Poisson laws and exterior stability for random alternating tensors

Source abstract

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 $\operatorname{Gr}(r,\mathbb {F}_q^N)$ in total variation, uniformly in $q$ and $m$. Consequently, the counting measure $Ξ_r$ 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.