Extreme least singular values of Gaussian row submatrices and a phase retrieval stability problem

Let $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ and $d_{\mathbb{F}}={\dim }_{\mathbb{R}}\mathbb{F}$. If $A_m\in {\mathbb{F}}^{N_m\times m}$ has independent standard Gaussian entries and $N_m\mathrm{/}m\to \gamma >1$, then $$\underset{\begin{array}{c}T\subset [N_m]\\ \mathrm{\mid }T\mathrm{\mid }=m\end{array}}{\min }{\sigma }_{\min }(A_{m,T})={\left(\frac{{\gamma }^{\gamma }}{(\gamma -1{)}^{\gamma -1}}\right)}^{-m\mathrm{/}{d}_{\mathbb{F}}+o_P(m)}\mathrm{.}$$ If $N_m=\gamma m+O(1)$, the convergence of $m^{-1}\log M_m^{\mathbb{F}}$ has probability error $O(m^{-1})$. In particular, at the real phase-retrieval threshold $N=2m-1$, $$\omega (A_m)=4^{-m+o_P(m)},$$ so the Gaussian Balan–Wang critical exponential base is $1\mathrm{/}4$.