The Kostlan–Shub–Smale random polynomials in the case of growing number of variables

Let $\mathcal{P}_n=\sum_{j}\mathcal{H}_{j}$ be the decomposition in $L^2(S^m)$ of the space of homogeneous polynomials of degree $n$ on $\mathbb{R}^{m+1}$ into the sum of irreducible components of the group $\mathrm{SO}(m+1)$. We consider the asymptotic behavior of the sequence $\nu_{n}(t)=\frac{\mathsf{E}(|\pi_{j}u|^{2})}{\mathsf{E}(|u|^{2})}$, where $t=\frac{j}{n}$, $\pi_{j}$ is the projection onto $\mathcal{H}_{j}$, and $\mathsf{E}$ stands for the expectation in the Kostlan-Shub–Smale model for random polynomials. Assuming $\frac{m}{n}\to a>0$ as $n\to\infty$, we prove that $\nu_{n}(t)$ is asymptotic to $\sqrt{\frac{4+a}{\pi n}}\,e^{-n(1+\frac{a}{4})(t-\sigma_{a})^{2}}$, where $\sigma_{a}=\frac12(\sqrt{a^{2}+4a}-a)$.