Motivic and cohomological stabilisation of the Quot scheme of points

We prove that the motive of the punctual Quot scheme $\mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})_0$ stabilises, when $n \to \infty$, to $[\mathrm{Gr}(d-1,\infty)]\cdot \sum_{i=0}^{r-1}\mathbb L^{di}$. We prove that the stable cohomology ring $\mathrm{H}^\ast(\mathrm{Quot}^d(\mathscr{O}^{\o