article

Title: Critical values of random analytic functions on complex manifolds

Authors: Renjie Feng and Steve Zelditch

Issue: Volume 63 (2014), Issue 3, 651-686

Abstract:

$We study the asymptotic distribution of critical values of random holomorphic sections $s_n\in H^0(M^m,L^n)$ of powers of a positive line bundle $(L,h)\to(M,\omega)$ on a general K\"ahler manifold of dimension $m$. By critical value is meant the value of $|s(z)|_{h^n}$ at a critical point where $\nabla_hs_n(z)=0$, where $\nabla_h$ is the Chern connection. The distribution of critical values of $s_n$ is its empirical measure. Two main ensembles are considered: \begin{enumerate}[label=(\roman*)] \item the normalized Gaussian ensembles so that $\mathbf{E}\|s_n\|^2_{L^2}=1$; \item the spherical ensemble defined by Haar measure on the unit sphere $SH^0(M,L^n)\subset H^0(M,L^n)$ with $\|s_n\|^2_{L^2}=1$. \end{enumerate} The main result is that the expected distributions of critical values in both the normalized Gaussian ensemble and the spherical ensemble tend to the same universal limit as $n\to\infty$, given explicitly as an integral over $m\times m$ symmetric matrices.$