Polygons and multi-product of eigenfunctions

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j{\}}_j$, which satisfy ${\mathrm{\Delta }}_g{e}_j=-{\lambda }_j^2{e}_j$. We study the following inner product of eigenfunctions $$⟨e_{i_1}{e}_{i_2}\dots e_{i_k},e_{i_{k+1}}⟩=\int e_{i_1}{e}_{i_2}\dots e_{i_k}\overline{e_{i_{k+1}}}dV\mathrm{.}$$ We show that, after a mild averaging in the frequency variables, the main ${\mathrm{\ell }}^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies ${\lambda }_{i_1},{\lambda }_{i_2},\dots ,{\lambda }_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where ${\lambda }_{i_1},{\lambda }_{i_2},\dots ,{\lambda }_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.