Critical values of random analytic functions on complex manifolds

Abstract

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical point of the norm. The metric and Kahler form endow the polynomials with a Hilbert space structure and we consider the associated Gaussian random polynomials and the spherical ensemble where ||s_n|| = 1 is chosen from Haar measure. Our main result is that the expected limit distribution of critical values as n tends to infinity in the spherical ensemble is universal (i.e. is independent of the choice of h), and we give an explicit formula for it. The limit distribution is the same as for suitably normalized Gaussian measures.