Poisson laws and exterior stability for random alternating tensors

Abstract

For fixed $k\ge 3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map ${\Theta }_N:{\Lambda }^k{\mathbb{F}}_q^N\to {\mathbb{F}}_q^m$. At the exact balance $m\left(\genfrac{}{}{0ex}{}{r}{k}\right)=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\mathrm{Gr}(r,{\mathbb{F}}_q^N)$ in total variation, uniformly in $q$ and $m$. Consequently, the counting measure ${\Xi }_r$ is asymptotically a Poisson point process, and its total mass $X_{N,r}$ is asymptotically Poisson. An exterior-rank stability theorem shows that near-extremal families decompose into Grassmann clusters with uniformly controlled span deficiency. We obtain quantitative rates and identify the first exterior-dependence scale. We also show that rare null $(r+1)$-spaces force high-order factorial-moment divergence, while the fixed-target bilinear cases $m=1,2$ exhibit non-Poisson critical behavior.

30 pages, 3 figures