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 $\mathrm{H}^\ast(\mathrm{Quot}^d(\mathscr{O}^{\oplus r}_{\mathbb{A}^\infty}), \mathbb Z)$ is graded-isomorphic to the ring $\mathbb Z[c_1,\ldots,c_{d}]/(c_d^r)$, where $\mathrm{deg}\,c_i = 2i$, answering in the positive a question of Pandharipande. Finally, we prove that the motive of the nested Hilbert scheme stabilises to the motive of the infinite flag variety and we compute the cohomology ring in the limit.