Border subrank via a generalised Hilbert-Mumford criterion

We show that the border subrank of a sufficiently general tensor in (Cⁿ)⊗ d is O(n1/(d-1)) for n → ∞. Since this matches the growth rate Θ(n1/(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.