Let Nn be an n × n complex random matrix, each of whose entries is an independent copy of a centered complex random variable z with finite nonzero variance σ2. The strong circular law, proved by Tao and Vu, states that almost surely, as n →∞, the empirical spectral distribution of Nn/(σn) converges to the uniform distribution on the unit disc in 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 − Nn/(σn) (where x ∈ ℂ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix M with operator norm at most n3/4−𝜖 and for all η ≥ 0, Pr sn(M + Nn) ≤ η ≲ nCη +exp(−nc), where sn(M + Nn) is the least singular value of M + Nn and C,c are positive 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. Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the ∞→ 2 operator norm of heavy-tailed random matrices.

中文翻译：

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强循环定律：组合观点

让ñn豆角，扁豆n × n复随机矩阵，其每一项都是居中复随机变量的独立副本z具有有限非零方差σ2. 由 Tao 和 Vu 证明的强循环定律指出，几乎可以肯定，如n →∞, 的经验光谱分布ñn/(σn)收敛于单位圆盘上的均匀分布ℂ. Tao 和 Vu 证明中的一个关键成分，它使用了加法组合学的深层思想，是控制随机矩阵的最小奇异值的下尾X一世 - ñn/(σn)（在哪里X ∈ ℂ是固定的），故障概率为反多项式。在本文中，使用一种简单而新颖的方法（特别是，不使用来自加法组合或任何网络参数的机器），我们表明对于任何固定的复矩阵米最多有算子范数n3/4-𝜖并为所有人η ≥ 0, 公关 sn(米 + ñn) ≤ η ≲ nCη +经验(-nC), 在哪里sn(米 + ñn)是的最小奇异值米 + ñn和C,C是正的绝对常数。我们的结果在常数范围内是最优的C,C由于 Tao 和 Vu，反指数型错误率比反多项式错误率有所提高。我们的证明依赖于由 Ferber、Luh、Samotij 和作者开发的逆 Littlewood-Offord 理论中所谓计数问题的解决方案，一种新颖的复杂反集中不等式，以及基于控制∞→ 2重尾随机矩阵的算子范数。