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 Xk and real numbers ak, determine the probability p that the sum ∑k ak Xk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1 / p.
Rudelson, M., & Vershynin, R. (2008). The Littlewood-Offord problem and invertibility of random matrices. Advances in Mathematics, 218(2), 600–633. https://doi.org/10.1016/j.aim.2008.01.010