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

Let $A=(a_{ij})$ be an $n\times n$ random matrix with i.i.d. entries such that $\mathbb{E}{a}_{11}=0$ and $\mathbb{E}{a_{11}}^2=1$. We prove that for any $\delta >0$ there is $L>0$ depending only on $\delta$, and a subset $\mathcal{N}$ of $B_2^n$ of cardinality at most $\exp (\delta n)$ such that with probability very close to one we have $$A(B_2^n)\subset \bigcup _{y\in A(\mathcal{N})}(y+L\sqrt{n}{B}_2^n)\mathrm{.}$$ As an application, we show that for some $L^{\mathrm{\prime }}>0$ and $u\in [0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies $\mathbb{P}\{s_n(A)\le \epsilon n^{-1\mathrm{/}2}\}\le L^{\mathrm{\prime }}\epsilon +u^n$ for all $\epsilon >0$. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.