The strong circular law: a combinatorial view

Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is an independent copy of a centered complex random variable $z$ with finite non-zero variance $\sigma^{2}$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(\sigma\sqrt{n})$ converges to the uniform distribution on the unit disc in $\mathbb{C}$. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix $xI - N_{n}/(\sigma\sqrt{n})$ (where $x\in \mathbb{C}$ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using tools from additive combinatorics or any net arguments), we show that for any fixed matrix $M$ with operator norm at most $n^{0.51}$ and for all $\eta \geq 0$, $$\Pr\left(s_n(M+N_n) \leq \eta \right) \lesssim n^{C}\eta + \exp(-n^{c}),$$ where $s_n(M+N_n)$ is the least singular value of $M+N_n$ and $C,c$ are absolute constants. Our result is optimal up to the constants $C,c$ and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. During the course of our proof, we extend the solution of the counting problem in inverse Littlewood-Offord theory, recently isolated by the author along with Ferber, Luh, and Samotij, from Rademacher variables to general complex random variables. This significantly improves on estimates for this problem obtained using the optimal inverse Littlewood-Offord theorem of Nguyen and Vu, and may be of independent interest.