Abstract:Let $M_n$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones. we establish quantitative lower bounds on the smallest singular value of the shifted matrices $M_n-z \mathbf{I}_n$ whenever $|z| \leq \sqrt{d}\, \log\log d$ and $ C \log n \leq d \leq n/2$ for some absolute positive constant $C$. As an application, we show that the empirical spectral distribution of the appropriately rescaled matrix $M_n$ converges in probability to the circular law provided that $\log^{2+\varepsilon} n \leq d=o(n)$ for some fixed $\varepsilon \in (0,1)$.
From: Tingzhou Yu [view email]
[v1]
Sun, 12 Apr 2026 03:59:07 UTC (606 KB)
[v2]
Tue, 11 Aug 2026 01:45:43 UTC (609 KB)
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