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

Abstract

Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea 11 2 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B 2 of cardinality at most exp(δn) such that with probability very close to one we have $$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$ . 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 $$\mathbb{P}\left\{ {{s_n}\left( A \right) \leq \varepsilon {n^{ - 1/2}}} \right\} \leq L'\varepsilon + {u^n}$$ for all e > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.