Border subrank via a generalised Hilbert-Mumford criterion

We show that the border subrank of a sufficiently general tensor in $({\mathbb{C}}^n{)}^{\otimes d}$ is $\mathcal{O}(n^{1\mathrm{/}(d-1)})$ for $n\to \mathrm{\infty }$. Since this matches the growth rate $\mathrm{\Theta }(n^{1\mathrm{/}(d-1)})$ for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest.