Poisson laws and exterior stability for random alternating tensors

For fixed $k\ge3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map $Θ_N:Λ^k\mathbb {F}_q^N\to\mathbb {F}_q^m$. At the exact balance $m\binom rk=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\operatorname{Gr}(r,\mat