On the smallest singular value of symmetric random matrices

Abstract We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian variable $\xi$ with mean 0 variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves result Vershynin, who obtained such bound $n^{1/2}$ replaced by $n^{c}$ small constant c $1/8$ $(1/8) - \eta$ (with implicit constants also depending $\eta > 0$ ). Furthermore, when is Rademacher variable, we prove O(\epsilon^{1/8} \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) The special case $\epsilon = recent Campos, Mattos, Morris, Morrison, which showed $\mathbb{P}[s_n(A_n) 0] O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in departure from previous two best bounds probability singularity matrices, had relied somewhat specialized involved combinatorial techniques, our methods fall squarely within broad geometric framework pioneered Rudelson suggest possibility principled approach to study singular spectrum matrices. main innovations work new notions arithmetic structure – Median Regularized Least Common Denominator (MRLCD) Threshold, natural refinements (RLCD)introduced should be more generally useful contexts where one needs combine anticoncentration information different parts vector.