Critical values of fixed Morse index of random analytic functions on Riemann surfaces

TL;DR

This paper derives the limiting distribution of critical values for random analytic functions with fixed Morse index on Riemann surfaces, providing explicit formulas for maxima and saddle points and showing topological invariance of second order terms.

Contribution

It offers explicit formulas for the distribution of critical values at fixed Morse indices and demonstrates topological invariance of higher-order terms on Riemann surfaces.

Findings

Explicit limiting distribution formulas for critical values at fixed Morse indices.

Computed distributions for local maxima and saddle points on Riemann surfaces.

Proved topological invariance of the second order term in the distribution expansion.

Abstract

This note is an addendum to 'Critical values of random analytic functions on complex manifolds, Indiana Univ. Math. J. 63 No. 3 (2014), 651-686.' by R.Feng and S. Zelditch (arXiv:1212.4762). In this note, we give the formula of the limiting distribution of the critical values at critical points of random analytic functions of fixed Morse index and we compute explicitly such limitings for local maxima and saddle values in the case of Riemann surfaces; we also prove that the second order term in the full expansion of the distribution of the critical values is topologically invariant on Riemann surfaces.