Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries

Let A = (a_ij) be an n × n random matrix with i.i.d. entries such that Ea₁₁ = 0 and Ea₁₁ ² = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B₂ ⁿ of cardinality at most exp(δn) such that with probability very close to one we have A(B2n)∪y∈A(N)(y+LnB2n). In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies P{ s_n(A) ≤ εn^{− 1 / 2}} ≤ L' ε+ uⁿ for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.