69,056 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
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
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
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