Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.

Your matrix $A = X^T X$ where $X$ is a random $m \times N$ matrix with a continuous distribution having a density. An $m \times m$ submatrix of $A$ is $Q^T A P = (XQ)^T XP$ where $P$ and $Q$ are $N \times m$ matrices each consisting of $m$ columns of the $N \times N$ identity matrix. $XP$ and $XQ$ are $m \times m$ submatrices of $X$. With probability $1$, any $m \times m$ submatrix of $X$ has rank $m$ (its determinant is a non-constant polynomial in the matrix entries, and since the distribution of $X$ has a density the value of this polynomial is almost surely nonzero). So with probability $1$, $(XQ)^T XP$ has rank $m$.