Average Nikolskii factors for random diffusion polynomials on closed Riemannian manifolds

For $1\le p,q\le \mathrm{\infty }$, the Nikolskii factor for a diffusion polynomial $P_{\mathbf{a}}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\mathbf{a}})=\frac{\mathrm{\parallel }{P}_{\mathbf{a}}{\mathrm{\parallel }}_q}{\mathrm{\parallel }{P}_{\mathbf{a}}{\mathrm{\parallel }}_p},P_{\mathbf{a}}(\mathbf{x})=\sum _{k:{\lambda }_k\le n}{a}_k{\phi }_k(\mathbf{x}),$$ where $\mathbf{a}=\{a_k{\}}_{{\lambda }_k\le n}$, and $\{({\phi }_k,-{\lambda }_k^2){\}}_{k=0}^{\mathrm{\infty }}$ are the eigenpairs of the Laplace-Beltrami operator ${\mathrm{\Delta }}_{\mathbb{M}}$ on a closed smooth Riemannian manifold $\mathbb{M}$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,{\sigma }^2)$ coefficients and obtain the exact orders. For $1\le p<q<\mathrm{\infty }$, the average Nikolskii factor is of order $n^0$ (i.e., constant), as compared to the worst case bound of order $n^{d(1\mathrm{/}p-1\mathrm{/}q)}$, and for $1\le p<q=\mathrm{\infty }$, the average Nikolskii factor is of order $(\ln n{)}^{1\mathrm{/}2}$ as compared to the worst case bound of order $n^{d\mathrm{/}p}$.