This is related to the probability density of the quantum harmonic oscillator. (The Hermite polynomials are the eigenstates of that problem.) For large $k$ the density $$P_k(x)=\frac{1}{\sqrt{2\pi}k!} [h_k(x)]^2 \, e^{-x^2/2},$$ normalized to unity, has the approximation $$p_k(x)=\frac{1}{2\pi}(k-\tfrac{1}{2}x^2)^{-1/2},\;\;|x|<2\sqrt{k}.$$ The boundaries $\pm 2\sqrt{k}$ are the classical turning points of the harmonic oscillator. In that context $E=k$ is the energy of the particle and $V(x)=\tfrac{1}{2}x^2$ is the harmonic potential in which it moves. The large-$k$ approximation is known as the WKB approximation. More generally, for an arbitrary potential $V(x)$ the density varies as $[E-V(x)]^{-1/2}$.
The plot compares $P_k$ and $p_k$ for $k=100$. The function $p_k$ is a smoothed version of $P_k$, suitable for averaging quantities that vary little on the scale of $1/\sqrt{k}$.
Alternatively, the asymptotic approximation can be obtained from the asymptotic expansion of the Hermite polynomials,
$$e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.$$