Small-ball bound for the distance of a Gaussian column to the span of the others: $\mathbb P\{\operatorname{dist}(a_j,H_j)^2\le u\}\le (eu/(N-n+1))^{(N-n+1)/2}$

Let $1\le n\le N$ and let $A$ be an $N\times n$ random matrix with independent standard normal entries, with columns $a_1,\dots,a_n$. Fix a column index $j$ and let $H_j=\operatorname{span}\{a_i:i\ne j\}$, with

Then for every real $u$ with $0\le u\le N-n+1$,

Conditionally on the other columns, $a_j$ is a standard Gaussian vector independent of $H_j$, a subspace of dimension at most $n-1$; the squared distance of $a_j$ to it dominates a $\chi^2$ variable with $N-n+1$ degrees of freedom, whose lower tail obeys the stated bound. Integrating over the other columns gives the claim.

Combined with the deterministic reduction $\sigma_{\min}(A)\le s\Rightarrow\exists j,\ \operatorname{dist}(a_j,H_j)\le\sqrt n s$ and a union bound over $j$, this yields the small-ball estimate $\mathbb P\{\sigma_{\min}(A)\le s\}\le n\,(e n s^2/(N-n+1))^{(N-n+1)/2}$.

Formalization Note. The law of $A$ is gaussianMatrix N n (a product over rows of product measures); the event is written with the explicit infimum, and the probability of a possibly non-measurable set is the outer measure (the set is in fact measurable). The dimension of $H_j$ may drop below $n-1$; the bound holds regardless since the degrees of freedom only increase.