97 research outputs found
A continuous variant of the inverse Littlewood-Offord problem for quadratic forms
Motivated by the inverse Littlewood-Offord problem for linear forms, we study
the concentration of quadratic forms. We show that if this form concentrates on
a small ball with high probability, then the coefficients can be approximated
by a sum of additive and algebraic structures.Comment: 17 pages. This is the first part of http://arxiv.org/abs/1101.307
Random doubly stochastic matrices: The circular law
Let be a matrix sampled uniformly from the set of doubly stochastic
matrices of size . We show that the empirical spectral distribution
of the normalized matrix converges almost surely
to the circular law. This confirms a conjecture of Chatterjee, Diaconis and
Sly.Comment: Published in at http://dx.doi.org/10.1214/13-AOP877 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Let denote a random symmetric by matrix, whose upper diagonal
entries are iid Bernoulli random variables (which take value -1 and 1 with
probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show
that is non-singular with probability for any positive
constant . The proof uses an inverse Littlewood-Offord result for quadratic
forms, which is of interest of its own.Comment: Some minor corrections in Section 10 of v
Circular law for random discrete matrices of given row sum
Let be a random matrix of size and let
be the eigenvalues of . The empirical spectral
distribution of is defined as \mu_{M_n}(s,t)=\frac{1}{n}#
\{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.
The circular law theorem in random matrix theory asserts that if the entries
of are i.i.d. copies of a random variable with mean zero and variance
, then the empirical spectral distribution of the normalized matrix
of converges almost surely to the uniform
distribution \mu_\cir over the unit disk as tends to infinity.
In this paper we show that the empirical spectral distribution of the
normalized matrix of , a random matrix whose rows are independent random
vectors of given row-sum with some fixed integer satisfying
, also obeys the circular law. The key ingredient is a new
polynomial estimate on the least singular value of
Random matrices: Law of the determinant
Let be an by random matrix whose entries are independent real
random variables with mean zero, variance one and with subexponential tail. We
show that the logarithm of satisfies a central limit theorem. More
precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf
{P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log
n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le
x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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