Stability of eigenmaps for random averaging Laplacians

Let $g(x)$ be a smooth sampling density on $\mathbb{R}$, and let random averaging Laplacians be constructed from points sampled according to $g$. Define

$$\mathcal{L}_{(g)}f:=-2\frac{g'}{g}f'-f”.$$

Stability conjecture. Eigenmaps of the random averaging Laplacians give numerically stable locally uniform approximations to the corresponding eigenmap of $\mathcal{L}_{(g)}$. This predicts locally uniform spectral stability for sampling from smooth distributions; the half-line case requires suitable boundary conditions, and a related multidimensional case is discussed in the paper.