Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2 , which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables X k and real numbers a k , determine the probability p that the sum k a k X k lies near some number v. For arbitrary coefficients a k of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.
REF studied the the small-deviation bound of the smallest singular value of a sub-gaussian matrix:
The Littlewood-Offord problem and invertibility of random matrices
Adv. Math
Mathematics