We also show that it is enough to
require boundedness from above of the -th moment, , of the
corresponding entries.Comment: 25 pages, a condition on one parameter was added in the statement of
Theorem 1.3 and Lemma 6.2, results unchange
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