Motivic and cohomological stabilisation of the Quot scheme of points

We prove that the motive of the punctual Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}^d({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r}{)}_0$ stabilises, when $n\to \mathrm{\infty }$, to $[\mathrm{G}\mathrm{r}(d-1,\mathrm{\infty })]\cdot {\sum }_{i=0}^{r-1}{\mathbb{L}}^{di}$. We similarly show that the Poincaré polynomial of the Quot scheme ${\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}^d({\mathcal{O}}_{{\mathbb{A}}^n}^{\oplus r})$ stabilises and we compute the limit in terms of the infinite Grassmannian. 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. These results provide affirmative evidence to a question of Pandharipande concerning the cohomology of Quot schemes on ${\mathbb{A}}^{\mathrm{\infty }}$.