IUMJ

Title: Complexity of random smooth functions on compact manifolds

Authors: Liviu Nicolaescu

Issue: Volume 63 (2014), Issue 4, 1037-1065

Abstract:

We prove a universal result relating the expected distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on an arbitrary compact Riemann $m$-dimensional manifold to the expected distribution of eigenvalues of a $(m+1)\times(m+1)$ random symmetric Wigner matrix. We then prove a central limit theorem describing what happens to the expected distribution of critical values when the dimension of the manifold is very large.