Much of the literature on random metrics constructs these as follows. Start from a reference metric $g_0$ on the compact Riemannian manifold $M$. The corresponding Laplacian $\Delta_0$ has eigenfunctions $\phi_j$ with eigenvalues $\lambda_j$. Draw a set $\{a\}$ of random numbers $a_j$, independently from a normal distribution, then the random metric is $$g_{\{a\}}=\exp\left(\sum_j e^{-\lambda_j}a_j\phi_j\right)g_0.$$ Each random metric is conformal to the reference $g_0$. The weight $e^{-\lambda}$ may be replaced by another function $F(\lambda)$ which decreases to zero when $\lambda\rightarrow\infty$.

These lecture notes discuss the spectral statistics of random metrics constructed in this way.

Much of the literature on random metrics constructs these as follows. Start from a reference metric $g_0$ on the compact Riemannian manifold $M$. The corresponding Laplacian $\Delta_0$ has eigenfunctions $\phi_j$ with eigenvalues $\lambda_j$. Draw a set $\{a\}$ of random numbers $a_j$, independently from a normal distribution, then the random metric is $$g_{\{a\}}=\exp\left(\sum_j e^{-\lambda_j}a_j\phi_j\right)g_0.$$ Each random metric is conformal to the reference $g_0$. The weight $e^{-\lambda}$ may be replaced by another function $F(\lambda)$ which decreases to zero when $\lambda\rightarrow\infty$.

These lecture notes discuss the spectral statistics of random metrics constructed in this way. (See also the course web site.)