Summary: 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 \(\sigma_{a}=\frac12(\sqrt{a^{2}+4a}-a)\).