asymptotics - Relatioship between the average of determinants of the following class of matrices and the determinant of the trivial one

pr.chance – High-probability upper-bound for $|(G^high G)^{-1} (1,ldots,1)|_2$, the place $G$ is gaussian random matrix with iid entries from $mathcal N(0,1/N)$ Answer

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pr.chance – High-probability upper-bound for $|(G^high G)^{-1} (1,ldots,1)|_2$, the place $G$ is gaussian random matrix with iid entries from $mathcal N(0,1/N)$

Let $G$ breathe an $N occasions n$ random matrix with iid entries from $mathcal N(0,1/N)$, with $n/N =: lambda in (0, 1)$, and let $u=(1,1,ldots,1) in mathbb R^n$.

Question. What is a upper-bound for $|(G^high G)^{-1} u|$ which holds with high-probability ?

Note {that a} trifling certain which holds w.p $1-e^{-Omega(N)}$ is given by
$$
|(G^high G)^{-1} u| le lambda_{max}((G^high G)^{-1})|u| = sqrt{n}/lambda_{min}(G^high G) = mathcal O(sqrt{n}), tag{1}
$$

since $lambda_{min}(G^high G) = Omega(1)$ w.p $1-e^{-Omega(N)}$.

Observation. Empirically, i’ve seen that the certain (1) is nice when $lambda$ is “small” and really lax when $lambda$ is “large”.

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