Average Nikolskii factors for random trigonometric polynomials

For $1\le p,q\le \mathrm{\infty }$, the Nikolskii factor for a trigonometric polynomial $T_{\mathbf{a}}$ is defined by $${\mathcal{N}}_{p,q}(T_{\mathbf{a}})=\frac{\mathrm{\parallel }{T}_{\mathbf{a}}{\mathrm{\parallel }}_q}{\mathrm{\parallel }{T}_{\mathbf{a}}{\mathrm{\parallel }}_p},T_{\mathbf{a}}(x)=a_1+\sum _{k=1}^n(a_{2k}\sqrt{2}\cos kx+a_{2k+1}\sqrt{2}\sin kx)\mathrm{.}$$ We study this average Nikolskii factor for random trigonometric polynomials with independent $N(0,{\sigma }^2)$ coefficients and obtain that the exact order. For $1\le p<q<\mathrm{\infty }$, the average Nikolskii factor is order degree to the 0, as compared to the degree $1\mathrm{/}p-1\mathrm{/}q$ worst case bound. We also give the generalization to random multivariate trigonometric polynomials.